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In this paper we point out an error in paper [2] and study the asymptotic behavior of the differential equation Lnx + f(t, x) = r(t). The results obtained are extensions of some of the results of [2].

A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = -" The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.

The incorrectness of Birkhoff's result, with which the paper [2] is concerned, was noted already by F. R. Gantmacher in 1954 in his book Theory of matrices (Russian); cf. the recent English translation of the second part of this, [l, pp. 175-176]. Example C in [2] is, in fact, the analogue of Gantmacher's counter-example when z= a> is taken instead of 2 = 0 as the singular point. Gantmacher does not, however, discuss the cases in which the result is correct; cf. [2, D]. Unfortunately, the writer became aware of Gantmacher's work only when it was too late to have paper [2] withdrawn or even to have a note added to it.

Among other results, it is proved that if a sequence {t 1l) of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all a-compact sets or all open sets, then there exists a maximal open set U such that ft,1 ( U) O-* { f t,) being the associated submeasures. In [5], J. D. Stein, Jr. considers some versions of Phillip's lemma and the Vitali-Hahn-Saks theorem for sequences of regular scalar-valued Borel measures on Hausdorff spaces. The measures he considers are bounded, regular, and finitely additive Borel measures which are easily seen to be countably additive (to prove this, first note that I pi, the variation of such a measure /L, is regular, bounded, and finitely additive [5, Lemma 1]) and so for a sequence {Bi) of Borel sets, B1j0, and > 0, C a sequence of compact sets (Ki), Ki c Bi and vi(Bi \ K1) some no and so I til(BiWO), an observation which enables one to prove his results easily, in more general forms, and under weaker conditions. Theory of submeasures developed in [1] will be used. Let G be an Abelian Hausdorff topological group, X a Hausdorff topological space, and yt a countably additive, regular, G-valued measure on X (by regularity we mean that for any Borel set B in X and a 0-nbd U in G, there exists a compact set C c B such that tt(K) C U, VK c B \ C, K Borel). If G is normed [1, p. 270], we have an associated submeasure ft, AK(B) = sup{14(A)I:A c B,A Borel) = sup{ I(C)I: C c B, Ccompact), which is finite [1, p. 279, Corollary 4.1 1], exhaustive, a-subadditive, and order-continuous [1, II]. It is also regular in the sense that given E > 0 and a Borel set B, 3 a compact set K c B such that i(B \ K) < (proof by contradiction, using exhaustivity). For a collection of measures or subReceived by the editors August 25, 1976. AMS (MOS) subject classifications (1970). Primary 28A25; Secondary 46G10.

must hold. By an elementary theorem of geometry, C3 /t2s3M3(s) = 1, and a similar argument shows that tc satisfies this same relation in the case where wx/2 1. Because of the restriction on tc only those values of s may be used for which sM(s) oo as s->1, it is not permissible to let s-*1. Added in proof. See review by E. Reich, Math. Rev. vol. 19 (1958) p. 736.

Let _Q(t) be the rational function field over the rationals, Q, let Q((t)) be the Laurent series field over Q, and let W be a group of odd order. We investigate the following question: does there exist a finite-dimensional division algebra D central over Q(t) or Q((t)) which is a crossed product for W ? If such a D exists, W is said to be Q(t)-admissible (respectively, Q((t))-admissible). We prove that if W is Q((t))-admissible, then W is also Q(t)-admissible; we also exhibit a Q(t)-admissible group which is not Q((t))admissible. Let K be a field and let ' be a finite group. ' is said to be K-admissible if there exists a division algebra D, finite dimensional and central over K, which is a crossed product for '. Equivalently, ' is K-admissible if there exists a division algebra D with center K having a maximal subfield L Galois over K with Gal(L/K) S '. Admissibility questions for K = Q, the field of rational numbers, have been studied extensively in the literature (e.g., [Sc] and [ SO2 ]). More recently, results have been obtained when K is an algebraic function field over some field Ko ([FSS] and [FS]). In this paper we study admissibility questions for groups of odd order when K is either the rational function field Q(t) or the Laurent series field Q((t)). We show for such groups that Q((t))admissibility implies Q(t)-admissibility but not conversely. We also construct examples of groups of odd order which are Q((t))-admissible but which have homomorphic images which are not Q((t))-admissible; by contrast, if K is a number field, a homomorphic image of a K-admissible group is necessarily K-admissible [Sc, Corollary 2.3]. We fix below most of the basic terminology and notation that we will employ throughout this paper. Let K be a field. By a K-division algebra we mean a division algebra having center K which is finite dimensional over K. We say that A/K is central simple if A is a simple algebra with center K which is finite dimensional over K. Suppose A/K is central simple. By Wedderburn's Theorem, A _ M, (D) where D is a K-division algebra; we refer to D as the division algebra component of A. The Schur index of A, ind(A), equals Received by the editors September 21, 1993. 1991 Mathematics Subject Classification. Primary 12E1 5.

In this paper we complete the proof of the fact that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two. This result was announced in our previous paper, but the proof there contained a gap and the construction given there should also be slightly modified. This correction is done in this paper.

This note is to correct a mistake in [1]. So, the notation, definitions, and references are those of that paper. If M is maximal subgroup of a finite group G and H/K is a chief factor of G supplemented by M we cannot say in general that H E I (M). For example, if G is the dihedral group of order 30, M a maximal subgroup of G isomorphic to Sym(3), and H = Soc(G), we have that H f I(M) since k(H) = H. (Compare this with the last paragraph of page 236 in [5].) This motivates that Proposition 1 in our paper does not hold. Changing that by Proposition 1* below, we prove below that the five theorems of the paper remain true.

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.

Recently, we introduced a space [Hfi(A);B] which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (The Hankel transformation of a Banach space-valued generalized function, Proc. Amer. Math. Soc. 119(1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of [Hf, x A ; B], it can be uniquely represented by an element of [H^A) ; B] and hence of [H^ ; [A ; B]]. This is motivated by a generalization of Zemanian (Realizability theory for continuous linear systems, Academic Press, New York, 1972) for the product space Dgn x V where F is a Frechet space. His work is based on the facts that the space D$n is an inductive limit space and the convolution product is well defined in Dr . This is not possible here since the space Hn(A) is not an inductive limit space. Furthermore, D(A) is not dense in Hf,(A). To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of A .

Let A be any p.i. algebra in characteristic zero. Then the GKdimension of finitely generated subalgebras is linearly bounded in the number of generators. Let A be any p.i. algebra in characteristic zero. For a, ..., ak E A we denote by (al, ... , ak) the subalgebra generated by these elements. By [I] the GK-dimension of (a,, . .. , ak) will be finite for any finite k . In this paper we will show how these dimensions depend on k. Namely, Theorem. For any p.i. algebra A there exists a linear function f(k) such that, for all a,,... ,ak E A, GKdim(al, ..., ak) < f(k). The main tool in proving this paper will be the following theorem of Kemer's ([4, Corollary 1], also proven in [3, Corollary 8]): Theorem. Let M, (E) denote the n x n matrices over the infinite-dimensional Grassmann algebra E. Let A be any ( characteristic zero!) p.i. algebra. Then for large n, A satisfies all of the identities of M,(E) . Now let U be the universal p.i. algebra for M (E) with canonical generators x1, x2, ... , and let Uk be (xl, ... , Xk), the subalgebra generated by xl, .. ., xk. We showed in [2] that GKdim Uk = (k 1)n2 + 1. Without resorting to that work, it is not hard to show that GKdim Uk is bounded by a linear function in k. Here is a sketch suggested by the referee: Let K be the algebra gotten by adjoining the commutative variables t(a) and the anticommuting variables e (a) to the field F, i, y = 1, ..., n a = 1, ..., k. For each a let Xa be the n x n matrix with (i, j)-entry t(a) + e (a) for each (i, j). Then Uk is the subalgebra of Mn(K) generated by ii X .. , Xk . Hence, GKdim Uk < GKdim Mn (K) . It is then not hard to see that GKdim Mn (K) = kn2. The proof of our theorem now follows. By Kemer's theorem A satisfies all of the identities of Mn(E) for some n. Hence, (a,, ... , ak) will be a homomorphic image of Uk, so GKdim(al, ..., ak) < GKdim Uk which is linear in k . Received by the editors July 23, 1992. 1991 Mathematics Subject Classification. Primary 1 6A38. Partially supported by NSF grant DMS-9100258. Pai ? 1994 American Mathematical Society 0002-9939/94 $1.00 + S.25 per page

