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The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.

In a paper by Z. Lu and J. Rowlett, it is shown that the eigenvalues of the weighted Laplacian can be approximated by eigenvalues of a naturally associated family of narrow graphs. In this paper, we generalize this result to the p p -Laplacian. Our approach features overcoming the nonlinearity of the p p -Laplacian when p ≠ 2 p\neq 2 , which is different from the Laplacian case.

In this paper we present a probabilistic construction of countable abelian p p -groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian p p -groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin’s theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability 1 1 , produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. … It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian p p -groups. Valuated abelian p p -groups are natural generalizations of abelian p p -groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian p p -groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all n > 3, there are n-point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

We prove the following theorems: (1) Suppose that f; 2W 2W is a continuous function and X is a Sierpiiiski set. Then (A) for any strongly measure zero set Y, the image f[X + Y] is an so-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (see [N], [NSW], [S]) in which properties (mainly, the algebraic sum) of certain singular subsets of the real line R and of the Cantor set 2' were investigated. Throughout the paper, by a set of real numbers we mean a subset of 2' and by "+" we denote the standard modulo 2 coordinatewise addition in 2W. Let us also assume that a "measure zero" (or "negligible") set always denotes a Lebesgue measure zero set. We apply the following definition of sets of real numbers. Definition 1. An uncountable set X is said to be a Luzin (respectively, Sierpin'ski) set iff for each meager (respectively, measure zero) set Y, XnY is at most countable. We say that a set X is of strong measure zero (respectively, strongly meager) iff for each meager (respectively, measure zero) set Y, X + Y 7& 2'. Remark 1. It is well known (see [M] for example) that every Luzin set is strongly measure zero. Quite recently J. Pawlikowski proved that each Sierpin'ski set must be strongly meager as well (see [P]). Let us recall that a set X is called an so-set (or Marczewski set) iff for each perfect set P one can find a perfect set QCP that is disjoint from X. M. Scheepers showed in [S] that for a Sierpin'ski set X and a strong measure zero set Y, X + Y is an so-set. Later, in [NSW] it was proven that this also holds when X is strongly meager. We have the following functional version of the M. Scheepers' result. Theorem 1. Let X be a Sierpin'ski set and let Y be a strong measure zero set. Assume also that f: 2' -* 2W is a continuous function. Then the image f[X + Y] is an so-set. Received by the editors July 16, 1998 and, in revised form, September 9, 1998 and March 10, 1999. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15.

In the space I r I_r of invariant r r -dimensional subspaces of a null system in ( 2 r + 1 ) (2r+1) -dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections φ : I r → I r \varphi :I_r\to I_r , for which both φ \varphi and φ − 1 \varphi ^{-1} preserve adjacency. In the present paper we show that the two conditions φ : I r → I r \varphi :I_r\to I_r is a surjection and φ \varphi preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.

In this paper the author proves that any two elements from one of the following classes of operators are completely isomorphic to each other. 1. {VN(Fn): n > 2}. The Ill factors generated by the left regular representation of the free group on n-generators. 2. {C* (Fn): n > 2}. The reduced C*-algebras of the free group on ngenerators. 3. Some "non-commutative" analytic spaces introduced by G. Popescu in 1991. The paper ends with some applications to Popescu's version of von Neumann's inequality.

In this paper, we show that there are infinitely many tunnel number two knots K such that the tunnel number of K#K' is equal to two again for any 2-bridge knot K'. INTRODUCTION Let K be a knot in the 3-sphere S3, and t(K) the tunnel number of K. Here the tunnel number of K is the minimum number of mutually disjoint arcs properly embedded in the exterior of K whose exterior is a handlebody. We call the family of such arcs an unknotting tunnel system for K. In particular, we call it an unknotting tunnel for K, if the family consists of a single arc. On behavior of the tunnel number of knots under connected sum, the most simple case is: Theorem 1 ([N], [Sc] and [MS]). Tunnel number one knots are prime. And in the previous paper, we have shown: Theorem 2 ([MI]). Let K1 and K2 be non-trivial knots in S3, and suppose t(Ki #K2) = 2. Then: (1) if neither K1 nor K2 is a 2-bridge knot, then t(KI) = t(K2) = 1 or, (2) if one of K1 and K2, say K1, is a 2-bridge knot, then t(K2) 1) and Kn the knot illustrated in Figure 1. To prove Theorem 3, we show that (1) : t(Kn) = 2, (2) : t(Kn#K) = 2 for any 2-bridge knot K and (3) : Kn and Kn, are mutually different types if n and n' are mutually different integers (> 1). Received by the editors April 10, 1993. 1991 Mathematics Subject Classification. Primary 57M25.

Let F, G E C[x, y]. If the Jacobian determinant of F and G is 1, then G is said to be a Jacobian mate of F. If, in addition, G has degree less than that of F, then G is said to be a younger mate of F . In this paper, a necessary and sufficient condition is given for a polynomial to have a younger mate. This also gives rise to a formula for the younger mate if it exists. Furthermore, a conjecture concerning the existence of a younger mate is shown to be equivalent to the Jacobian conjecture. Throughout this paper, F and G will be polynomials in C[x, y] where C denotes the field of complex numbers. We say that F and G satisfy the Jacobian hypothesis if their Jacobian determinant is one, i.e., Fx Gy Fy Gx = 1 . In this case, we also say that G is a Jacobian mate of F. Furthermore, if the x-degree (resp. y-degree, total degree) of G is less than that of F, then G is said to be a younger mate of F relative to the x-degree (resp. y-degree, total degree). For instance, x + y has younger mates y and -x relative to the x-degree and the y-degree, respectively, but has no younger mate relative to the total degree. This paper was motivated by the Jacobian conjecture which asserts that if F has a Jacobian mate G, then (F, G) is an automorphism pair. In Section 1, it is shown that a younger mate is unique (up to an additive constant) and universal, i.e., if a Jacobian mate G of F exists, then any other mate of F can be expressed as G plus a polynomial in F. In Section 2, the problem of existence of a younger mate of F is reduced to the case where F is monic in both variables. In Section 3, a necessary and sufficient condition for the existence of a younger mate and a formula for a younger mate provided one exists are given. Finally, in Section 4, a conjecture concerning the existence of younger mates is formulated and shown to be equivalent to the Jacobian conjecture. Received by the editors July 6, 1993 and, in revised form, January 18, 1994. 1991 Mathematics Subject Classification. Primary 13B25, 13F20, 14E09, 16W20.

Upper bounds for the Betti numbers of generalized Cohen-Macaulay ideals are given. In particular, for the case of non-degenerate, reduced and irreducible projective curves we get an upper bound which only depends on their degree. 0. INTRODUCTION Let I be a homogeneous ideal of a polynomial ring S = K[xl, xn] over a field K, R = S/I, M := (xI, ..., xn), m = MR and e = e(I) :=e(R) the multiplicity of R/I. I is said to have a property P if R has this property P. It is a classical question to give upper and lower bounds for the Betti numbers, ,8i of S/I. A well-known conjecture due to Buchsbaum-Eisenbud says that ,61(S/I) > (n) for 0-dimensional ideals, and very recently Valla has given sharp upper bounds for the case of C.M. ideals (see [V]). The goal of this paper is to extend Valla's result to generalized C.M. ideals, i.e. ideals whose local cohomology modules Hm(R) are of finite length for all i < dim(R). As in [H], the key point is to reduce the computation to the case of C.M. ideals. Now we give a brief description of the paper. In ? 1, we fix notations and recall some results needed later on. In ?2, in order to prove our main result (Theorem 2.6), we first reduce to the case of 0-dimensional ideals and then we extend Valla's bounds to arbitrary (not necessarily non-degenerate) 0-dimensional ideals. As a consequence and related to Buchsbaum-Eisenbud's conjecture we get the upper bound f3 (S/I) < (n7)e for the Betti numbers of any homogeneous 0-dimensional ideal I. In the last section, applying our results we obtain upper bounds for the Betti numbers of the homogeneous ideal of some special projective schemes. In particular, for the case of non-degenerate, reduced' and irreducible projective curves, C, we get an upper bound which only depends on the degree of the curve C. Received by the editors September 30, 1993 and, in revised form, January 10, 1994. 1991 Mathematics Subject Classification. Primary 1 3D02. The first author was supported by a grant of CRM, Institut d'Estudis Catalans. The second author was partially supported by DGICYT PB91-0231-C02-02. ? 1995 American Mathematical Society

