Search

In [10] Patera and Sharp conceived a new relation, subjoining, between semisimple Lie algebras. Our objective in this paper is twofold. Firstly, to lay down a mathematical formalization of this concept for arbitrary Lie algebras. Secondly, to give a complete classification of all maximal subjoinings between Lie algebras of the same rank, of which many examples were already known to the above authors.The notion of subjoining is a generalization of the subalgebra relation between Lie algebras. To give an intuitive idea of what is involved we take a simple example. Suppose is a complex simple Lie algebra of type B2. Let be a Cartan subalgebra of and Δ the corresponding root system. We have the standard root diagramInside B2 there lies the subalgebra A1 × A1 which can be identified with the sum of and the root spaces corresponding to the long roots of B2.

A finite line transitive affine planeAis a finite plane which admits a collineation groupGacting transitively on the set of all lines ofA. Wagner [11] has shown thatAis a translation plane and Hering [9] recently investigated the structure ofAunder the assumption thatGhas a composition factor isomorphic to a given nonabelian simple group. The purpose of this paper is to show that if the number of points on a line ofAis odd, and ifGcontains no Baer involutions, then the hypothesis of Hering's Main Theorem holds.

The notion of pure subgroups is due to Prufer [7]. It has proven extremely useful in establishing structural properties of abelian groups. In a recent paper [9], Waterhouse introduced the concept of a pure subfield of a purely inseparable extension. Let L be a purely inseparable modular extension of k, and let K be an intermediate field. K is called pure if K and k(Lpn) are linearly disjoint over k(Kpn) for all n. Waterhouse used this concept to establish the existence of basic subfields [9].

Letdenote theNth Fermat curve. The period lattice ofF(N)is contained with finite index in the product of certain latticesLr,s(see [6]), and to this inclusion of lattices there corresponds an isogeny of the Jacobian ofF(N)onto a product of abelian varieties. The purpose of this paper is to determine when two factors in this product are isogenous overC, and whether they are absolutely simple.

A method for constructing the product of two Schur functions was stated, but not proved in the most general case, by Littlewood and Richardson [1] in 1934. This method, which came to be known as the Littlewood-Richardson rule, was later proved completely by Robinson [2] in 1938. In this proof, Robinson describes an operation on a finite sequence of positive integers. It is this operation, set in a more general context, that is the subject of this paper.

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D1 in the inverse image of N under p. The choice of D1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ1. Let π : D → D1 be the fiber bundle obtained by exponentiating the normal bundle of D1 in D. Let μ be the volume form of D1.

This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2). To define a Shimura variety S one needs an algebraic group G over Q together with certain supplementary data [2]; the set S(C) of complex points on the variety then appears as a double coset space

This paper concerns relative complexity of an algebraic theory T and its affine part A, primarily for theories TR of modules over a ring R. TR, AR and R itself are all, or none, finitely generated or finitely related. The minimum number of relations is the same for T R and AR. The minimum number of generators is a very crude invariant for these theories, being 1 for AR if it is finite, and 2 for TR if it is finite (and 1 ≠ 0 in R). The minimum arity of generators is barely less crude: 2 for TR} and 2 or 3 for AR (1 ≠ 0). AR is generated by binary operations if and only if R admits no homomorphism onto Z2.

The theory of Fourier transforms of tempered distributions as developed by Laurent Schwartz [17] is quite simple and elegant and has wide variety of applications, but there does not exist a corresponding neat and simple theory for the Hilbert transform of generalized functions (distributions) having wide applications. One of the objectives of this paper is to develop such a theory for the Hilbert transform of generalized functions and indicate its applicability to a variety of problems.In problems of physics sometimes we need to find harmonic functions u(x, y) in the region y > 0 whose limit as y → 0+ does not exist in pointwise sense but does exist in the distributional sense. The theory of Hilbert transform of generalized functions that we are going to develop will provide an answer to the existence and uniqueness of this problem.

A subset S of a topological space X is z-embedded in X in case each zero-set of S is the restriction to S of a zero-set of X. (A zero-set is the set of zeros of a real-valued continuous function.) For the basic theory of z-embedding, see [3] and [4] (and see [4] for a comprehensive bibliography of relevant papers).