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class Co on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction T of class CO on Hilbert space is unicellular if and only if T has one-point spectrum and every invariant subspace for T is cyclic (Theorem 6). (3) A contraction T of class Co on Hilbert space is unicellular if and only if T has one-point spectrum and all invariant subspaces of T are hyperinvariant subspaces of T (Theorem 8). The aim of this paper is to discuss the unicellularity of contractions of class Co. We shall generalize the unicellularity of operators acting on finitedimensional spaces to that on infinite-dimensional spaces. Throughout this paper, we denote by H a complex, separable, and infinitedimensional Hilbert space. The related concepts and symbols can be found in [4]. Definition 1. Let H1 be a nontrivial invariant subspace of T. H1 is called a Banach reducible subspace of T if there exists another nontrivial invariant subspace H2 of T such that

Let G be a finite group and let it be a set of primes. A subgroup H of G is called 7r-quasi-normal in G if H permutes with every Sylow p-subgroup of G for every p in i . In this paper, we investigate how 7r-quasinormality conditions on some subgroups of G affect the structure of G. All groups considered are finite. The purpose of this paper is to investigate the influence of it-quasi-normality conditions on some subgroups of a finite group. it denotes a set of primes. Let G be a group, and let H be a subgroup of G. H is called 7:-quasi-normal in G if H permutes with every Sylowp-subgroup of G for every p in it; H is called' S-quasi-normal in G if H permutes with every Sylow subgroup of G; H is called quasi-normal in G if H permutes with every subgroup of G. Lemma 1. Let A, B, and C be subgroups of the group G. If A and B permute with C, then the subgroup (A, B) permutes with C [1, Hilfssatz 1, p. 207]. Lemma 2. Assume A < M < G and N a G. If A is it-quasi-normal in G, then A is 7r-quasi-normal in M and AN/N is it-quasi-normal in G/N [1, Hilfssatz 3, p. 207]. Lemma 3. Let A and B be subgroups in the group G. If A is it-quasi-normal in G and AB = BA, then A n B is it-quasi-normal in B [1, Hilfssatz 4, p. 207]. Lemma 4. Let A be a subgroup in the group G. If A is S-quasi-normal in G, then AaaiG [1, Satz l, p. 209]. Lemma 5. Assume A < G and P e Sylp (G) for every prime in p in it. If A is it-quasi-normal in G, then A n P e Sylp(A). Proof. Since AP is a subgroup of G, by the Sylow theorems, it follows that AnP e Sylp(A). Lemma 6. Let A be a maximal it-quasi-normal subgroup in G. Then one of the following statements is true: (a) A is a maximal normal subgroup in G. Received by the editors June 6, 199^0 and, in revised from, July 2, 1991. 1991 Mathematics Subject Classification. Primary 20D99, 20D40, 20F1 7; Secondary 20F1 6.

Talagrand [ 10] gives an example of a Banach space with weak topology which is not a Radon space, independently of their weight. This result gives an answer to a question formulated by Schwartz [9]. In this paper, following the papers of Drewnowski and Roberts [1] and Talagrand [10], we prove that the classical space (lo /c0 , weak) is not a Radon space. Introduction. A Hausdorff topological space E is said to be a Radon space if every finite Borel measure is a Radon measure; i.e., (A) = {8(K): K c A, K compact} for each Borel subset A of E. We shall say that a cardinal a is of measure zero (resp. nonmeasurable) if there is not a real-valued, diffuse, nontrivial measure (resp. {0, 1 }-valued), on the power set of a set A with cardinal a. The weight (density character) of a topological space E is the smallest cardinal such that there exists in E a dense subset A with this cardinal. A topological space E has the a-property of Lindelof, a a transfinite cardinal, if for each family (Gi)icE of open subsets of E there exists J c I such that card(J) < a and Uic Gi = Uij Gi* The smallest cardinal a such that E has the a-property of Lindelof is called the L-weight of E. Likewise, E is a Flock space if for every well-ordered family (Gi)LEI of open sets, with Ha = Gc\ Ufl

While the main theorem of our paper [DG] is correct, the paper contains two small errors. We are grateful to R. Cauty for pointing them out. The first error concerns Lemma 7, which was offered as a second method of proving the principal theorem under special hypotheses. This Lemma 7 is false: a counterexample is to be found in Corollary 3.3 of [M]. Thus the short ?3 of our paper entitled "A variant" should be disregarded. The second error concerns Theorem B. Our proof does indeed prove

The space of boundary values of Riess systems on the unit ball of RI is decomposed under the action of the special orthogonal group SO(n). The corresponding irreducible representations are explicitly determined. Two applications of this result are given. Introduction. In [3 and 4] we considered Rn-valued functions on the unit sphere Sn-1 of Rn and discussed the generalization of the results of Marcel Riesz on the conjugate function to this setting. This was done with the aid of a theory of singular integral operators developed in [3]. It is well known that, just as in the case of the circle and the real line, the Fourier transform carries singular integral operators of the type considered in [3] into multiplier operators. In the present paper we shall describe concretely the Fourier analysis of Rn-valued functions on sn-1, and we shall identify the multiplier corresponding to the Riesz transform. In ?2 we shall apply our Fourier analysis to prove that, in analogy with the complex analytic case, the restriction of the generalized Cauchy-Riemann equations to Sn-1 characterizes the tangential parts of the boundary values of Riesz systems. As it will tum out from our discussion, this is really a result about harmonic forms, and is essentially contained in [21; our proof is an analog of the classical method of solving the Dirichlet problem for the ball by means of an expansion into spherical hamonics. Throughout this paper we restrict our attention to the case n > 5. The cases n = 3,4 require some modifications corresponding to modifications occurring in the Branching Theorem; the case n = 2 is the classical case of the unit circle. 1. Decomposition of a representation. We write L = L2(Sn-1, R) and we consider the representation T of the special orthogonal group SO(n) given for g E SO(n), f E L, and x' E Sn-1 by (1.1) (Tgf)(x') = gf(g-lx'). If we interpret the elements of L as sections of the trivial vector bundle Sn-1 X Rn with SO(n) acting in the natural way on both factors, then T is just the action of SO(n) on the sections. Our first goal is to decompose T into irreducible constituents. We define, using the dot to denote the standard inner product on RI, LT. = {f E L|f(x') = x' = 0, x' E Sn-1,} LNor = {f E Llf(x') = e0(x')x', ep E L2(Sn-1, R)}. Received by the editors February 10, 1981 and, in revised form, September 14, 1981. 1980 Mathematics Subject Ciaeafication. Primary 43A85; Secondary 42B15, 42B20. 'Research partially supported by NSF grants.

Suppose A is a C*-algebra and B is a Banach algebra such that it can be continuously imbedded in B(H), the Banach algebra of bounded linear operators on some Hilbert space H. It is shown that if 9 is a compact algebra homomorphism from A into B, then 9 is a finite rank operator, and the range of 9 is spanned by a finite number of idempotents. If, moreover, B is commutative, then 9 has the form 9(x) = Xi(x)El + ** + Xk(x)Ek, where El, . . ., Ek are fixed mutually orthogonal idempotents in B and X1,.. ., Xk are fixed multiplicative linear functionals on A. Introduction. Suppose A is a commutative, semi-simple, unital Banach algebra. In [8] H. Kamowitz proved that if 0 is a compact endomorphism on A, A' is the set of all multiplicative linear functionals on A, and 0* is the adjoint of 0, then nf*f(A') is finite. One consequence of this result is a characterization of compact endomorphisms of C(X) [8, Corollary 2.2]. In this paper we characterize compact homomorphisms in a more general setting where they are defined from a C*-algebra into a Banach algebra that has a continuous imbedding in B(H). The existence of a nonzero compact endomorphism on a Banach algebra implies the existence of a nonzero proper closed two-sided ideal in that algebra, as S. Grabiner has shown in [4]. Throughout this paper "homomorphism" will mean an "algebra homomorphism." We call a set of idempotents {ei: i E I} mutually orthogonal if ejej = 0, whenever To prove our main result we will need the following extension lemma. LEMMA. Let A be a C*-algebra without identity, and let 0 be a compact homomorphism from A into a Banach algebra B. Then, there exists a compact homomorphism 0 from the C*-unitization C e A of A into B which extends 0. PROOF. Let (e,) be a bounded approximate identity of A with sup' le, I1 i x) = 0(x), 0(x)e = lim0(x)0(e,>i) = lim0(xe,>i) = 0(x). Received by the editors November 6, 1986 and, in revised form, March 9, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46K05, 46L05, 47B05; Secondary 43A65, 43A75.