Let G 0 {G_0} be a connected Lie group with Lie algebra g 0 {g_0} and let h be a Lie subalgebra of the complexification g of g 0 {g_0} . Let C ∞ ( G 0 ) h {C^\infty }{({G_0})^h} be the annihilator of h in C ∞ ( G 0 ) {C^\infty }({G_0}) and let A = A ( C ∞ ( G 0 ) h ) \mathcal {A} = \mathcal {A}({C^\infty }{({G_0})^h}) be the annihilator of C ∞ ( G 0 ) h {C^\infty }{({G_0})^h} in the universal enveloping algebra U ( g ) \mathcal {U}(g) of g. If h is the complexification of the Lie algebra h 0 {h_0} of a Lie subgroup H 0 {H_0} of G 0 {G_0} then A = U ( g ) h \mathcal {A} = \mathcal {U}(g)h whenever H 0 {H_0} is closed, is a known result, and the point of this paper is to prove the converse assertion. The paper has two distinct parts, one for C ∞ {C^\infty } , the other for holomorphic functions. In the first part the Lie algebra h ¯ 0 {\bar h_0} of the closure of H 0 {H_0} is characterized as the annihilator in g 0 {g_0} of C ∞ ( G 0 ) h {C^\infty }{({G_0})^h} , and it is proved that h 0 {h_0} is an ideal in h ¯ 0 {\bar h_0} and that h ¯ 0 = h 0 ⊕ v {\bar h_0} = {h_0} \oplus v where v is an abelian subalgebra of h ¯ 0 {\bar h_0} . In the second part we consider a complexification G of G 0 {G_0} and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that A ( O ( G ) h ) = U ( g ) h \mathcal {A}(\mathcal {O}{(G)^h}) = \mathcal {U}(g)h if and only if G / H G/H has many holomorphic functions. This is the case if G / H G/H is a quasi-affine variety. From this we get that if H is a unipotent subgroup of G or if G and H are reductive groups then A ( C ∞ ( G 0 ) h ) = U ( g ) h \mathcal {A}({C^\infty }{({G_0})^h}) = \mathcal {U}(g)h .

The theorem is proved that a continuum that has only finitely many arc components (hence an arcwise connected continuum) and contains an n-od for every integer n must contain an infinite-od. FitzGerald [1, Corollary 4.8, p. 157] showed that a locally connected continuum that is an n-od for every positive integer n is an oc-od. Heath obtained the same conclusion by replacing locally connected with a hereditary decomposable condition (see Theorem A). In the same paper [2, Example 2, p. 477] she constructed a continuum that is an n-od for every positive integer n but is not an oc-od. Nall in [3, p. 245], generalizing a result of Sorgenfrey, gave sufficient conditions in order for a continuum to contain an oc-od. In a conversation with the author, Nall and Hagopian raised the question of whether an arcwise connected continuum that contains an n-od for every n must contain an oo-od. The purpose of this paper is to give an affirmative answer to this question. In fact, the answer is yes under the weaker hypothesis that the continuum has only finitely many arc components. In the case of infinitely many arc components, however, the answer is no, and a continuum is constructed with an infinite number of arc components that contains an n-od for every n but no oc-od. A continuum is a On-continuum if no subcontinuum separates it into more than n components. A continuum X is an n-od if there exists a subcontinuum H, called the hub, such that X\H has at least n components. If X\H has an infinite number of components then X is an oo-od. Note that a continuum X is a On-continuum if and only if X is not an (n + 1)-od. Example. There exists a continuum with an infinite number of arc components that contains an n-od for every n but does not contain an oc-od. Let X be a continuum irreducible between a and b that admits a monotone upper semicontinuous map f onto [0, 1] such that (i) fl (0) = a, f'l(1) = b, (ii) f'(y) is degenerate if y I k for n = 3, 4, ..., (iii) fl(n) is a simple n-od of diameter l for n = 3,4, ..., and (iv) f'(-) Received by the editors September 3, 1991; presented to the Annual Spring Topology Conference at California State University, Sacramento on April 11, 1991. 1991 Mathematics Subject Classification. Primary 54F20; Secondary 54B15.

In On a theory of computation and complexity over the real numbers ..., Bull. Amer. Math. Soc. 21 (1989), 1-46, Blum, Shub, and Smale investigated computability over the reals and over ordered rings in general. They showed that over the reals, output sets of machines are recursively enumerable (i.e., halting sets of machines). It is asked in the aforementioned paper which ordered rings have this property (which we abbreviate O = R . E . O = R.E. ). In Ordered rings over which output sets are recursively enumerable, Proc. Amer. Math. Soc. 112 (1991), 569-575, Michaux characterized the members of a certain class of ordered rings of infinite transcendence degree over Q \mathbb {Q} satisfying O = R . E . O = R.E. In this paper we characterize the subrings of R \mathbb {R} of finite transcendence degree over Q \mathbb {Q} satisfying O = R . E . O = R.E. as those rings recursive in the Dedekind cuts of members of a transcendence base. With Michaux’s result, this answers the question for subrings of R \mathbb {R} (i.e., archimedean rings).

This paper continues the study of the inductiveness of posets in terms of fixed apexes and points. The author proves some new equivalents of the Axiom of Choice, i.e., Zorn’s lemma. These statements are of fixed apex type and fixed point type theorems. The paper includes comments about these theorems and presents new characterizations of inductiveness and quasi-inductiveness of posets in terms of fixed apexes and fixed points.

In a recent paper [BSS], L. Blum, M. Shub, and S. Smale developed a theory of computation over the reals and over commutative ordered rings; in § 9 \S 9 of [BSS] they showed that over the reals (and over any real closed field) the class of recursively enumerable sets and the class of output sets are the same; it is a question (Problem 9.1 in [BSS]) to characterize ordered rings with this property (abbreviated by O = R.E. here). In this paper we prove essentially that in the class of (linearly) ordered rings of infinite transcendence degree over Q \mathbb {Q} , that are dense (for the order) in their real closures, only real closed fields have property O = R.E.

The aim of this paper is to present an example of a bounded balanced domain of holomorphy in C" (n > 3) with continuous Minkowski function that is not Kobayashi-finitely-compact. INTRODUCTION It is known [6] that if G c C" is a bounded Reinhardt-domain of holomorphy with 0 C G then G is finitely-compact with respect to (w.r.t.) the Carath6odorydistance cG, i.e., all cG-balls are relatively compact subsets of G w.r.t. the usual topology. In a more general case, if G = Gh = {z C Cn: h(z) < 1} is a bounded balanced domain of holomorphy with continuous Minkowski function h, then G is finitely-compact w.r.t. the Bergman-distance bG [4]. On the other hand, the continuity of h is a necessary condition for G = Gh to be finitely-compact w.r.t. the Kobayashi-distance kG [5, 1]. In this paper we give an example of a bounded balanced domain of holomorphy G = Gh C C3 with continuous h that is not kG-finitely compact and therefore, not cG-finitely compact. This answers a question formulated by J. Siciak in [7]. In particular, the example shows that, in general, there is no comparison of type bG < CkG for bounded balanced domains of holomorphy with continuous Minkowski function. DEFINITIONS AND STATEMENT We repeat some of the notions that will be needed in the sequel. Definition. A domain G c Cn is called balanced' iff whenever z C G and E C, 11?

Let E -X be an elliptic modular surface and S the tangential ruled surface of a projective embedding of X. The divisor that collects the involutions of the elliptic fibers of E is precisely the branch locus of E -S (at least generically). In this paper, we present two theorems that characterize this divisor in terms of the action of the group of modular automorphisms. These results extend work of D. Burns [1]. 0. INTRODUCTION AND STATEMENTS OF THE MAIN RESULTS The purpose of this paper is to present two theorems that together extend a result of D. Burns [1, Theorem 2]. This extension gives a rather complete picture of the invariant-theoretical question addressed there. We believe that these results will prove useful in the construction of elliptic modular surfaces with prescribed finite group as automorphisms. Thus, suppose that F is a torsion-free subgroup of finite index in F(1) SL2 (Z), and let X = X(F) be the usual modular curve attached to F [cf. 3 for details on modular curves]. There is an elliptic surface E = E(F) fibered over X such that, whenever F < A < F(l), the finite group ' := A/F acts on E in a very nice way [cf. I or 4]. In particular, -1 E F(1) induces an involution of E, and the quotient S S/ (-1) is birationally ruled over X. In [1], Burns constructs intrinsically a minimal model S of S (at least for almost all F). This geometric model S is a ruled surface over X, and Burns shows [ 1, Theorem 1] that extrinsically S is the tangential ruled surface of any projective embedding of X. Hence, to construct E explicitly from X and ' one should pinpoint the branch locus of the map E -* S. One component of this branch locus is the zero-section Z c S. A second component is the image & of a smooth 3-fold section bE c E. The divisor class of & is D := 3Z + 3Kx + KrX, where a canonical divisor Kx and the Received by the editors March 28, 1989 and, in revised form, December 19, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 14J27; Secondary 144D25, 14J50. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