In this paper we introduce a new approximation operator of the Arato-Renyi type and study its properties. Special cases of our operator are the power-series, of W. Meyer-König and K. Zeller (see [9]) and the generalized Berenstein power series introduced by A. Jakimovski and D. Leviatan in [5] and analyzed by them in [6].

In [4-8], the author has obtained several counterexamples to conjectures involving chain conditions, normality conditions, completeness, and the existence of point countable bases in Moore spaces. Each of these examples was obtained by constructing, by various means, a Moore space based on another space X0. In this paper, the author unifies these construction techniques and states some of the relationships between the original spaces and the derived Moore spaces.

In [2, Theorem 3.2, p. 429] and [3, Theorem 2.1, p. 155], the author established two induction theorems which gave rise to a series of fundamental results in spectral and rearrangement inequalities. In particular, the classical inequalities of Hardy-Littlewood-Pólya [4, Theorem 108, p. 89] and Pólya [6] were derived and conditions for equalities obtained (see [2, Theorem 3.8, p. 433] and [3, Theorem 2.6, p. 157]). In [5, Theorem 6, p. 651 ; and Theorem 20, p. 569], Fischer and Holbrook also gave alternative conditions for equalities to hold in the aforesaid inequalities. In this paper, we show that the result of Fischer and Holbrook can be proved by induction using an inductive rearrangement theorem which turns out to be a stronger version of the induction theorems given in [2, Theorem 3.2, p. 429] and [3, Theorem 2.1, p. 155].

This paper is devoted to the study of convergence and variation of operator spectra with respect to the distance G of Gokhburg and Markus [5] for subspaces and linear operators in a Banach space. We use the convention of Kato [7] and refer to convergence with respect to G as generalizedconvergence.

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.

(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one haswith k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements in and is the normed valuation belonging to . In particular M/L is unramified at all divisors of primes q ∉ S ∪ {p}.

In this paper we consider representations of weakly regular algebras with permutable congruences and a Boolean orthogonality as direct products of orthoprime algebras. Our theorems generalize results of Abian [2; 3] and Speed [24] which characterize direct products of integral domains, and results of Abian [1] and Chacron [7] which characterize direct products of division rings.

In this paper, we are concerned with the following problem: Let S be a finite set and Sm* ⊂ 2S a collection of subsets of S each of whose members has m elements (m a positive integer). Let f be a real-valued function on S and, for p ∊ Sm*, define f(P) as Σs∊pf (s). We seek the minimum (or maximum) of the function f on the set Sm*.The Traveling Salesman Problem is to find the cheapest polygonal path through a given set of vertices, given the cost of getting from any vertex to any other. It is easily seen that the Traveling Salesman Problem is a special case of this system, where S is the set of all edges joining pairs of points in the vertex set, Sm* is the set of polygons, each polygon has m elements (m = no. of points in the vertex set = no. of edges per polygon), and f is the cost function.

This paper is devoted to the study of sets of four unequal integers ni such that the twelve quotients ni/nj(i ≠ j) of pairs of distinct members sum to zero. (Without the restriction i ≠ j the condition would be equivalent to (Σ ni)(Σ l/ni) = 0, of no great interest.) Constructions for these sets are given, and relations between them are studied. It is found that each set belongs to an orbit of six related sets, and that each such orbit is related to four neighbours, each of which is another orbit of six sets. A study is made of the graph formed by assigning a node to each orbit of six solutions and joining it to the nodes assigned to its four neighbours. This appears to comprise one component containing an infinity of cycles together with an infinite forest of infinite trees.

Given a finite group G, the Frattini subgroup of G, Φ(G) is defined to be the intersection of all the maximal subgroups of G. Of late there have been several attempts to consider generalizations of Φ(G). For example, Gaschutz [7] and Rose [13] have investigated the intersection of all non-normal, maximal subgroups of a finite group. Deskins [6] has discussed the intersection of the family of maximal subgroups of a finite group whose indices are co-prime to a given prime. In [4-5, 12] we have considered the investigation of the family of all maximal subgroups of a finite group whose indices are composite and co-prime to a given prime. We have obtained several results about the family . In this paper which is a sequel to [4] we prove some further results about this family indicating the interesting role it plays especially when G is solvable or p-solvable. First we recall the main definition from [4].