In this paper it is shown that for each inner function u, there exists a singular inner function S which is divisible in ffx + C by all positive powers of u. Introduction. In this paper, we continue the study of division in H°° + C begun by Guillory and Sarason. We let H°° denote the space of boundary functions for bounded analytic functions in the open unit disk D and C denote the space of continuous, complex valued functions on 3D. We let Va denote the usual Lebesgue space with respect to Lebesgue measure. It is well known that H°° + C is a closed subalgebra of L°°. The space H°° (or Hx + C) will be identified with its analytic (or harmonic) extension to D. C. Guillory and D. Sarason began the study of division in Hx + C by determin- ing a criterion for deciding whether an Hx + C function is divisible by all positive powers of a unimodular H°° + C function (3). In the same paper, the question of finding, for each inner function u, a singular inner function which is divisible in Hx + C by all positive powers of u, is posed. We shall answer this question affirmatively. The techniques used to prove this are a combination of the techniques used in (1 and 3). As in (1), our main tools are interpolating Blaschke products and the Chang-Marshall Theorem. A sequence {zn} of distinct points in D is called an interpolating sequence if there exists 8 > 0 such that

It is shown that continua which are arcwise connected and contain no simple closed curves have the fixed point property for homeomorphisms, answering in the affirmative a question of Bing. The proof uses measure theoretic techniques. Given a homeomorphism h of a compact metric space X onto itself, a probability measure is constructed on X which is invariant under h. Introduction. A continuum (compact, connected, metric space) X is said to be l-arcwise connected if given any two points x, y E X, x y, there is a unique arc in X whose endpoints are x and y. This is equivalent to saying that X is arcwise connected and contains no simple closed curves. It is well known that such spaces need not have the fixed point property (see [6, p. 884]). In [1, p. 126, Question 6] Bing asks whether such spaces have the fixed point property for homeomorphisms. The object of this paper is to answer this question in the affirmative. The proof uses measure theoretic techniques. The paper is divided into two sections. In the first, the requisite analysis is developed. The second section is devoted to the proof of the fixed point theorem. 1. If X is a continuum and fi is a (complete) regular Borel measure on X, then the si-measurable subsets of X will always include the analytic sets (see [4, p. 482]). This section is devoted to showing that the arc components of any continuum are analytic (and hence si-measurable) and to producing a probability measure on an arbitrary continuum X which is invariant under a given homeomorphism of X onto itself. Definition 1.1. Let X be a compact metric space. Then 2x will denote the space of all closed subsets of X with the Hausdorff metric (see [4, p. 407]) and C(X) will denote the subspace of 2x consisting of all subcontinua of X. 2x is compact (see [5, pp. 45, 47]) and C(X) is closed in 2x (see [5, p. 139, Theorem 14]). Definition 1.2. A subset A of a complete separable metric space is said to be analytic if it is the continuous image of a Borel subset of some complete Received by the editors July 17, 1974. AMS (MOS) subject classifications (1970). Primary 54F20; Secondary 54E50, 54H05, 28A70.

A special case of a conjecture of R. Douglas is solved by an elementary argument using K0-theory. Let F be a subgroup of the additive reals R and let F+ = {x E F: x > O}. Douglas [2] defines a one-parameter semigroup of isometries to be a homomorphism x ~-4 V of F+ into the set of isometries on some Hilbert space H (i.e. x V = VyV? for x ,y E F+ and V0 = 1 ). Denoting by AF(VX) the C*-algebra generated by all VT (x E F+) and calling the map x 4 V nonunitary if no Tx is unitary except VJ = 1 , he shows that if x 4 VT and x W are nonunitary one-parameter semigroups of isometries on F then the algebras AF(TV) and AF(KV) are canonically isomorphic. Thus one can speak of Ar (isomorphic to AF( Vy) ) and of its commutator ideal CrF. Douglas shows that CF is simple, and that if Iand F2 are subgroups of R, then Ar and ArF are *-isomorphic iff ri and I2 are isomorphic as ordered groups. He obtains other interesting results on these algebras and conjectures that CF and C-, are *-isomorphic implies that Iand I2 are isomorphic as ordered groups. In this paper we show that CF is an AF-algebra for F a subgroup of Q (the additive rationals) and that in this case we have KO(CF) = F where K0( ) denotes the K0-group of Cr. (For a good account of K0-theory see Goodearl [3].) From this we deduce that Douglas' conjecture is true for subgroups of Q. Douglas was led to investigating these algebras Ar in the context of a generalized Toeplitz theory. The author has shown they satisfy a certain universal property which can facilitate their analysis, and he has generalized them by associating with every ordered group G a C*-algebra which reflects both order and algebra properties of G. The results presented here are part of an ongoing investigation of this more general theory, of which the author intends to give a fuller account in a forthcoming paper. Let H be a separable infinite-dimensional Hilbert space, and let U be the unilateral shift on H. We denote by C the C*-subalgebra of B(H) generated by U, and by K the commutator ideal of C. (The commutator ideal of a Received by the editors August 31, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L80. (? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

A C*-algebra is constructed which is separable, simple, nuclear, nonunital, and contains no nonzero projections. Some results on automorphisms of AF algebras are also obtained. A C*-algebra is said to be projectionless if it contains no projections other than 1 (if present) and 0. It has long been an open question whether there exists a projectionless simple C*-algebra (see [13, p. 18], [6, 1.9.6], [8, p. 81], [14, p. 242]). In this paper we construct a projectionless simple separable nuclear nonunital C*-algebra. It is quite possible that the methods of this paper can be modified to yield a projectionless simple unital C*-algebra. It is conjectured that the C*-algebra generated by the regular representation of the free group on two generators (known to be simple and unital) is projectionless. 1. Outline of construction. The general method of construction is motivated by the construction of the Bunce-Deddens weighted shift algebras [5] as described by Green [12, p. 248]. The algebra A is constructed as an inductive limit of C*-algebra An, each of which is a continuous field algebra on a circle T with a constant simple fiber B with the embedding j,,: A, -->A,,+1 inducing the "twice around" map z -> z2of T onto T. The algebra B will be the (unique) simple unital AF algebra whose ordered group KO(B) is isomorphic to the additive group of real algebraic numbers [10, 2.2]. B has the following properties: (1) B has a unique normalized trace , which is faithful. (2) If p and q are projections in B, then p q if and only if (p) = T(q). (3) If X is any algebraic number with 0 B such that f(l) = a(f(O)). Prim(A(a)) = {J,: 0 e PROPOSITION 1.1. A(a) is projectionless if (and only if) a(1) #6 1. Received by the editors December 13, 1978. AMS (MOS) subject classifications (1970). Primary 46L05. ? 1980 American Mathematical Society 0002-9939/80/0000-01 60/$02.25