A well-known result of Utumi asserts that a two-sided continuous two-sided artinian ring is quasi-Frobenius. In this paper we extend Utumi's result to quasi-continuous rings. Introduction and definitions A well-known result of Utumi (7) asserts that a two-sided continuous two- sided artinian ring is quasi-Frobenius. In (4) this result was extended to two- sided continuous rings with ACC on essential left and right ideals. In (3), mo- tivated by a result of Carl Faith on self-injective rings, it was shown that a two-sided continuous ring with ACC on left annihilators is quasi-Frobenius. There are examples of two-sided artinian one-sided continuous rings which are not quasi-Frobenius (see (4)). In this paper we extend Utumi's result to quasi- continuous rings. Throughout this paper all rings considered are associative with identity and all modules are unitary /?-modules. We write J(M), Z(M), Soc(M), and E(M) for the Jacobson radical, the singular submodule, the socle, and the injective hull of rM , respectively. For any subset X of R, Ir(X) represents the left annihilator of X in R. Consider the following conditions on a module rM : (Cl) Every submodule of M is essential in a summand of M. (C2) Every submodule isomorphic to a summand of M is itself a summand. (C3) If Mx and M2 are summands of M with Mx n M2 = 0, then Mx ® M2 is a summand of M. M is called continuous if it satisfies conditions (Cl) and (C2), quasi-continuous if it satisfies (Cl) and (C3), and a CS-module if it satisfies condition (Cl) only. It is easy to see that (C2) implies (C3), but the converse is not true in general. Thus, every continuous module is quasi-continuous. The ring of integers Z is

In this paper we prove that the spatial numerical range of a given operator on a separable Banach space is pathwise connected. In 1977 Luna [1] gave a partial answer to a question of Bonsall and Duncan [2] by proving that the spatial numerical range of a bounded linear operator is pathwise connected in the case of a reflexive Banach space. Later in 1985 Weigel [3] simplified considerably Luna's proof. In the present paper we give a nearly elementary proof of the same fact in the case of a separable Banach space. Let E be a complex Banach space, and let E* be the conjugate space of E. We set, for x E E, D(x) = {y E E*; IIyII = lIxii and y(x) = 11xI12}. The set-valued mapping x t D(x) is usually called the duality mapping of E. The set D(x) is a nonempty convex and w*-compact subset of E*. The graph of D is defined to be the set F=FE= {(xy); yED(x)}. Theorem 1. Let E be a separable Banach space. Then the graph FE of the duality mapping of E is a pathwise connected set. Proof. Let (xl, Yi), (X2, Y2) E F. Without real loss of generality we may assume that llxiii = 11x211 = 1. Let V = V{xi, x2} be the subspace of E generated by xl, x2, and let s(X) be the unit sphere of a Banach space X. It is sufficient to prove that the set G = {(x, y); x E s(V), y E D(x)} is pathwise connected. Claim 1. G is compact. Let ((xi, Yi))iE, be a net in G. There is a subnet J of I such that (Xj)jEJ converges in norm to some element x. We can choose a subnet K of J for which (Yk)kEK converges in w*-topology to some y. We must show that y E D(x) . But we have 0 < iIY(X)-1I| < lii ?y(X) -Yk(X)ii + iiyk (X)-Yk (Xk) l0. Received by the editors April 1, 1993. 1991 Mathematics Subject Classification. Primary 47A12; Secondary 46B20.

Suppose that T is a bounded linear operator on a complex Banach space X. If T2(X) is closed, T(X) n N(T) is finite dimensional, and S is a bounded linear operator on X such that S is invertible, commutes with T, and has sufficiently small norm, then T S is upper semi-Fredholm. Throughout this paper X will denote a complex Banach space. We write Y(X) for the set of all bounded linear operators on X. For T E Y(X), we denote by N(T) the kernel and by T(X) the range of T. The operator T is called upper semi-Fredholm if T(X) is closed and dim N(T) < 00. We write a(T) for the spectrum of T. It is well known that the resolvent RA(T) = (AIT) -1 is a holomorphic function of A for points A in the resolvent set C \ a(T). The aim of this paper is the following generalization of the "punctured neighborhood theorem" for upper semi-Fredholm operators: Theorem 1. Suppose that T E A(X), T2 has closed range, and T(X) n N(T) is finite dimensional. Then: (a) T S is upper semi-Fredholm whenever S E S(X) is invertible, TS = ST, and 11S11 is sufficiently small. Furthermore, we have dim N(T S) = dim (N(T) nn Tn (X))a (b) If 0 is a boundary point of a(T), then 0 is a pole of the resolvent of T. For the proof of Theorem 1 we need some additional notation and a preliminary lemma. Let T E Y(X). We write a(T) and fl(T) for dim N(T) and codim T(X), respectively. The operator T is called lower semi-Fredholm if fl(T) is finite (in this case T has closed range, by [4, Satz 55.4]). T is called semi-Fredholm if T is upper or lower semi-Fredholm. T is Fredholm if both a(T) and ,@(T) are finite. The index of a semi-Fredholm operator T is defined by ind(T) = a(T) fl(T). Received by the editors November 9, 1992 and, in revised form, August 2, 1993. 1991 Mathematics Subject Classification. Primary 47A 10, 47A53, 47A55.

Let X be a universal cover of a finite connected graph. A uniform lattice on X is a group acting discretely and cocompactly on X. We provide a counterexample to Bass and Kulkarni's Deformation Conjecture (1990) that a discrete subgroup F < Aut(X) could be deformed, outside some F-invariant subtree, into a uniform lattice. A uniform tree X is, by definition, the universal cover of a finite connected graph Y. A group F < Aut(X) is a uniform X-lattice if F is discrete (i.e., every vertex stabilizer Jx for x E VX is finite, where VX is the set of all vertices of X) and the quotient graph F\X is finite. In this case Y = FI\X, where rI = IHI (Y) is a free group acting freely on X. In particular G\X is finite, where G = Aut(X), and H is then a discrete cocompact subgroup (i.e., uniform lattice) in the locally compact group G. The basic theory of such uniform three lattices was developed in BassKulkarni [BK]. In that paper they obtained many important results. It was shown there that if X is a locally finite tree and G = Aut(X), then in order for X to be uniform one needs not only the finiteness of G\X but also a 'unimodularity condition', that the locally compact group G be unimodular; this condition also has a combinational interpretation. When X is uniform they further showed that there is a uniform lattice F < G such that F\X = G\X, that every free uniform lattice is conjugate to a subgroup of F, and that every uniform lattice F is conjugate to one commensurable with F. Recall that ro and IF are said to be commensurable, denoted IFO . F,1, if the index [Fi: FOnFI] is finite for i=0, 1. In [BK], one finds the Deformation Conjecture. This asserted, roughly, that a discrete subgroup F < G could be 'deformed', outside some F-invariant subtree, into a uniform lattice. This conjecture was proved in [BK] whenever G\X is a tree or a loop. In this paper, we provide a counterexample to the Deformation Conjecture. Let X be a uniform tree, G = Aut(X). Definition. Let F < G and let Y c X be an F-invariant subtree. By a deformation of F outside of Y we mean a monomorphism h: F -G such that Received by the editors March 15, 1993 and, in revised form, April 21, 1993. 1991 Mathematics Subject Classification. Primary 05C25, OSCOS. i 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

In this paper we prove the containment I ( n m ) ⊂ M ( n − 1 ) m I m I^{(nm)}\subset M^{(n-1)m}I^m , for a radical ideal I I of s s general points in P n \mathbb {P}^n , where s ≥ 2 n s\geq 2^n . As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least 2 n 2^n points in P n \mathbb {P}^n .

Let $\mathcal{B}_r$ be the $(r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $\mathcal{V}$. Irreducible $\mathcal{B}_r$-modules were used to construct irreducible Whittaker modules in [MZ2] and irreducible weight modules with infinite dimensional weight spaces over $\mathcal{V}$ in [LLZ].In the present paper, we construct non-weight Virasoro modules $F(M, \Omega(\lambda,\beta))$ from irreducible $\mathcal{B}_r$-modules $M$ and $(\mathcal{A},\mathcal{V})$-modules $\Omega(\lambda,\beta)$. We give necessary and sufficient conditions for the Virasoro module $F(M, \Omega(\lambda,\beta))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also we also give the isomorphism criterion for two $F(M, \Omega(\lambda,\beta))$., Comment: The introduction was revised

The main result of this paper states that the Krull dimension of a fully bounded Noetherian ring containing an uncountable central subfield is bounded above by its global dimension, provided that the latter is finite. The proof requires some results on projective dimensions and on localization (Corollary 4 and Theorem 11, respectively), which may be of independent interest. If P is a prime ideal in a Noetherian ring R, then P is contained in a unique clique, X, a subset of Spec(u) defined below (Definition 6). Now in some circumstances, the set C(X) of elements of R regular modulo every element of X is an Ore set in R, and the localized ring Rx obtained by inverting the elements of C(X) has certain desirable properties. In this case, X is said to be classical (Definition 7). We prove in Theorem 8 that if R is a Noetherian ring of finite global dimension whose cliques are classical, then the classical Krull dimension of R is bounded above by its global dimension. Generalizing work of B. J. Mueller and A. V. Jategaonkar (16, 13), we show that if R is a Noetherian fully bounded ring containing an uncountable set F of central units such that the difference of two distinct elements of F is still a unit, then all cliques in Spec(ii) are classical (Theorem 11). This applies, in particular, if R has an uncountable central subfield, as in the abstract. Using Theorems 8 and 11 in their general forms, K. R. Goodearl and L. W. Small have shown that the inequality of Krull and global dimensions is true for all Noetherian P. I. rings (10). In this paper, all modules are right modules unless it is indicated otherwise. If M is a module, then r(M) is the right annihilator of M and (if appropriate) l(M) is the left annihilator of M. A ring is Noetherian if it satisfies the ascending chain condition on right and left ideals. If P is a prime ideal of R, then R/P is right bounded if every essential right ideal of R/P contains a nonzero two-sided ideal. A ring R is fully bounded Noetherian (abbreviated FBN) if it is Noetherian and for every prime ideal P, R/P is both right and left bounded. Part of this research was done while the first author held a visiting position at the University of Washington. He is grateful to that institution for its hospitality. The research of the second author was supported in part by a grant from the NSF. 1. Homological results. LEMMA 1. Let R be a ring, A a right R-module, and M a maximal co-Artinian ideal of R. Then for every nonnegative integer n, ExtR(A,R/M) — 0 if and only