Let be a family of subsets of a topological space X. We do not require to be a covering of X, nor do we assume that the members of are necessarily open. In this paper we shall assume that is of a special sort, which we call Σ-Finite. We show that a Σ-Finite family is both locally finite and star-finite, and in particular that an open covering of X is Σ-Finite if and only if it is star-finite. We then prove that every Σ-Finite family is ᓂ-discrete, so that in particular, every star-finite open covering of X is (ᓂ-discrete. There seems to be some applications of this fact.

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).

Besides the simple ones, there are several other kinds of C*-algebras which it has proved interesting to try to classify. For instance, a large body of results relates to the extensions of one given C*-algebra, possibly simple, by another. Extrapolating in this direction, we have considered the class of C*-algebras which can be decomposed in the strongest possible nontrivial sense in terms of their simple subquotients, and such that these simple subquotients in turn are as uncomplicated as possible.We have found that the classification of these C*-algebras, namely, the lexicographic direct sums of elementary C*-algebras, is to a large degree tractable, and yet involves an interesting new invariant in the antiliminary case, which is the case of no minimal ideals. Even the postliminary case, which is the case that the ordered set of simple subquotients satisfies the decreasing chain condition, is not without interest as an extension of the case of finitely many simple subquotients, analysed in the earlier papers [1] and [2].

With the aid of a theorem about the Julia points of a function meromorphic in the unit disk, this paper strengthens a theorem of K. Meier. As a consequence a stronger form of Plessner's Theorem is seen to hold which contains a theorem of E. F. Collingwood. An additional consequence is a stronger form of Meier's analogue to Plessner's Theorem.

1. Introduction. Throughout this paper, G is a connected linear algebraic group over an algebraically closed field whose characteristic is denoted p. For any closed subgroup H of G, denotes the Lie algebra of H and H0 denotes the connected component of the identity of H.A Borel subalgebra of is the Lie algebra of some Borel subgroup B of G. A maximal torus of is the Lie algebra of some maximal torus T of G. In [4], it is shown that the maximal tori of are the maximal commutative subalgebras of consisting of semisimple elements, and the question was raised in § 14.3 as to whether the set of Borel subalgebras of is the set of maximal triangulable subalgebras of .

An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel functionIts behaviour at infinity is described by an asymptotic expansionThe purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

The material in this paper is taken from the author's doctoral dissertation [2]. We will use the terminology and notation of [3]. Let us recall those terms which will be needed here.We define a restricted linear space (RLS) as a finite set of p elements, called points, of which q subsets, called lines, are distinguished so that the following axioms hold:(RLS-1) Any two distinct points u, v belong to exactly one common line uv.

The following well known inequality was first proved by Bernstein [2].THEOREM A. If pn(x) is a polynomial of degree n, such that |pn(x)| ≦ 1 for –1 ≦ x = +1, then1The dominant n(1 – x2)–;1/2 is best possible only at the zeros of the Tchebychev polynomialbut the bound is precise at every interior point as far as the exponent of n is concerned.Theorem A was extended to the case of higher derivatives by Duffin and Schaeffer in [4]. In that paper they make extensive use of the oscillation property of the polynomial Tn(x) and of the related function

In the first part of the paper, given a continuous map f from a Hausdorff topological space X onto a Hausdorff topological space Y, we consider the reciprocal map f* from Y into the collection of closed subsets of X, which maps y ∈ Y to . is endowed with the pseudotopological structure of convergence of closed sets. We will use the filter description of this convergence, as defined by Choquet and Gähler [2], [5], which is equivalent to the “topological convergence” of sets as introduced by Frolík and Mrówka [4], [10]. These notions in fact generalize the convergence of sequences of sets defined by Hausdorff [6]. We show that the continuity of f* is equivalent to the openness of f. On f*(Y), the set of fibers of f, we consider the pseudotopological structure induced by the closed convergence on .

Let A be a compact operator on a separable Hilbert space . The aim of this paper is to investigate the relationship between the weak closure of the algebra of polynomials in A (denoted by U(A)) and its invariant subspace lattice Lat A.

We are interested in the question of the decideability of the classical knot problem. A knot is the embedded image of a circle S1 in Euclidean 3-space E3. If L1, L2 are knots, then L1 ≈ L2 if there is an orientation-preserving homeomorphism h: E3 ⟶ E3 with h(L1) = L2. By the “knot problem” we mean: given two arbitrary tame knots L1, L2 (a knot is tame if it is equivalent to a polygonal knot), decide in a finite number of steps whether L1 ≈ L2. The object of this paper is to show that the knot problem is ‘'stably equivalent“ to a problem of deciding membership in the double cosets of a distinguished subgroup K2n of the classical braid group B2n [1].