Let G denote the universal-covering of the DeSitter group and C*(G) the group C*-algebra of G. In this paper we use the extension theory of C. Delaroche to describe the structure of C*(G). Introduction. Let G denote the universal-covering of the DeSitter group and C*(G) the group C*-algebra of G, i.e., the enveloping C*-algebra of the involutive Banach algebra LX(G) (see (2)). The main goal of this paper is to give a complete description of the structure of C*(G). Briefly, the main result is that C*(G) is isomorphic to the restricted product of certain C*-algebras whose structures have concrete descriptions given by the extension theory of C. Delaroche (1). In §1 of this paper we summarize the classification of the irreducible unitary representations of G given by J. Dixmier (3) and the character formulas for these representations given by T. Hirai (6). We refer to (3) or (9) for all information concerning the structure of G. In §2 we investigate the behavior of the irreducible characters and then follow the method of J. M. G. Fell (4) to describe the topology on G. An important step in this program is that of proving a Riemann-Lebesgue lemma for G. This we also do in §2. In §3 we determine the structure of C*(G). Since there are an infinite number of points where G fails to be Hausdorff, the methods of (1) do not apply directly. However, we are able to express C*(G) as the restricted product of certain C*-algebras each of which is describable via Theorem VI.3.8 of (11 When G = SL(2, C), the structure of C*(G) was first described by Fell (5) and later by Delaroche (1). For G = SL(2, R), the structure of C*(G) was determined by Milicic (7) by using methods similar to those of Fell in the SL(2, C) case. For the remaining Lorentz groups, one should be able to use the parameterization of G given by Thieleker (10), the character formulas given by Hirai (6), and the Delaroche extension theory to obtain results similar to those in this paper. This problem reduces to knowing the topologi- cal behavior at the "ends" of the complementary series representations

(i) lim(m ) supla n n N}= 0, and (ii) the bounded convergence field is an algebra. Since then we learned that essentially the same matrix appeared in the 1951 paper of G. A. Garreau L2], who used it as an example of a matrix which sums a sequence lx n to 0 iff 1jxnj} is (C, 1) summable to 0. The connection of this matrix to multiplicative summability is noted by G. M. Petersen L3], L4], where reference to Garreau's paper is made.

In E. G. Effros's paper [4] given at the 1980 Kingston Conference of the American Mathematical Society, he stated that "an example of a finite but not stably finite C*-algebra has yet to be found." This paper seeks to give an example of such an algebra by using a simple application of the duality between Kand Ext-theory arising from the work of Brown, Douglas and Fillmore (see, for example, [2 and 3]). Throughout this paper the C*-algebras will be separable and unital, except in the obvious cases such as X, the algebra of compact operators on an infinite-dimensional, separable Hilbert space XV. 2 will denote the algebra of all bounded, linear operators on JX, so that X is the only ideal (= closed, two-sided ideal) of Y. The quotient is 12, the Calkin algebra. Denote by U( -V) the group of all unitary operators in Y. Finally, for each positive integer n let M,1 be the algebra of complex valued, n x n matrices. Let A be a separable, unital C *-algebra. We say that A is finite if for every pair of elements x, y in A, xy= 1 implies yx= 1. A is stably finite if every finite-dimensional matrix algebra over A is finite; that is, if Mn 0 A is finite for each positive integer n. It is now time to introduce the notion of K-theory. Again, let A be a separable, unital C*-algebra. We can define a semigroup J(A) as the set of projections in YX A with direct sum as addition, modulo equivalence of projections. Following [6] this equivalence of projections can be defined in three different ways (equivalence, *-equivalence and unitary equivalence), but as Kaplansky shows [6, Chapter 3, in particular Theorem 28], these all give the same relation on a stable algebra. Alternatively, J(A) can be defined as equivalence classes of idempotents in )r A with direct sum as addition. Again, the two relevant types of equivalence (equivalence and invertible equivalence) turn out to be the same, and this version of J(A) is the same as the last [6, Theorem 27]. For e an idempotent in )r A, denote by [e]j the class of e in J(A). Following J. L. Taylor [9, Chapter 6], let KO(A) be the Received by the editors Jaunary 11, 1984 and, in revised form, October 3, 1984. 1980 Mathematics Subject Classificcation. Primary 46L05.

In this paper we give an alternative proof, without reference to Urysohn's lemma, of the metrization theorem of Bing [2], Nagata [6], and Smirnov [8] via the theory of symmetric spaces as developed by H. Martin in [5]. A symmetric d on a point set X is a function Xx X-+[O, oo) satisfying (1) d(x, y)=O if and only if x=y, and (2) d(x, y)=d(y, x). A topology T on X is said to be determined by d provided that for every subset U of X, U belongs to Tif and only if it contains an e-sphere N(p; 8) (= {x: d(p, x) O whenever KrnF= 0, K is compact, and F closed, then X is metrizable. This theorem strengthened an earlier theorem of A. V. Arhangel'skil [1], who introduced the notion of symmetric spaces. Martin achieves a proof of Theorem 1 by showing that X must satisfy the hypotheses of Mrs. Frink's theorem [3], a classical result in metrization theory. As a corollary of Theorem 1, Martin (and Arhangel'skil) obtains the theorem of S. Hanai and K. Morita [4], and A. H. Stone [9] on the metrizability of perfect images of metric spaces. The purpose of this paper is to obtain the metrization theorem of Bing [2], Nagata [6], and Smirnov [8] as a consequence of Theorem 1. It is interesting to note that Urysohn's lemma is never used in this approach, as was the case in the approach used by D. Rolfsen in [7]. More specifically, let us assume that X is a regular, T1 space with a a-locally finite base = U?= n, where 4! is locally finite and c2 4n+1, n> 1 Received by the editors May 8, 1973 and, in revised form, August 10, 1973. AMS (MOS) subject classifications (1970). Primary 54E35, 54E25; Secondary 54DI0, 54D20.

The semidirect product of a group A by a group B is necessarily nilpotent only in the case A and B are p-groups for the same prime p, A is nilpotent of bounded exponent, and B is finite. In an earlier paper Morley has established an upper bound on the class of a nilpotent semidirect product of an abelian p-group of bounded exponent by an arbitrary finite p-group. In this paper this result is improved by considering a direct product decomposition for B and also by extending the results to give a new upper bound on the class in the most general case. The standard wreath product of A by B is a nilpotent semidirect product of relatively large class in the case A and B satisfy the conditions above, and this new bound improves the known results on the class of these wreath products.

It is well known that a real-valued, bounded, Baire class one function of a real variable is the derivative of its indefinite integral at every point except possibly those in a set which is both of measure zero and of first category. In the present paper, a bounded, Darboux, Baire class one function is constructed to have the property that its indefinite integral fails to be differentiable at non-cr-porous set of points. Such functions are then shown to be "typical" in the sense of category in several standard function spaces. We shall denote by B1, DB1, bB1, and bDB1 the spaces of Baire class 1 func- tions, Baire 1 Darboux functions, bounded Baire 1 functions, and bounded Baire 1 Darboux functions, respectively, all defined on the interval (0,1) and all equipped with the topology of uniform convergence. The word "typical" in the title refers to any property which holds for most elements in a space of functions in the sense of category; i.e., the collection of functions not possessing the property is of first category in the space. Perhaps a better title for the present paper would be, "Another typical property of DB1 functions" for many such properties are known. A virtually complete catalog of such results can be found in the survey article (2) by Ceder and Pearson. In the process of writing (4) the present authors became curious about the "size" of the set of points at which bounded functions in certain classes can fail to be the derivatives of their indefinite integrals. Clearly, for functions in bBx this exception must be both of measure zero and of first category. In (4) we were especially concerned with finding circumstances under which this set would be a-porous. (The concept of a er-porous set was introduced in (3) by E. P. Dolzenko. The porosity of a set E of real numbers at the point x on the real line is the value where l(x,r,E) denotes the length of the largest open interval contained in the intersection of the complement of E with the interval (x - r, x + r). The set E is porous if it has positive porosity at each of its points, and it is cr-porous if it is a countable union of porous sets. Thus, cr-porous sets are of both measure zero and first category. Dolzenko showed that cr-porous sets are the natural exceptional sets for certain types of boundary behavior for complex functions defined, for example, in the upper half plane. More recently, cr-porous sets have been found to play a useful role in describing behavior of real functions.) In (4) we showed that any function in the subclass of bB1 consisting of bounded approximately symmetric functions is the derivative of its indefinite integral except at a cr-porous set of points. As noted in (4), it is easy to see that an arbitrary function in bB1 need