A p-ring (or generalized Boolean ring) P is a ring of fixed prime characteristic p in which ax=a for all a in P. In this paper P is partially ordered by a relation which is a generalization of the usual Boolean order. A subset S of P is then called quasiorthogonal if ab(a-b)=O for all a, b in S. It is shown that P is injective in the category of p-rings if and only if every quasiorthogonal subset has a supremum under this partial order. Sikorski [4] has shown that in the category 9 of Boolean rings the injective objects are the complete Boolean rings. The purpose of this paper is to present a generalization of this result to the category 9 of p-rings, where 9 is understood to be the category with objects rings P of fixed prime characteristic p in which aP=a for all a e P and with morphisms the usual ring homomorphisms. If P is a p-ring, then the set B(P) of idempotents of P is a Boolean ring under the multiplication of P and the new addition defined by aE3b=a+ b-2ab. Batbedat [2] has used B to establish an isomorphism between b? and R. (See also Stringall [6].) Hence, the injective objects in 9 are simply those P for which B(P) is a complete Boolean ring. In this paper the irnjective objects in b? will be characterized by a kind of completeness of a particular partial order that is an extension of the usual partial order on the Boolean ring of idempotents. DEFINITION ]. For a, b E P, a_b if and only if aP-1b=a. It is easily shown that _ is a partial order on P (see, e.g., Abian [1]), but in general (P, _) is not a lattice. It is, however, a lower semilattice with meet defined by aAb=a-a(a-b)P-1. A simple calculation shows that a(bAc)=abAac and aAO=O for a, b, c e P. Foster [3] has shown that every element a of P is uniquely expressible as a sum of elements {ej(a): jE Z.} of B(P). In particular, v-1 (1) a = Zjej(a) i=l Presented to the Society, January 26, 1973 under the title Injectivity in the category of p-rings; received by the editors February 27, 1973. AMS (MOS) subject classifications (1970). Primary 06A70; Secondary 06A40, 06A23.

A set-theoretic lemma is introduced and various applications are given, including: (1) a result of Erdos and Hajnal on the coloring number of a graph; (2) game characterizations of the coloring number of a graph; (3) K. Alster's result that a point-countable collection of open, compact scattered spaces has a point-finite clopen refinement; (4) normal, locally compact, metacompact spaces which are scattered of finite height are paracompact. 1. Introduction. In this paper, a purely set-theoretic lemma is presented which appears to be the "common part" of K. Alster's result (A) that a point-countable collection of compact scattered spaces has a point-finite clopen (open and closed) refinement, and a result of Erdos and Hajnal (EH) on the coloring number of a graph. The author has found this lemma to be quite useful (see (G2j, for example). In this paper, we will show how the lemma can be used to prove the above results, and we will also give some other applications. One such application is a "game characterization" of the coloring number of a graph, from which the Erdos-Hajnal result easily follows. Other applications are partial answers to the question of whether metalindelkf spaces are preserved by closed maps, and Tall's question of whether normal, locally compact, metacompact spaces are paracompact. 2. The lemma. In this section, we state and prove the main lemma. In the lemma, 2M is the set of all subsets of M, and Aa / A means that, for some ordinal K

If ( Q , q ) (Q,q) is a Steiner quadruple system and x x is any element in Q Q it is well known that the set Q x = Q ∖ { x } {Q_x} = Q\backslash \{ x\} equipped with the collection q ( x ) q(x) of all triples { a , b , c } \{ a,b,c\} such that { a , b , c , x } ∈ q \{ a,b,c,x\} \in q is a Steiner triple system. A quadruple system ( Q , q ) (Q,q) is said to have at least n n nonisomorphic associated triple systems (NATS) provided that for at least one subset X X of Q Q containing n n elements the triple systems ( Q x , q ( x ) ) ({Q_x},q(x)) and ( Q y , q ( y ) ) ({Q_y},q(y)) are nonisomorphic whenever x ≠ y ∈ X x \ne y \in X . Prior to the results in this paper the maximum number of known NATS for any quadruple system was 2. The main result in this paper is the construction for each positive integer t t of a quadruple system having at least t t NATS.

The strong lower limit and the weak upper limit of a net of von Neumann subalgebras on which the conditional expectations exist with respect to a fixed faithful normal state are defined. The limits coincide if and only if the corresponding conditional expectations converge strongly. 1. Preliminaries. Let M be a a-finite von Neumann algebra and (p a faithful normal state on M. By the GNS construction it can be considered that M is acting on a Hilbert space H and there exists a cyclic separating vector 'D E= H with (p(x) = (D I xD) for every x E M. Denote by M* the space of all a-weakly continuous linear functionals on M. That is, M* is the predual of M. For a von Neumann subalgebra N of M, if there exists a projection 8 of norm one from M onto N with (p o 8 = (p ? is called the conditional expectation onto N [3, 6]. 1?. The conditional expectation onto N exists if and only if at(N) = N for every t E R, where { a, } is the modular automorphism group on M with respect to (P. 2?. If the conditional expectation 8 onto N exists, then ?(x)?D = PxD for every x E M, where P is the orthogonal projection of H onto ND. Throughout this paper we fix a net { Na } of von Neumann subalgebras of M and assume that the conditional expectation 8a onto Na exists for each a. The orthogonal projection of H onto Ha = NaD is denoted by Pa. In the recent paper [5] we proved that if { Na } is increasing (resp. decreasing), then the conditional expectation 8 onto VaNa (resp. laNa) exists and 8a(X) --8(x) strongly for every x E M and f 0 -?a f ? E, in norm for everyf E M*. In this paper we shall introduce the notion of the strong limit of { Na } and show that the limit exists if and only if the corresponding {fea } converge strongly. The following are elementary but will be useful below. 30. For any uniformly bounded net { xy } in M and x E M, xy -x strongly (resp. weakly) if and only if x yxD strongly (resp. weakly) in H. 40. Let { Py } be a net of orthogonal projections of H, and P an orthogonal projection of H. For any t E H, if -y* Pt weakly, then it does strongly. Received by the editors January 23, 1984 and, in revised form, July 16, 1984. 1980 Mathena(dtics Subject Classification . Primary 46L10.

In this paper, a necessary ring-theoretical criterion is given for a finite-rank torsion-free abelian group to have the cancellation property. This generalizes results obtained by L. Fuchs and F. Loonstra [5] for the rank-one case and resolves the cancellation problem for strongly indecomposable groups. Introduction. An abelian group G has the cancellation property if for any two abelian groups H and K, G E H G E K implies that H K. Examples for the failure of cancellation are abundant, even among torsion-free abelian groups of finite rank; B. Jonsson [6, 7]. On the other hand, the cancellation property does hold for finitely generated abelian groups (E. A. Walker [8]), and the torsion-free rank-one groups that satisfy cancellation have been completely determined by L. Fuchs and F. Loonstra [5]. In this note, and in a subsequent paper, these fragmentary results will be systematized. Here we establish a necessary cancellation criterion for torsion-free abelian groups of finite rank. THEOREM A. Let G be a torsion-free abelian group of finite rank satisfying the cancellation property. Write G = B @ C, where B is free and C contains no free summand. Then for each n e Z \ (0), every unit of E(C)/nE(C) lifts to a unit of E(C). The question arises of whether or not the above criterion is also sufficient. To this end, the cancellation property will be studied in the context of two related notions, substitution and stable range. An abelian group G has the substitution property if for any abelian group A = G$ E H = G2 e K and G1 G2 c G, there exists a G3 c A such that A = G3 E H = G3 E K. A ring R has 1 in the stable range if, for any fl, g1, f2, g2 E R with flgl + f2g2 = 1, there exists an h E R such that f1 + f2h is a unit of R. Clearly, substitution implies cancellation. The connection between substitution and stable range properties, which holds in a more general setting, is given by R. B. Received by the editors February 16, 1984. 1980 Mathematics Subject Classification. Primary 20K15.

In recent papers, the author has proved that 4 4 -webs W(4,2,2) {\text {W(4,2,2)}} of codimension 2 and maximum 2 2 -rank on a 4 4 -dimensional differentiable manifold are exceptional in the sense that they are not necessarily algebraizable, while maximum 2 2 -rank 2 2 -codimensional d d -webs W(d,2,2), d > 4 {\text {W(d,2,2),}}d > 4 , are algebraizable. Examples of exceptional isoclinic webs W(4,2, 2) were given in those papers. In the present paper, the author proves that a polynomial nonisoclinic 3 3 -web W(3,2,2) {\text {W(3,2,2)}} cannot be extended to a nonisoclinic 4 4 -web W(4,2,2) {\text {W(4,2,2)}} and constructs an example of a nonisoclinic 4 4 -web W(4,2,2) {\text {W(4,2,2)}} of maximum 2 2 -rank.

We show that pairwise disjointness implies multiple disjointness for metric, weakly mixing regular minimal flows with an abelian phase group. A result on the disjointness of graphic minimal flows is also included. In a recent paper Auslander and Markley studied graphic flows and multiple disjointness for integer actions. Their major result is the following: Given a family of graphic minimal sets (Xi, Ti), Ti a homeomorphism of Xi, and nonzero integers a(i), if the flows (Xi, Ta(i)) are pairwise disjoint, then the product of all the flows is minimal. A related result is that, given a graphic flow (X, T), then (X, Ti) and (X, Tn) are disjoint whenever m :$ +/ n. In this paper we generalize the related result to Zn actions. We intended to generalize the major result to Rn actions, but found rather surprisingly that the result held for metric, weakly mixing regular minimal flows with abelian phase group, a substantially stronger result. The techniques used should be useful for proving generalizations of other results about graphic minimal flows as well as related results in [KN76] and [W75]. I would like to thank Ed Ihrig for useful conversations, Joe Auslander for catching an error in my original manuscript, and the referee for allowing me to include an improved result in the revised copy. Preliminaries. Standing assumptions. We assume that a flow (X, T) has a compact phase space and locally compact abelian phase group. Let J be the set of idempotents in the universal minimal set for T. DEFINITION. A flow (X, T) is a regular minimal flow iff it is a minimal flow such that, for any pair of points x, x' with (x, x') an almost periodic point in (X, T) x (X, T), there exists an automorphism h of (X, T) with h(x) = x'. DEFINITION. A minimal flow (X, T) is weakly mixing minimal iff it has no nontrivial equicontinuous factor. When X is metric this is equivalent to (X, T) x (X, T) having a point with dense orbit. DEFINITION. A flow (X, T) is totally minimal iff (X, S) is minimal for all syndetic subgroups S of T. DEFINITION. A flow (X, T) is graphic iff it is a weakly mixing minimal flow, and Xu is a single orbit for all idempotents u in J iff it is a minimal flow such that, for any pair of points x, x' with (x, x') an almost periodic point in (X, T) x (X, T), there exists a t in T with xt = x'. Note a graphic flow is a regular minimal flow and is totally minimal by 2.32 of [GH]. First we consider when pairwise disjointness implies multiple disjointness. Received by the editors October 10, 1984 and, in revised form, September 16, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H20. ?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

The paper contains a new proof of the fact that the Hahn-Banach majorized extension theorem for linear operators is valid iff the range ordered space is conditionally complete. The proof is based on quite different principles than the original proof of Bonnice, Silverman and To. The key element is a reformulation of the extension problem in terms of linear selections of special convex-valued mappings called fans. Let Y be a real linear space ordered by a cone K (which may be a wedge). Consider the following two properties: Hahn-Banach extension property (HBEP). For any linear space X, any linear subspace L c X, any sublinear mapping P: X -> Y and any linear operator B: L -* Y such that Bx Y such that Ax < P(x), Vx E X, and Ax = Bx, Vx EE L; Least upper bound property (LUBP). Any set Q c Y bounded from above has a least upper bound; in other words, there is an element sup Q E Y (not necessarily unique) such that y S sup Q for any y E Q and y < z for all y E Q implies sup Q < z. THEOREM A. An ordered space Y has the Hahn-Banach extension property if and only if it has the least upper bound property. This theorem has a long history which probably started (as far as more than one dimensional Y is concerned) with Kantorovic's paper [6] where the implication LUBP =X HBEP was proved for X being a normed space and P an abstract norm in X. We refer to [3] for a proof of the implication in a general situation. Silverman and Yen [9] (see also [3]) proved that HBEP implies LUBP if K is linearly closed (intersection of K with any line is closed in the natural topology of the line). Finally, Bonnice and Silverman [1], [2] and To [10] proved that HBEP itself implies linear closedness of the positive cone and hence LUBP. Received by the editors August 18, 1979 and, in revised form, December 10, 1979 and July 7, 1980. 1980 Mathematics Subject Classification. Primary 46A40, 47B55, 47D20.

It is well known that an indecomposable right ideal decomposition of a ring is not necessarily unique. In this paper we show that the reduced right ideals of such a decomposition are unique up to isomorphism and the remainder of the decomposition forms the unique MDSN. In the main theorem we use triangular matrices to prove that a ring with an indecomposable decomposition is basically composed of a nilpotent ring, a ring (containing a unity) with an indecomposable decomposition which equals its MDSN, and a direct sum of indecomposable reduced rings with unity. Generally, nilpotent elements detract from the arithmetic structure of a ring. Various methods (e.g. several types of radicals) have been developed to isolate the nilpotency of a ring. In this paper, we continue the study of the (right) MDSN, the unique minimal direct summand (i.e. idempotent generated right ideal) containing the nilpotent elements of the ring, which was begun in [1] and [2]. Throughout this paper, all rings will be associative; R denotes a ring with unity; N(X) is the set of nilpotent elements of X. As background from [2], we note the following: (1) the MDSN is a semicompletely prime two-sided ideal (i.e. if x' E MDSN then x E MDSN) which equals the intersection of all direct summands containing the set of nilpotent elements of the ring; (2) if R has a MDSN then so does R[x]; (3) not every ring has a MDSN [2, p. 715], however for rings without a MDSN there is an essential direct sum which approximates the decomposition obtained for rings with a MDSN; (4) in a reduced ring (i.e. without nonzero nilpotent elements) every idempotent is central, and a direct sum of reduced rings is reduced; (5) the complement of the MDSN is a reduced direct summand which is unique up to isomorphism. LEMMA 1. Let e E R such that e is a unity on eR. Then: (i) (1e)R and (1 e)Re are two-sided ideals of R; (ii) (1 e)R(l e) = R(l -e); (iii) (1e)R = ( -e)Re E3 R(l e) (left ideal direct sum) hence R= eR @D (I e)Re E R(l -e) (additive group direct sum); (iv) R is ring isomorphic to

A graph product is the fundamental group of a graph of groups Amongst the simplest examples are HNN groups and free products with amalgama- tion. The conjugacy problem is solvable for recursively presented graph products with cyclic edge groups over finite graphs if the vertex groups have solvable conjugacy problem and the sets of cyclic generators in them are semicritical. For graph products over infinite graphs these conditions are insufficient: a further condition ensures that graph products over infinite graphs of bounded path length have solvable conjugacy problem. These results generalise the known ones for HNN groups and free products with amalgamation. 1. Introduction. Groups which are graph products (fundamental groups of graphs of groups in the terminology of Bass and Serre) have attracted considerable attention, in view of their utility in combinatorial group theory. In particular, decision problems for HNN extensions and free products with amalgamation—the simplest graph products—have been of interest. The conjugacy problem for graph products is generally unsolvable (Miller (10)), but Lipschutz (9) gives conditions ensuring solvability of the conjugacy problem for free products with cyclic amalga- mated subgroups and Hurwitz (7) for HNN extensions with cyclic associated subgroups. This paper considers the conjugacy problem for graph products with cyclic edge groups, as part of the more general conjugacy problem for (Brandt) groupoids. Note that the defining graph here is slightly different from that considered in the Bass-Serre theory, so that an edge group there is regarded as a group at a source vertex in the terms of this paper. The semicriticality conditions of Lipschutz and Hurwitz for cyclic generators are amended and extended below. It is then shown that under the anticipated general- isation of their conditions, the conjugacy problem is always solvable only for graph products over finite graphs. This restriction is due to the fact that additional graph-theoretic decision problems arise naturally in this context. For graph products over infinite graphs, a further condition is given which ensures that the conjugacy problem is solvable for graph products over graphs with either bounded path length or else finitely many sources and an infinite cyclic group at each.