The Segal conjecture regarding the Burnside ring and stable cohomotopy of a finite group G is reduced for the case G = Z2X Z2 to a statement about Ext groups. This statement has since been proved by H. Miller, J. F. Adams and J. H. C. Gonawardena. The Segal conjecture states that for any finite group G, there is an isomorphism (G: A (G) -> 7T(BG) from the completed Burnside ring to the zeroth stable cohomotopy group of its classifying space. The conjecture was proved for cyclic groups in [4, 2, and 9]. In this paper we reduce the conjecture for Z2 X Z2 to a statement, (1), about Ext groups. This Ext statement was conjectured by Davis in [1] based upon extensive calculations and is proved by Adams, Gunawardena, and Miller in [7]. Let P = Z2[x, x'] be made into a module over the mod 2 Steenrod algebra A as in [5], and for S < k < n < xo let Pk' be the subquotient of P which is nonzero in degree k through n, inclusive. If n = x or k = -x, they will usually be omitted from the notation. The suspension liM of a graded module M is defined by (7Jm),+j= Mi. STATEMENT 1. There is an epimorphism of A -modules EP (g) Ep_ 1 Z2 (D EPO which induces an isomorphism in ExtA (, Z2). In fact p(sxa 0 sx b) = (?~I(?~~'(+ a+,)Sa+b+ I) +(SX () SX = ((a0+l)(b0+l) (a+ b+l)SX ) As mentioned above, Statement 1 is proved in [7]. The main result of this paper is THEOREM 2. Statement 1 implies that CZ2 XZ2 is an isomorphism. The proof of Theorem 2 mimicks [4]. The main part is to use Statement 1 to calculate 'u?(RP' x RP ) via the Adams spectral sequence. We begin by deducing from (1) the Ext groups relevant to [RPI A RP,I, S0I, the group of stable homotopy classes of maps. DEFINITION 3. If M is a (left) A-module, let DM denote the dual module, made into a left A-module using the antiautomorphism X; i.e., (DM)k = HomZ2(M k, Z2) with (9(+)(m) = ((X(O)m). Received by the editors January 30, 1981. 1980 Mathematics Subject Classification Primary 55Q10; Secondary 55T15.

We consider the set [w}w and the cofinality of the set 'cA assuming that some cardinals are endowed in total measures. Introduction. A cardinal ,c is real-valued measurable if there exists a iccomplete atomless probability measure on P(c), the set of all subsets of the cardinal Kc. The status and the philosophy concerned e.g. real-valued measurable cardinals have been detailed and presented in a survey paper by A. Kanamori and M. Magidor [KM]. We shall concentrate mainly on two problems: the cofinality of sets of functions with respect to eventual domination and some combinatorics on w, both assuming the existence of some total measures. T. Jech and K. Prikry [JP] showed, assuming 2W is real-valued measurable, that the cofinality of the set of all functions from w1 into w equals 21. We extend and complete their result. We show (Theorems 1 and 3, ?2), under the same assumption about 2', that if Kc w. It is well known that under Martin's axiom, 2W cannot be real-valued measurable. The reason is, that under Martin's axiom the cofinality of the set of all functions from w into w equals 2W while assuming 2W is real-valued measurable, the cofinality is < 2W (see also Theorem 4, ?2). We give some other reasons for which some cardinals cannot carry total measures. We show (Corollary 2, ?1) that if there exists a maximal Kc-tower on w, then Kc is not real-valued measurable cardinal. It has been shown by S. Hechler [H] that each regular cardinal Kc, w < K < 2W can be (consistently) the length of some maximal Kc-tower on w. Throughout the paper we use standard set-theoretical notation. For example [w]W is used to denote the set of all infinite subsets of the least infinite ordinal W. All undefined terms can be found in [J]. 1. Let W denote the set of all functions from w into w. For two arbitrary functions f, g c WW we set f

In this paper we present some results concerning the automatic order boundedness of disjointness preserving operators on Riesz spaces (vector lattices). Let L and M be Archimedean Riesz spaces. The linear mapping T from L into M is called disjointness preserving if Tf I Tg whenever f I g in L. Observe that a positive linear mapping T is disjointness preserving iff T is a Riesz homomorphism. The purpose of this note is to prove a basic result (Theorem 2) concerning disjointness preserving operators, which has many applications to the automatic order boundedness problem for disjointness preserving operators. First of all it provides a short and simple proof of a recent result of Yu. A. Abramovich [1] to the effect that a disjointness preserving operator T with the additional property that infn (I Tfn I + I Tgn 1) = 0 in M whenever fn, gn -O 0 (r.u.) in L, is order bounded. We note that a similar result for band pxeserving operators on Archimedean Riesz spaces was proved by S. J. Bernau [6] (recall that the linear operator T from L into itself is called band preserving if f I g implies that Tf I g). Furthermore, Theorem 2 has as an immediate corollary that any order bounded disjointness preserving operator T can be written as the difference of two Riesz homomorphisms, a result due to M. Meyer [10]. Finally we shall use Theorem 2 to show that any disjointness preserving operator from L into M is order bounded on some order dense ideal in L, whenever L is uniformly complete and M satisfies some additional conditions (e.g. if M is a normed Riesz space). In particular, it follows from a combination of Proposition 6 and Theorem 8 that any band preserving operator on a Banach lattice is order bounded (and hence norm bounded), which is a result of Yu. Abramovich, A. I. Veksler and A. V. Koldunov [2]. For terminology used and properties of Riesz spaces not explained or proved in this paper, we refer to [4 or 9]. We start with a lemma. LEMMA 1. Let L be an Archimedean Riesz space and let n be a fixed natural number. If 0 < u < e in L, then there exist 0 -Pk E L (k = 0, 1,... ,2n) and 0 < v, w E L such that (i)koPk = U, Received by the editors March 10, 1983 and, in revised form, June 13, 1983. 1980 Mathematics Subject Classification. Primary 47B55; Secondary 06F20.

This note establishes internal criteria on a category C and a separator 1: in C which characterize the condition that the >-induced covariant hom-functor hx: C Set is (epi, mono-source)-topological. Introduction. Hoffmann [4] showed how topological functors (of Herrlich [2]) may be recovered from factorizations of sources and sinks in the domain category. This process was extended to (E, M)-topological functors in our paper [1], where we claimed, without elaborating details, that the results could be used to internally characterize the condition that a covariant hom-functor h1: C Set is (epi, monosource)-topological. In this note, we establish such a characterization which, in fact, depends on the supporting results of that paper. Our references are only intended to be immediately relevant rather than exhaustive, and our terminology is generally that of [1, 2, 3]. We note the following concepts before stating our main result: DEFINITIONS. Let C be a category, l an object in C and e: X -p Y a morphism in C. Further, let h : = C(, -), the covariant hom-functor induced by E. (1) C is said to admit 2-disjoint coproducts iff for every coproduct sink (ui: Xi IIXi), and morphism y: l -. IIXi, there exist i E I and x: -* Xi such that y = ui o x, i.e. iff (h1ui)1 is an epi-sink. (2) The morphism e: X -* Y is said to be 2-coextendible (cf. [3]) iff for every y: l Y, there exists x: l -Xsuch thaty = e o x, i.e. iff h1e is onto. We have the following characterization: THEOREM. Let C be a category with an object l such that there exists at most a set of nonisomorphic objects C with h:C = 0. Then the following conditions are equivalent: (1) h1: C -Set is an (epi, mono-source)-topologicalfunctor. (2) C and l satisfy the following four conditions: (a) C is a co-complete category; (b) l is a separator; (c) for any set S, every morphism 2 S l is a copower injection; (d) C admits 2-disjoint coproducts and every regular epi in C is 2-coextendible. PROOF. (1) implies (2): (2)(a) follows from the fact that Set is co-complete and h. lifts colimits; Received by the editors May 6, 1981. 1980 Mathematics Subject Classification. Primary 1 8A20, 1 8A22, 1 8A30, 1 8A32, 1 8A40, 1 8B99, 1 8D30.