In this paper, it is shown that the only (nontrivial) finitely generated rotation invariant ideals in H? are z"H? for some positive integer n. Using results about function algebras, it is shown that other rotation invariant ideals exist. Rotation invariant ideals of other subalgebras of L?? are also studied. Let D and AD be the open unit disc and its boundary, respectively, in the complex plane C. Let L? be the Banach algebra of all essentially bounded Lebesgue measurable functions on AD with the essential supremum norm. If f is a function in L?' and A is a point in AD, let fA denote the L0 function defined by fA (z) = f (Az). Let B denote a closed subalgebra of L?. An ideal I of the algebra B is a rotation invariant ideal if it is a proper (nonzero) closed ideal of B such that, whenever f is a function in I and A is a point in AD, the function fA is also in I. In this paper we discuss the existence of rotation invariant ideals of subalgebras of L? containing H? (the space of bounded analytic functions on D). Let C denote the space of continuous, complex valued functions on AD. It is known [7] that a closed subalgebra of L? properly containing H? must contain H? + C. Using a theorem of Axler [1] we construct rotation invariant ideals of every closed subalgebra of L? properly containing H?. The ideals z'H' are the obvious rotation invariant ideals of H?. In [5], S. Power asked whether there are any others. Using results about function algebras, we shall show that there exist rotation invariant ideals other than z'H??. Let QC denote the largest C-* algebra contained in H? +C and QA =QCnH'. In [9], many analogues of known theorems for the disc algebra A are proven for QA. We shall use these to describe rotation invariant ideals in QA and QC. Throughout this paper many identifications are made. The set of all nonzero multiplicative linear functionals of a closed subalgebra B of L? is called the maximal ideal space of B and is denoted M(B). With the weak-* topology, M(B) is a compact Hausdorff space. There is one-to-one correspondence between maximal ideals of B and kernels of elements of M(B) . Each element in M(H?) has a unique norm preserving extension to a linear functional on LOO. Thus we identify M(B) with a closed subset of M(H?). We think of D as a subset of M(H?). We shall identify a function in L? with its harmonic extension to D. We also identify LOO with C(M(LOO)), the space of continuous functions on M(L?) and QC with C(M(QC)). I thank S. Axler and T. Wolff for valuable discussions. Received by the editors September 17, 1984. 1980 Mathematics Suject Ciasification. Primary 46H10, 46J15, 46J20. 'Partially supported by a grant from the National Sciences Foundation. (?)1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

We give a necessary and sufficient condition for a one-dimensional homology class of a graph manifold to be represented by a graph link. The purpose of this paper is to give a necessary and sufficient condition for a one-dimensional homology class of a graph manifold to be represented by a graph link. For this, we define the Jaco-Shalen-Johannson complex of a graph manifold (?1), using the so-called torus decomposition theorem due to Jaco-Shalen and Johannson (see also Waldhausen [2]), and the main result is stated as follows (?2): THEOREM. Let M be a graph manifold prime to S1 X S2 and p: M -T WM the natural map to the Jaco-Shalen-Johannson complex of M. Then an element a of H1(M; Z) can be represented by a graph link if and only if p*o(a) = 0 in Hl(WM; Z). If the ambient manifold M is not prime to S1 X S2, the homotopy class of the map p: M -T WM is not unique in general, and this makes the statement complicated (?3). The proof of these results is based on the study of global graph links in [3]. 1. Preliminaries. Throughout this paper, manifolds are compact, oriented, of dimension three and with toral boundary, and links are oriented. A manifold M is a graph manifold if there is a family of disjointly embedded tori in M such that each connected component of the manifold obtained by cutting M along these tori is the total space of an S1-bundle over a surface. A link is called a graph link if its exterior is a graph manifold. We say that a manifold is prime to S1 X S2 if its prime decomposition does not contain S1 X S2. See Jaco [1] and Yano [3] for other terminology. To state our result, we need the following theorem due to Jaco-Shalen and Johannson (see Jaco [1] and also Waldhausen [2]). THEOREM 1.1 (JACO SHALEN, JOHANNSON). Let M be a Haken manifold which is either closed or with incompressible boundary. Then there exists a unique Seifert submanifold E c M up to ambient isotopy such that (1) E is maximal, and (2) if E' is a Seifert manifold pair distinct from (S3, 0), (S2 X SI, 0), (D2 X I, aD 2 X I) or (S1 X D2, 0), then every nondegenerate map f: E' -> (M, aM) is homotopic to fo such that fo(E') c E. The E above is called the characteristic Seifert manifold of M. Received by the editors September 19, 1983 and, in revised form, March 14, 1984. 1980 Mathematics Subject Classification. Primary 57M99, 57N10.

We show that a finitely generated projective extension B of a commuta- tive complex unitary Banach algebra A induces an open mapping it between the carrier spaces. We next prove that if tr is a local homeomorphism then B contains an inertial subalgebra. Finally we present a necessary and sufficient condition for B to be uniform if A is. 1. Introduction. Throughout this paper Banach algebra means a commutative Banach algebra over the complex field C with an identity element 1. If A is a Banach algebra, then its carrier space, endowed with the Gelfand topology, is denoted by M(A) and its Shilov boundary by o(A). For each a G Av/e denote by â the Gelfand transform of a. The Jacobson radical of A is written R(A). From now on B will denote a finitely generated projective extension of a fixed Banach algebra A. We suppose B to be endowed with Magid's norm under which B is also a Banach algebra (10, Theorem 4, p. 138). We call it the projection of M(B) onto M(A) induced by the inclusion of A into B. The above situation is illustrated by the following example: take B = A(x)/(a(x)) where a(x) is a monic polynomial in A(x) (the so-called Arens-Hoffman extensions of A). For such extensions, the structure of the projection it as well as the study of the properties that B can inherit from A have been the main subject in a series of papers ((4), (7), (8)). The purpose of this paper is to establish some results along these lines for the more general situation. To state the main results it is convenient to introduce a definition. If \p G M(B), = w(i|/), and m^ = Ker

For any given nonnegative matrix A, this paper develops a notion of the kth measure of column indecomposability of A. The behavior of this measure on products of matrices is investigated. These results are then applied in developing several results on nonhomogeneous Markov chains. Let n > 1 be an integer and N = ,2, ..., n.Let A be an n x n nonnegative matrix. If A contains no r x s 0-submatrix with r + s = n, then A is said to be fully indecomposable. This notion of full indecomposability arises often in the study of nonnegative matrices and has attracted the attention of much research. In [2], this combinatorial notion was quantified by developing a notion of the measure of full indecomposability. This measure is defined as Uk (A) = mi~n ( m~ax~ a.) IkA-RI+ICIn=n-k iEERJE c a for k = 0, 1, . . ., n 2 with R, C C N where S denotes the cardinality of the set S. Further, let Uk(A) = Un42(A) for k > n 2. Some use of this quantitative measure was demonstrated in [1], [2], and [3]. This use included the developing of bounds on Perron eigenvalues and eigenvectors of a nonnegative matrix and the development of several results on nonhomogeneous Markov chains. Let m > 1 be an integer and M = {1, 2, . . ., m). The purpose of this paper is to extend the notion of full indecomposability to arbitrary nonnegative matrices and to give a quantitative measure for this extension. For this, let A be an m x n nonnegative matrix. Set , = min ( max a.) IRI+ICI=n-k iEER,JEC ' for k = 0, 1, . . ., n 2 where R C M and C C N. Further set k = for k > n 2. In this case, Puk is simply mini j aU. Set c = minj(maxi aU). Then we define Ck (A) = min{c, Ik}. For A an m x 1 column matrix we define Ck (A) Received by the editors July 7, 1976. AMS (MOS) subject classyiications (1970). Primary 15A48; Secondary 15A51.

A formula is given for the first coefficient of the Conway polynomial of a link in terms of its linking numbers. A graphical interpretation of this formula is also given. Introduction. Suppose that L is an oriented link of n components in 53. Associated to L is its Conway polynomial V?(z), which must be of the form VL(z) = z-1[a0 + alz2+ ■■■+amz2m\. Let VL(z) = VL(z)/z"~1. In this paper we shall give a formula for a0 = VL(0) which depends only on the linking numbers of L. We will also give a graphical interpretation of this formula. It should be noted that the formula we give was previously shown to be true up to absolute value in [3]. The author wishes to thank Hitoshi Murakami for bringing Professor Hosakawa's paper to his attention. We shall assume a basic familiarity with the Conway polynomial and its properties. The reader is referred to [1, 2, 4, 5 and 6] for a more detailed exposition. The fact that VL(z) has the form described above can be found in [4 or 6], for example. 1. A formula for V?(0). Suppose L [Kv K2,... ,Kn) is an oriented link in S3. Let ltj = lk(AT,, Kj) if i +j and define /„ = -jLu-ii+ihjDefine the linking matrix ££, or S^(L), as JSf= (/, ). Now JSPis a symmetric matrix with each row adding to zero. Under these conditions it follows that every cofactor =S?;y of £? is the same. (Recall thatSetj = (-l)'+ydet MtJ, where Mi} is the (i, j) minor of &.) Theorem 1. Let L he an oriented link of n components in S3. Then V?(0) = 3?ij, where ^j is any cofactor of the linking matrix Jif. Proof. Let F be a Seifert surface for L. We may picture F as shown in Figure 1.1. Let {ai} be the set of generators for Ha[F) shown in the figure and define the Siefert matrix V = {vii) in the usual way. Namely, vii} = lk(a,+, af), where a,+ is obtained by lifting ai slightly off of F in the positive direction. Then if a, n a, = 0 we have vi, = vii = lk(a,, uj). If ai n Oj # 0, then {;', j) = {2k — 1,2/c) for some Received by the editors August 14, 1984 and, in revised form, December 28, 1984. 1980 Mathematics Subject Classification. Primary 57M25.