The main purpose of this paper is to show that for some continuous function-f and any preassigned, countable and dense set D of real numbers there exists a measurable function d which takes on every real value in every interval such that the range of f + d is D. A real valued function f defined on an interval I is said to have the intermediate value property if whenever xl and x2 are in I, and y is any number between f(xl) and f (x2), there is a number X3 between xl and x2 such thatf(x3) = y. Because of Darboux's work on this property, one now usually calls a function having the intermediate value property a Darboux function. A function f is Darboux if and only if f maps any connected subset of I onto a connected set. Since the sum of Darboux functions may fail to be a Darboux function, some mathematicians have examined how badly it can fail. Bruckner and Ceder [1] have recently shown that corresponding to each Darboux functionf which is not constant on any subinterval, there exists a function d which takes on every real value in every subinterval of I such that the range of f + d is a preassigned countable and dense set. Certainly d is a Darboux function and the range of f + d is totally disconnected. The technique used by Bruckner and Ceder in constructing the function d does not imply the measurability of d. Indeed Bruckner and Ceder have shown that d must be nonmeasurable whenever f is absolutely continuous. They asked whether d had to be nonmeasurable if we weaken absolute continuity of f to continuity of f. This paper answers this question affirmatively for certain continuous functions f. This is a consequence of the following. THEOREM. Let g be a Darboux function which is not constant on any subinterval of its domain I such that the set A = {oa: a E& R,g 1(a) is perfect) Received by the editors March 2, 1976. A MS (MOS) subject classifications (1970). Primary 26A 15; Secondary 26A21, 28A20.

All rings under consideration are Prufer domains or valuation domains. We characterize the set of basic ideals and the set of Cideals in an arbitrary valuation ring. Basic ideals were introduced in 1954 by Northcott and Rees. The concept of a C-ideal is, in a sense, directly opposite to that of a basic ideal. We then prove that a necessary and sufficient condition for every ideal in a domain D to be basic is that D be a one-dimensional Pruifer domain. Introduction. An ideal B is a reduction of the ideal A if B C A and BAn = An+l for some positive integer n. An ideal is basic if it has no proper reductions and is a C-ideal if it is not a reduction of any larger ideal. The first two of the above definitions appeared originally in a paper by Northcott and Rees [3]. The setting of that paper was a local ring with an infiniteresidue field. They established a nice description of basic ideals in those rings: an ideal is basic if and only if it can be generated by analytically independent elements. It was also shown that an ideal B which does not consist entirely of zero divisors is a reduction of the ideal A if and only if the elements of A are analytically dependent on B. This condition was also shown to b-e equivalent to the existence of an ideal C with BC = AC. These last two results would thus describe C-ideals in that setting. In [2], we investigated reductions of ideals in arbitrary commutative rings. In Noetherian rings, basic ideals were characterized to the extent that they are characterized in local rings. It was also shown that elements of the principal class, a generalization of analytic independence, generate basic ideals but not conversely. The basic properties of C-ideals were also examined. We also showed that a domain is Pruifer if and only if each finitely generated ideal is basic [2, Theorem 6.5, p. 62]. It was also shown that in a one-dimensional Priifer domain, every ideal is basic; that is, the basic ideal property holds [2, Theorem 6.1, p. 61]. We also observed that the basic ideal property need not hold for all Prufer domains. These facts suggest that in a Priufer domain of dimension greater than one, there are ideals which are not basic. We establish this in Theorem 10. Received by the editors March 11, 1974 and, in revised form, August 12, 1974. AMS (MOS) subject classifications (1970). Primary 13A15, 13F05; Secondary 13C05.

Let p be a prime and let n > 1. A necessary and sufficient condition that the classifying space BU(n) is p-equivalent to the product of nontrivial spaces is that p does not divide n. Let U(n) denote the Lie group of unitary n x n matrices, and let U = lim U(n). In this paper we study the classifying space BU(n) and determine those primes at which this space is equivalent to a product. The result is quite different from the infinite case. Recall that when we pass to the limit there are two types of splitting that occur. The first requires no localization; BU BT1 x BSU. The proof of this splitting is elementary, of course, but it does use the H-structure on BU. The second type of splitting is truly p-primary. At each prime p, BU splits into a product of p irreducible spaces p BU c-p JJ B(2n,p). n=1 This was first proved by Peterson [6]. A thorough account of this splitting is also given in Zabrodsky's book [8]. The main result of this paper is THEOREM. If 1 < n < ox, then BU(n) is irreducible at p if and only if p divides n. If p does not divide n, then BU(n) p BT1 x BSU(n) and both factors are irreducible. Most of the work in our proof involves showing that when p divides n, the unstable algebra H* (BU(n); F) is indecomposable over the Steenrod algebra. In other words, it cannot be expressed as the tensor product of two nontrivial unstable A*-algebras. Here A* denotes the Steenrod algebra modulo the two-sided ideal generated by the Bockstein coboundary. Our proof uses reflection groups and the methods and results of Adams and Wilkerson [2]. I would like to thank my advisor, C. W. Wilkerson, for his help and encouragement. 1. A*-algebras. Let H* and E* be A*-algebras. We say that E* is a retract of H* if there are A*-maps sr such that 7r *i=1EReceived by the editors December 28, 1984. 1980 Mathematics Suject Cliafifation. Primary 55P45. Key wods and phraes. Classifying spaces, Steenrod operations, modular representations. (?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 307 This content downloaded from 157.55.39.59 on Mon, 17 Oct 2016 04:52:19 UTC All use subject to http://about.jstor.org/terms

Local capacity is introduced (the usual notion of the capacity is global). It is proven (see Theorem 1) that some classes of metric spaces, naturally defined in terms of local capacity, contain a space which can be mapped onto any other member of its class without any stretching. Such a theorem would fail if local capacity is replaced by the usual notion of (global) capacity. Using Theorem 1 and simple properties of Met (X, Y) (see ?2) it follows immediately that for every class of compact metric spaces with uniformly bounded diameters and capacities there exists a compact space which contains an isometric image of any space from the class (in general this universal space cannot be found within the class). Introduction. Today it is well known that the global notions of metric capacity are useful in the theory of approximation, functional analysis and some other branches of mathematics (geometry of numbers, topology, ...). Capacity of a metric space X can be defined as a function e -* Se(X), > 0, where S (X) is the smallest cardinal number of a cover of X with the subsets of diameter < e. In this paper a local notion of metric capacity is introduced. There are some reasons to expect that such local notions should be very often even more convenient than global ones. The goal of this paper is purely metric. In ? 1 we prove that every class of complete metric spaces with uniformly bounded diameters and local capacity contains a member which can be mapped onto any other one without increasing any distance (see Theorem 1 below). The following simple example shows that such a theorem would be false for global capacity. Let X = {xI, x2, X3, X4} with metric d(xl, xi) = 2 for i= 2, 3, 4 and d(xi, xj) = 1 for 2 < i < j < 4, and Y = {YI ,Y2,Y3,Y4} with metric d(y1,y2) = d(y3,y4) = 1 and d(yi,yj) = 2 for i = 1 or 2 andj = 3 or 4. Both X and Y can be covered by two sets of diameter < 1 and by four sets of diameter < 2. But if a metric space Z is a union of two subsets of diameter