In this paper we give a shorter and much more elementary proof of a theorem which describes the structure of certain localisations of the enveloping algebra of a completely solvable Lie algebra. Such a localisation is shown to be a twisted group algebra where the group is free abelian of finite rank and the coefficient ring is a polynomial extension of a Weyl algebra. Introduction. Let g be a completely solvable Lie algebra over a field k of characteristic zero, P a prime ideal of the enveloping algebra U = U(g), E the semicentre of U/P (see ? 1 below) and (U/P )E the corresponding quotient ring. (The interest of (U/P)E is that any simple U-module with annihilator P is naturally a (U/P )E-module and (U/P )E is a simple algebra.) If g is nilpotent then E is the centre of U/P and (U/P)E is a Weyl algebra An, n > 0, where An = K[yl, * .. * yng a/ ayl X .. a/ ayn 1 the "ring of differential operators with polynomial coefficients". In [4] it was shown that if g is completely solvable then (U/P)E may be regarded as a ring of differential operators in which the multiplication has been altered by a 2cocycle. In [5] the cohomology group involved was determined and it followed readily that such a "twisted" ring of differential operators had a much more elementary presentation as a "group algebra" of a free abelian group in which the group elements induce automorphisms on the coefficient ring. This group algebra is constructed (see ? 1) from the data (V, 6, G) (and is denoted by ?(V, 6, G)) where V is a finite-dimensional vector space, 8 is an alternating bilinear form on V and G is a finitely generated subgroup of the additive group of the dual space V*. The proof that (U/P)E is isomorphic to 6T(V, 6, G) as given in [4] and [5] is rather complicated. In particular, the proofs in [4] depend on results on smash products from [3]. In this paper we give an elementary proof that (U/P )E (V, 8, G ), which is completely independent of [3]. In order to make the whole argument intelligible we briefly sketch those Received by the editors September 8, 1976. AMS (MOS) subject classifications (1970). Primary 17B35; Secondary 16A08.

n The collection of all closed linear subspaces of H invariant under A (i.e. invariant under every a E A) is denoted by Lat A. A weakly closed algebra A c B(H) is reductive [8] if 1 E A, and Lat A = Lat MA. A linear subspace K c H is paraclosed [4] if there exist a Hilbert space HO and a bounded linear operator Q: Ho+H such that QHo = K. The collection of all paraclosed subspaces of H, invariant under A is denoted Latl/2A. A weakly closed algebra A c B(H) will be called parareductive if 1 E A, and Latl/2A = Latl/2MA. In this paper (Theorem 2.1) we show that if A is a parareductive algebra then A is a von Neumann algebra. In order to prove this result, in' ? 1, we give some new results on paraclosed operators. Finally, in ?3 we prove a result, announced (without proof) in [7]. Since the paper was written a proof of Theorem 2.1 in the separable case has appeared [1]. Our approach covering the possibly nonseparable case is entirely different and in fact simplifies Azoff's proof. I am indebted to the referee for calling my attention to Azoff's paper and for the suggestion that Theorem 2.1 can be formulated in this general form.

We introduce a method for constructing Boolean algebras from trees which preserves some of the trees' properties. The method is used to produce a very simple construction for rigid Boolean algebras and to construct a weakly homogeneous Boolean algebra without homogeneous factors. 0. Introduction. In this paper we present two constructions: a weakly homogeneous Boolean algebra that answers a question of Rubin, and what we feel is the simplest construction of a rigid Boolean algebra. A rigid Boolean algebra is one which has no nonidentity automorphism. We begin our construction of a rigid Boolean algebra by taking a tree T satisfying (a) the height of T is w; (b) T has a single root; (c) for all distinct a, ,3 E T, the number of immediate successors to a is different from the number of immediate successors to ,8; and (d) the number of immediate successors is always regular. We form " wedges". Sa = (/3 E T: a < ,B}, for each a E T, and close the set of wedges under finite unions and complements. The result is our algebra. Many constructions of rigid Boolean, algebras have been given. In [vDMR] there is a short history. van Douwen, Monk and Rubin ask for a "natural" construction of a rigid Boolean algebra. We offer our construction as a candidate. van Douwen, in [vD] has a similar construction. He begins with a tree T satisfying (a) and (c) and with the property that for all a E T, icK, the number of immediate successors to a, satisfies KNO = K.. He then topologizes the tree by taking certain infinite unions of wedges. The result is a rigid 0-dimensional compact space. The algebra of closed-and-open sets is the desired rigid Boolean algebra. A Boolean algebra B is homogeneous if for any nonzero b E B, the set (a E B: a < b} viewed as a Boolean algebra is isomorphic to B. It is weakly homogeneous if for all distinct nonzero a, b E B there exist nonzero b, < b, a, < a such that the algebras consisting of {c: c < b,} and (c: c < a,} are isomorphic. We answer a question of Rubin in [R] by constructing a weakly homogeneous Boolean algebra that has no homogeneous factors. Received by the editors January 27, 1982 and, in revised form, October 25, 1982. 1980 Mathematics Subject Classificatiotn. Primary 06E99. 'These results are contained in Chapter 2 of the author's Ph. D. dissertation, prepared under the direction of J. D. Monk at the University of Colorado. The author expresses his gratitude to Professor Monk. 2This paper is dedicated to Merry Havens. ((1983 American Mathenatical Socicty 0002-9939/82/(XXX)-0385/$02.(X)

In a recent paper F. Treves studied a model of complexes of pseudodifferential operators in an open set of R", establishing necessary and sufficient conditions for its semiglobal solvability. In the present paper, the authors give necessary and sufficient conditions for the global solvability of an analogous complex defined on an orientable, compact smooth manifold without boundary. 1. Introduction and statement of the theorem. Let B be a ^-dimensional (v > 1), compact, connected, orientable C°° manifold without boundary. We will denote by A a linear selfadjoint operator, densely defined in a complex Hilbert space H, which is unbounded, positive and has a bounded invsise^-1. (We may think of A as being, for instance, (1 - Ax)s on R", or a selfadjoint extension of \DX\.) We will use the scale of "Sobolev spaces" Hs (for s E R), defined by A : if s > 0, Hs is the space of elements u of H such that Asu E H, equipped with the norm \\u\\s = M^mIIo. where || ||q denotes the norm in H = H°; if s < 0, Hs is the completion of H for the norm \\u\\s = WA'uWq. The inner product in Hs will be denoted by ( , )s. Whenever s E R, m E R, Am is an isomorphism (for the Hilbert space structures) of Hs onto Hs~m. By H°° we denote the intersection of the spaces Hs, equipped with the projective limit topology, and by H~°° their union, with the inductive limit topology. Since for each s E R, Hs and H~s can be regarded as the dual of each other, so can i/00 and H~°° : with their topologies, they are the strong dual of each other. We denote by C°°(fi; H°°) the space of C°° functions in fi valued in H00. It is the intersection of the spaces C7(s;// ) (of the ^/'-continuously differen- tiable functions defined in fi and valued in H^ as the nonnegative integers y, k, tend to +00. We equip C°°(fi; Hx) with its natural C°° topology. We will denote by

It is shown that not every Boolean algebra is the Boolean part of a nondiscrete relation or cylindric algebra, but every nonatomless Boolean algebra is. Solutions of Tarski's Cube Problem for nondiscrete relation and cylindric algebras are given. Introduction. Relation algebras (RAs) and cylindric algebras (CAs) are Boolean algebras (BAs) endowed with additional structure. The question naturally arises: Which BAs are Boolean parts (i.e. Boolean reducts) of RAs or CAs? This question can be considered as a representation problem for BAs, too: Which BA can be represented as a BA of some binary relations on a set U closed under composition, inversion, and containing the identity relation on U (this is the RA case); and which BA can be represented as a BA of n-ary relations closed under, roughly, first-order definability, i.e. containing with every relation Rj,..., Rk the relations first-order definable (with n variables) from them, too? RA and CA,a denote the classes of RAs and CAs of dimension a respectively. There is a trivial way of turning every BA into a RA or a CAa; these RAs and CAs are called "Boolean" and "discrete" ones (cf. [7, p. 276 and 5, 1.3.11]). By [5, 1.2.14], every BA is the Boolean part of a nondiscrete CA1. J. D. Monk in 1981 and independently R. Laver in 1984 asked the question: Which BAs are Boolean parts (reducts) of nondiscrete CA ,s, a( > 2? Theorems 1 and 2 give partial answers. To CAa5s there correspond another BA, the so-called zero-dimensional part 3b W of a CA, W. By [5, 2.4.35], every BA is the zero-dimensional part of a hereditarily nondiscrete CA a (for all a > 0). In the last part of the paper we use this fact (together with Ketonen's solution for BAs) to give a full solution of Tarski's Cube Problem for CA,as as well as for RAs (of course we mean nondiscrete CA,as and RAs here). See Theorems 3 and 4. This solves that part of Problem 2.4 of [5] which was still left open in [6, p. 127]. The definitions of RA and CA.a, both originating with Tarski, can be found e.g. in [5, 7, 8, 10] and in [5,6] respectively. Below we recall that part of their definitions that will be needed in the present paper. Received by the editors April 2, 1985 and, in revised form, August 19, 1985 and February 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03G05, 03G25; Secondary 03E15, 06E99. C1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