It is shown that on Lp[O, 1] all bounded linear operators which are Hermitian in the Calkin algebra B(LP)/C(4), must be of the form "Hermitian plus compact". That is, essentially Hermitian operators have the form, real multiplier plus compact. 1. Let X denote an infinite dimensional complex Banach space and B(X) the corresponding space of bounded (resp. compact) linear operators on X. The Calkin algebra associated with X is given by A(X) = B(X)/C(X). Many papers recently have dealt with variations of the following lifting question: Given that a coset T + C(X) in A (X) has a certain property, does the coset "lift" to an operator T + K, K E C(X), having the same property? For example, Stampfli [8] has shown that, if X is a separable complex Hilbert space, for every operator T E B(X) there is a compact operator KT so that the Weyl spectrum of T and the spectrum of T + KT are equal. In fact for most lifting theorems X is a separable infinite dimensional complex Hilbert space. Recently however, in an attempt to consider more general Banach spaces, these authors have proved that if X = Ip, then Hermitian elements in the Calkin algebra lift to the form "Hermitian plus compact". In this paper the above result is extended to the case X = Lp[O, 1] (hereafter referred to as L.): namely, the essentially Hermitian operators on B(Lp), 1 1, 4' is unique.) So given A,, 4'(t) = sgn 4'(t) I4(t) IP where sgn it = e-is if it = pei9. Finally, if 4 and 4' are unit vectors in L. then 4;8'(Qk>) = = f (t)4'(t) dt. We begin with a result proved in [1, Lemma 1], for T E B(Ql). The proof carries over to B(4L) with only one minor change. Whereas in the proof of [1] there existed a projection P E 6Y and unit vectors + and 4' in Lp for which = supp e q PTP-' , due to the fact that T and either P or P' were compact operators, here, it may only be assumed that for given E > 0, there exists a This content downloaded from 157.55.39.205 on Sun, 04 Dec 2016 04:55:31 UTC All use subject to http://about.jstor.org/terms ESSENTIALLY HERMITIAN OPERATORS 73 projection P E 'P and unit vectors 4 and 4' satisfying + > supp pE IPTP'Ilp. Nevertheless the following result still holds. LEMMA 1. Let T E B(LP), 1 < p < xo,p #'2. Then sup IIPTP'll < cpri(T) < so

Given a nonnegative Cl-function p(x) on a Riemannian manifold R, denote by BO(R) the Banach space of all bounded C2-solutions of Au = pu with the sup-norm. The purpose of this paper is to give a unified treatment of BO(R) on the Wiener compactification for all densities p(x). This approach not only generalizes classical results in the harmonic case (p 0), but it also enables one, for example, to easily compare the Banach space structure of the spaces BO(R) for various densities p(x). Typically, let ,8(p) be the set of all p-potential nondensity points in the Wiener harmonic boundary A, and Cp(A) the space of bounded continuous functions f on A with ft A 8(p) 0. Theorem. The spaces Bp(R) and Cp(A) are isometrically isomorphic with respect to the sup-norm. Throughout this paper R is an orientable Riemannian C?-manifold of dim > 2, and p(x) is a nonnegative Cl-function on R. Denote by Bp(R) the space of bounded C -solutions u on R of the elliptic equation Au = pu, where Au is the Laplacian of u on R. As one studies bounded harmonic functions on the Wiener compactification, the space B p(R) has been investigated on the so-called Wiener p-compactification (cf. Loeb and Walsh [2], Wang [91). However, their consideration restricts one to construct different compactifications for different densities p(x). The purpose of the present paper is to give a unified treatment of the spaces B (R) on the Wiener compactification R * for all densities p(x). p This approach, for instance, enables one to easily compare the linear space structure of the spaces B p(R) for various densities p(x). Typically, let ,8(p) be the set of p-potential nondensity points x in the Wiener harmonic boundary A (see below for its definition), and C p(A) the space of bounded Received by the editors May 31, 1973 and, in revised form, October 24, 1973. AMS (MOS) subject classifications (1970). Primary 30A48.

M. Andre has used the property Hn, namely the vanishing of certain homology groups, and the deviation On to characterize some classes of rings. In the present paper the author establishes an inequality on the deviations and obtains a Nagata criterion for Hn-locus and its openness for quotients of complete intersection rings and excellent rings. The upper-semicontinuity for Sn is also proved for the same classes of rings. Introduction. We study a property of local rings (A, m, K) introduced by M. Andre, namely the vanishing of Hn(A, K, K) which gives regularity (resp. complete intersection) for n = 2 (resp. n = 3). At first we prove (Theorem 1.7) a property for the deviations An introduced by M. Andre which resembles a result of L. L. Avramov (cf. [Av]) on deviations en which appear in [G-L]. By an inequality on these deviations An (Theorem 2.3) we can prove a Nagata criterion for Hn-locus and its openness for some class of rings (Corollaries 3.5 and 3.7). Then we obtain upper-semicontinuity for An on excellent rings, for n 7zk 1, and on quotients of complete intersection rings, for n > 3. The previous inequality becomes equality in some particular cases (Propositions 4.1 and 4.2), so we can prove that An is constant on locally closed sets on every quotient of complete intersection rings. We conclude this paper showing the openness of 83(1)-locus on a class of rings, where 83(1) is a property "near" to complete intersection, and a sort of Nagata criterion for An upper-semicontinuous. We wish to thank Silvio Greco for his useful suggestions on this paper. 1. All rings considered in this paper are unitary, commutative and noetherian. A local ring A with maximal ideal m and residue field K will be often denoted by (A, m, K). Let P be a property of local rings, the P-locus of a local ring A is the set Up(A) consisting of those p E spec(A) such that A. has property P. We want to prove the openness of Up(A) for some homological property P. First we recall a few definitions that will be used in the following. Received by the editors March 12, 1979 and, in revised form, July 31, 1979. 1980 Mathematics Subject Classification. Primary 13D99. 'The present paper was written while the author was supported by a C.N.R.-N.A.T.O. fellowship at Brandeis University. i) 1980 American Mathematical Society 0002-9939/80/0000-0502/$03.25 201 This content downloaded from 207.46.13.114 on Thu, 26 May 2016 06:27:53 UTC All use subject to http://about.jstor.org/terms

Let A be a ring with identity. It is well know that a group ring A[G] with a finite group G is Quasi-Frobenius (QF) iff A is QF. Using the concept of Frobenius extensions introduced by F. Kasch [4], we shall obtain a similar result for QF-3 rings in this paper, namely, A[ G] is left QF-3 iff A is left QF-3. Here a ring is called left QF-3 if it has a minimal faithful left module, that is, a faithful left module which is isomorphic to a direct summand of every faithful left module. Further we shall show that in case A is a G-Galois extension of the fixed subring A G relative to a finite group G of ring automorphism of A in the sense of [7], A is left QF-3 iff A G is left QF-3. It should be noted that A and A G are not always left QF-3 even if A G and A are left QF-3, respectively and that in case A /A G is finite G-Galois, A is QF whenever A G iS QF but the converse is not necessarily true. Throughout this paper, all rings, all modules, all subrings and all ring homomorphisms are assumed to be unitary. We follow the notation of [10] unless specified otherwise. For A -A'-bimodules A MA" A NA, the notation A MA IA NA' denotes the fact that AMA' is isomorphic to a direct summand of a direct sum N(') of finitely many copies of ANA,. A module M is said to be cofinitely generated (co-f.g.) in case for every set { Mi; i E I } of submodules of M if the intersection fl ,Mi = 0, then there exist il,... , in in I such that Mn kMi = 0 (see [11]). If a module M is f.g. projective, co-f.g. injective and faithful, then M will be called a *-module for convenience. If a ring A is left QF-3, then a minimal faithful left A-module is a *-module, and conversely if A has a left *-module, then A is left QF-3 (see [2, Theorem 1]). Let A iD B 3 1A be a ring extension. We say A is a Frobenius (resp. a left QF) extension of B if BA is f.g. projective and if AAB -(resp. j) AHom(BA, BB)B, and a right QF extension is defined symmetrically (see [4] and [8]). If A is a Frobenius extension of B, then there exist a B-B-homomorphism h of A to B and r1, . .. , r"; 11, . . ., In in A such that x = E:2rih(lix) = Eih(xri)li for all x in A, and conversely (see [9]). When this is the case, we shall call such a system (h; li, ri)in a Frobenius system.

Let F be a locally convex Frechet space. F is said to have the Radon?Nikodym property if for every positive finite measure space (Ql,E ,u ) and every s,t-continuous vector measure m Z; F of bounded variation, there exists an integrable function f: Q -F such that m(S) = fsf(co)d,u(co), for every S E Z. Maynard proved that a Banach space has the Radon-Nikodym property iff it is an s-dentable space. It is the purpose of this paper to give the following analogous characterization: A Frechet space F has the Radon-Nikodym property iff F is s-dentable. 0. Introduction. In [8], Maynard obtained some equivalent geometric conditions for the average range of a vector measure in the characterization of Rieffel's Radon-Nikodym theorem [11, Main theorem, p. 4661. Based on these results, Maynard [9, Theorem 2.2] recently extended Rieffel's [12, Theorem 1] condition on the dentability of the average range to s?dentabil= ity of the average range. It was shown in [2], [7] that all of these results can be extended to locally convex Frechet spaces; see ? 2. As a consequence, the geometric characterization of Frechet spaces having the RadonNikodym property will be proved in ?3 below. 1. Preliminaries. Let (Q, E, jx) be a positive finite measure space, where Q is an abstract set, E is a o-algebra of subsets of Q, and ji is a real-valued measure defined on E. Without loss of generality, one can assume that E is jx-complete. Let V S G l11(S) > O}. Throughout this paper, let F be a locally convex Frechet space, and I U I " be a fundamental decreasing sequence of closed absolutely n n=1 Received by the editors June 14, 1973 and, in revised form, November 12, 1973. AMS (MOS) subject classifications (1970). Primary 28A45, 46G10; Secondary 46A05

A compact, orientable, Riemannian manifold of dimension n is considered, with the Laplace operator acting on the exterior 2-forms of the manifold. Examining the spectrum, Sp 2 {\text {Sp}^2} , of the Laplace operator acting on 2-forms, the question is raised whether Sp 2 {\text {Sp}^2} exerts an influence on the geometry of the Riemannian manifold. To answer this question, after some preliminaries, two compact, orientable, equispectral, i.e., having the same Sp 2 {\text {Sp}^2} , Riemannian manifolds are considered in §3. (We note, in particular, that equispectral implies that the two manifolds are equidimensional.) Assuming further that the second Riemannian manifold has constant sectional curvature, the paper exhibits all the dimensions, commencing with 2, for which the two Riemannian equispectral manifolds have the same constant sectional curvature. In particular, this implies that for certain dimensions, which are explicitly stated, the Euclidean n-sphere is completely characterized by the spectrum, Sp 2 {\text {Sp}^2} , of the Laplacian on exterior 2-forms. Next, two compact, orientable, equispectral, Einsteinian manifolds are considered. (Again, equispectral implies equidimensional.) Assuming that the second Einsteinian manifold is of constant sectional curvature, the paper exhibits all the dimensions for which the two Einsteinian equispectral manifolds have equal constant sectional curvature. In particular, taking the second manifold to be the standard Euclidean sphere, the paper classifies Einsteinian manifolds, which are equispectral to the sphere, by calculating all the dimensions for which the Einsteinian manifold is isometric to the sphere. In short, if one of the Einsteinian manifolds is the sphere, then for certain dimensions, equispectral implies isometric. In §4, compact, equispectral, Kählerian manifolds are considered, and additional conditions are examined which determine their geometry. Studying two compact, equispectral, Kählerian manifolds, and again assuming that one of the manifolds is of real, constant, holomorphic, sectional curvature, the paper exhibits all the dimensions for which the two manifolds have equal real, constant, holomorphic, sectional curvatures. As a particular case, the paper classifies all the dimensions for which complex projective space, with Fubini-Study metric, is completely characterized by the spectrum, Sp 2 {\text {Sp}^2} , of the Laplacian acting on exterior 2-forms. The calculations were performed by utilizing an electronic computer.

I. N. Herstein has proved that any Jordan derivation on a 2- torsion free prime ring is a derivation. In this paper we prove that Herstein's result is true in 2-torsion free semiprime rings. This result makes it possible for us to prove that any linear Jordan derivation on a semisimple Banach algebra is continuous, which gives an affirmative answer to the question posed by A. M. Sinclair in (5). Preliminaries. Throughout this paper all rings will be associative. Let R be a ring. The center of R will be denoted by Z(R). We shall write (a, b) for ab — ba. A ring R is said to be 2-torsion free, if whenever 2a — 0, with a e R, then a = 0. A ring R is called a prime ring if aRb = (0) implies a = 0 or b = 0. A ring R is called a semiprime ring if aRa = (0) implies a = 0. Let R be any ring. An additive mapping ': R —y R is called a derivation if (ab)' = a'b + ab' holds for all pairs a,b e R. An additive mapping ': R —> R is called a Jordan derivation if (o2)' = a'a + aa' holds for all a e R. Obviously, every derivation is a Jordan derivation. The converse is, in general, not true. A well-known result of I. N. Herstein (2) states that every Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of this result can be found in (1). The main purpose of this paper is to present a generalization of Herstein's result. More precisely, we shall prove that every Jordan derivation on a 2-torsion free semiprime ring is a derivation. In particular, every Jordan derivation on a 2-torsion free semisimple ring is a derivation, which generalizes a result of A. M. Sinclair (see (5)). From the fact that every linear derivation on a semisimple Banach algebra is continuous, and from our generalization of Herstein's result, it follows immediately that every Jordan derivation on a semisimple Banach algebra is continuous, which gives an affirmative answer to the question posed by A. M. Sinclair in (5). In the last part of the paper two characterizations of 2-torsion free prime rings are obtained.

In a recent paper, F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E), the Banach space of continuous functions from a compact Hausdorff space Q to E, has the Dieudonne property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of l1. In this paper we give a positive answer to their question. As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonne property if and only if C(Q, E) has the same property. If E is a Banach space and Q is a compact Hausdorff space, then C(Q, E) will stand for the Banach space of the E-valued continuous functions on Q under the supremum norm. A Banach space E is said to have the Dieudonne property if for every Banach space F, any bounded linear operator T: E -> F that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. In [3] F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E) has the Dieudonne property and they asked whether the same result is true when replacing the assumption that E* is separable by supposing only that 11 does not embed in E. In this paper we give a positive answer to their question. Recall that a topological space (X, -y) is said to be Polish if it is homeomorphic to a separable complete metric space and it is said to be analytic if it is the continuous image of a Polish space. A subset A of a topological space (X, y) is said to be coanalytic if its complement (X\A, y) is analytic. Finally A is said to be PCA if it is the continuous image of a coanalytic space. The notations and terminology used and not defined can be found in [5, 8, or 10]. In the proof of Lemma 3 we need the following two results. THEOREM 1 (M. SREBRNY [9]). Let X and Y be two analytic spaces and let F be a multivalued function from X to the subsets of Y, such that its graph is PCA and for which one can prove that for every x c X, F(x) :8 0 using only the axioms of ZFC. Then there exists a universally measurable map f: X -> Y such that f (x) c F(x) for every x E X. THEOREM 2 (I. ASSANI [1, 2]). Let E be a separable Banach space. The set of weakly Cauchy sequences is a coanalytic subset of EN. Received by the editors January 5, 1985. 1980 Mathematics Subject Classification. Primary 46G10, 46B22.

In 1947, R. Robinson conjectured that if f is in S, i.e. a normalized univalent function on the unit disk, then the radius of univalence of [zf(z)]'/2 is at least -. He proved in that paper that it was at least .38. The conjecture has been shown to be true for most of the known subclasses of S. This author shows through use of the Grunski inequalities, that the minimum lower bound over the class S lies between .49 and .5. Introduction. Let C denote the class of analytic functions on the unit disk U = { z: Iz I 1 }. Let S denote the univalent functions f in C normalized by f(0) = 1 f'(O) = 0. Denote by K, S*, C, and Sp the standard subclasses of S consisting of functions that are convex starlike, close to convex and spirallike respectively. For a subclass X (possibly a singleton) of ?, let rs(X) denote the minimum radius of univalence over all functions f in X. We use corresponding notation for the other subclasses of S. For example rs*(X) denotes the minimum radius of starlikeness over all functionsf in X. For a functionf in S define the operator IF: S -C 6 by rf = (zf) I . In 1947 R. Robinson [10] considered the problem of determining rs[I(S)]. Robinson observed that for each f in S, [17(f)]' + 0 for Izi .38. There have been a number of papers (e.g. [2], [3], [6], [7], [8]) on the connection between the operator r and various subclasses of S. In these papers it has been shown that rK[F(K)] = rs*[F(S*)] = rc[F(C)] = rsp[F(Sp)] = and that F preserves Rogosinski's class of typically real functions (not necessarily univalent) up to Izi