In this paper the Riesz theory for compact linear operators in a normed vector space is considered from the point of view of how far the axiom of choice is involved. Special attention is drawn to the theorem, by which for the operator I A, A being compact, the index vanishes and the nullspace has a closed algebraic complement. It is shown that this can be proved without making use of the axiom of choice. Let X be a normed vector space over the scalar field K of the real or complex numbers, and A: X -+ X be a compact linear operator. Then the following three theorems of the Riesz theory concerning the linear operator S := I -A are achieved by methods not using the axiom of choice (and so they were originally obtained by F. Riesz in 1918 [3]). THEOREM 1 (FIRST RIESz THEOREM). The nullspace of the operator S (as well as the nullspaces of all powers of S) is a finite-dimensional subspace. THEOREM 2 (SECOND RIESZ THEOREM). The range of the operator S (as well as the ranges of all powers of S) is a closed subspace. THEOREM 3 (THIRD RIESZ THEOREM). For the operator S, there exists a uniquely determined nonnegative integer r, called the Riesz number of S, satisfying the conditions1 (1) {O} = N(S?) c N(Sl) c ... c N(Sr) = N(Sr+1) (2) X = R(SO) D R(Sl) D D R(Sr) = R(Sr+l) (3) X = N(Sr) e R(Sr). For the further Riesz theory, however, essential use is made of the axiom of choice: So the coincidence of the dimension of the nullspace N(S) with the codimension of the range R(S) is nowadays proved in a very elegant manner by the index theorem, while the existence of a closed algebraic complement to N(S) is assured by means of the dual system formed by the underlying normed vector space and its (topologic) dual space [2]. In the following we will show that for these two statements the axiom of choice can be avoided.2 We start with some preparatory lemmas which will be proved here for completeness though they might be found, in one or another form, in the literature. Received by the editors January 26, 1986. The contents of this paper have been presented to the meeting on Mehods and Techniques in Mathewatical Physics, held December 1-7, 1985, at the Mathematisches Forschungsinstitut Oberwolfach, Federal Republic of Germany. 1980 Mathematics Subjhect Classi Secondary 04A25. 1The nullspace and the range of an operator are denoted by N(.) and R(.), respectively. 2Recently Colton and Kress [1, pp. 16-22] gave an approach to the Fredholm theory for compact linear operators with respect to a dual system where the axiom of choice is not needed. This result can be reobtained in a considerably simplified manner by means of the present paper. (?)1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Given a holomorphic fibering with fibre P, we compare the cohomological Brauer group of the base to that of the total space. This allows us to prove that the geometric Brauer group of any symmetric product of a Riemann surface coincides with the cohomological one. Grothendieck has introduced [GROT] the notion of a geometric Brauer group. Given a variety X, this group Br(X) classifies, roughly speaking, Pa-bundles over X modulo those bundles of the form P(E) for some vector bundle E. In this paper we work in the category of complex manifolds (Grothendieck's setting was, of course, scheme-theoretic). Given any manifold Z, we put Br'(Z):= H 2 ( Z, Oz)tors (torsion part of H 2(Z, (*)). The Brauer group Br(Z) can be identified with a subgroup of Br'(Z) and a fundamental question is: does the equality Br(Z) = Br'(Z) hold? The aim of this paper is to give a positive answer to this question in the following cases (Theorems 1 and 2, respectively): (1) When Z is a Pa-bundle over a manifold X with the property Br(X) = Br'(X); (2) When Z is a symmetric product C(n) of a compact Riemann surface. ?1 is devoted to some preliminary material. In ??2 and 3 we prove the two theorems mentioned above. It is my pleasant duty to thank A. Fujiki for his judicious comments and the friendly interest he took in this work. 1. We work in the category of holomorphic manifolds. A Pr _-bundle over the manifold X is a holomorphic submersion 7r: P -* X such that all fibres of ST are isomorphic to Pr_1. Such a map is automatically a locally trivial fibration with structure group PGL(r, C). Hence the set of isomorphism classes of such Pr l-bundles, denoted by Projectr1( X), is in natural bijection with H'( X, PGL(r, ()). We have an exact sequence of sheaves of (noncommutative) groups on X, 1 -C9* -> GL(r, -x) PGL(r, -x) 1, Received by the editors June 11, 1984 and, in revised form, September 26, 1984. 1980 Mathenmatics Subject Classification. Primary 14C30; Secondary 32L05, 14F25, 14H15. ?)1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

In the author's paper [1 ] it is shown that if a sequence of orientable n-dimensional generalized closed manifolds (abbreviated n-gcm) converge (n-1)-regularly to an n-dimensional set, then the limit set is also an orientable n-gcm (see Definitions 1 and 2). In this paper a similar result is obtained for manifolds with boundary. Throughout the paper we assume that our sets are imbedded in a compact Hausdorff space S and that the cycles are Cech cycles with coefficients in an arbitrary field which we will omit from our notation for a cycle. All of the basic homology theory needed is in [7] and a knowledge of it will be assumed. We shall use the notation { A } --+A to mean that the sequence of sets A iCS converges to A CS as a limit (see p. 10 of [5]). We shall consider convergence only when all of the sets Ai are closed, and shall not explicitly state this henceforth; the limit set, as is well known, will always be closed. We shall use small latin letters for points, large ones for sets, and script letters for finite collections of open sets which are coverings of S. We shall use "U" for point set union or sum, "n" for intersection, reserving + and - for the group operations. By VAA we shall mean the subcomplex of the nerve of V whose vertices are elements of the covering V that have a nonvacuous intersection with A. If every element of V is contained in some element of V, we shall say that V is a refinement of V, and shall write V >V, or V = V(V); we shall use the notation' IIH. to denote a simplicial projection from the nerve of V into the nerve of V. By U C C V we mean that UC V, and by V>? we mean that the closure of the elements of V form a refinement of V, and by V> *V, we mean that the stars of the elements of V form a refinement ofV. If a cell a of a covering V has a nucleus (the intersection of all the open sets representing the vertices of a) that meets a set B, we say that af is on B; if all the cells with nonzero coefficients in a chain Cr ofZ V are on B, we say that Cr is on B. If the nucleus of a is in the open set Q, we say a is in Q. We shall use

Let d be a divisor of a positive integer n. Then d is a unitary divisor if d and nld are relatively prime, and d is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. An integer is bi-unitaty perfect if it equals the sum of its proper biunitary divisors. The purpose of this paper is to show that there are only three bi-unitary perfect numbers, namely 6, 60 and 90. A divisor d of an integer n is a unitary divisor if d and nld are relatively prime. A divisor d of an integer n is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. Let a(n) be the sum of the divisors of n, let ?*(n) be the sum of the unitary divisors of n, and let ?**(n) be the sum of the bi-unitary divisors of n. We say that N is unitary perfect if a * (N) = 2N. Subbarao and Warren [2] showed that 6, 60, 90 and 87360 are the first four unitary perfect numbers; Wall reported [3] that 146,361,946,186,458,562,560,000 = 2183.547.11.13.19.37.79.109.157.313 is also unitary perfect and later showed [4] it to be the next such number after 87360. Subbarao [1] has conjectured that there are only finitely many unitary perfect numbers. We say that N is bi-unitary perfect if a* * (N) = 2N. The purpose of this paper is to show that the first three unitary perfect numbers, i.e., 6, 60 and 90, are the only bi-unitary perfect numbers. One easily verifies that ov* is multiplicative and that if p is prime and e> 1, then rY**(pe) = ry(pe) = (pe+l l)/(p 1) if e is odd, and or**(pe) = (pe+l l)/(p 1) -pe/2 if e is even. Hence a**(n)?a(n) with equality if and only if every prime which divides n does so an odd number of times. It also follows immediately that a* * (n) is odd if and only if n is 1 or a power of 2; consequently, each odd prime power unitary divisor of n contributes at least one factor 2 to a**(n). Received by the editors June 10, 1971. AMS 1970 subject classifications. Primary lOA20; Secondary lOA99.

In a previous paper [7], we studied the structure of HID spaces. In this paper, we consider the behavior of HID spaces under monotone mappings. The principal result of this paper is that, given an arbitrary compact metric space Y, there is an HID space X and a monotone map f: X-) Y. We also show that an arbitrary HID space can be mapped monotonically onto a space of any preassigned dimension, and that, given an HID space X and a positive integer n, there is an n-dimensional space Y and a monotone map f: Y->X. R. H. Bing showed in [2 ] that each nondegenerate monotone image of a pseudo-arc is a pseudo-arc. The results of this paper show that no similar monotone invariance property holds for spaces of dimension greater than 1. In this paper, all spaces will be compact metric spaces (compacta). We will be dealing with the Hilbert cube, which we regard as being the product of a countably infinite collection of straight line intervals IX = 11 X 12 X 13 X ... , where Ij = [-1/2i, 1/2'].

In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zero-free analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed. Let C denote the unit circumference in the z-plane and D its interior. The term C-polynomial shall mean a polynomial all of whose zeros lie on C. In [6] the first author unified the results of MacLane [5] and Chui [2 ] by proving THEOREM A. Let f(z) = 1 +c+c2z 2+ * * * be a zero-free holomorphic function in D. Then there exists a sequence of C-polynomials Pn(z) assuming the value one at z =0 which converges to f(z) uniformly on every closed subset of D. If in addition M-supizi