One often encounters problems that are difficult as they are, but become manageable when translated to a different category. Thus very often, problems on Boolean algebras are answered by first transferring them to problems on Boolean spaces. (See, for example, [7]). It is with this spirit that we approach in this paper two problems on Boolean algebras. These problems are two decades old, and are considered to be outstanding problems in the field.

Let dndn–1 … d0 be the b-ary representation of a positive integer N. Call the polynomial obtained from N base b. In the case the base is 10, f(x) will be called the polynomial obtained from N. Pólya and Szegö attribute the following theorem to A. Cohn [2, b. 2, VIII, 128]:THEOREM 1. A polynomial obtained from a prime is irreducible.This theorem was generalized in two different ways by John Brillhart, Andrew Odlyzko, and myself [1]. One way was by proving the theorem remains true regardless of the base being used. The second way was by permitting the coefficients of f(x) to be different from digits. Thus, for example, if , where 0 ≦ dk ≦ 167 for all k, and if f(10) is prime, then f(x) is irreducible. In this paper, Theorem 1 will be generalized in another way by considering composite N.

Throughout this paper the term ring will denote a commutative ring with unity and the term integral domain will denote a ring having no nonzero divisors of zero. The set of all prime ideals of a ring R can be viewed as a topological space, called the prime spectrum of R, and abbreviated Spec (R), where the topology used is the Zariski topology [1, Definition 4, § 4.3, p. 99]. The set of all prime ideals of R can also be viewed simply as aposet - that is, a partially ordered set - with respect to set inclusion. We will use the phrase the pospec of R, or just Pospec (/v), to refer to this partially ordered set.

The grading is by integers modulo 2 and we refer to it as 2-grading. For the definition of 2-graded Lie algebras L and their properties we refer the reader to the papers [1; 2; 3]. All algebras considered here are finite-dimensional over a field F of characteristic zero.

Throughout this paper T will denote a bounded linear operator which is defined on a Banach space and whose spectrum lies on a rectifiable Jordan curve J .The operators having some growth conditions on their resolvents have been the subject of discussion for a long time. Many sufficient conditions have been found to ensure that such operators have invariant subspaces [2 ; 3 ; 7 ; 8 ; 12 ; 13; 14; 21; 27; 28; 29], are S-operators [14], are quasidecomposable [9], are decomposable [4 ; 11], are spectral [7 ; 10 ; 15 ; 17], are similar to normal operators [16 ; 23 ; 25 ; 26], or are normal [15 ; 18 ; 22]. In this line we are going to show that many such operators are decomposable.

In this paper the representation type of the class of pointed irreducible coalgebras is studied. We refer the reader to [4] for the basic definitions. A coalgebra is of bounded representation type if there is a bound on the dimension of finite dimensional indecomposable comodules. In Section 1, we show that the representation type is dependent upon the size of the space of primitives. Indeed, a pointed irreducible coalgebra is of bounded type if and only if it is finite dimensional and the space of primitives is onedimensional, i.e. if and only if it is a coalgebra spanned by a finite sequence of divided powers.

Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.

Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .

This paper is concerned with some extensions of the Bures metric d defined on the set of normal states of a W*-algebra [2]. Each subgroup G of the automorphism group of leads naturally to a semi-metric dG. (See Definition 1.1 below.) When G is the identity group dG = d.

In this paper we generalize a theorem of Lyndon's [7], which states that a one-relator group G = F/(r) (F is free and r Ç F) has cohomological dimension cd (F/(r)) ≧ 2 if and only if the relator r is not a proper power in F. His proof relies on the Identity Theorem and recently he has shown [8] how a generalized version of this theorem and a generalized version of the Freiheitsatz can be simultaneously obtained by the methods of combinatorial geometry. These generalizations refer to a situation where the free group F is replaced by a free product of subgroups of the additive group of real numbers.

In this paper we consider matrix representations of compact groups over the field of the complex numbers. We shall deal mainly with finite groups.The Kronecker product of two irreducible representations σ1and σ2of a groupis in general a reducible representation of. The explicit reduction of such a product to irreducible representations σ3can be performed by means of a unitary matrix, the elements of which are called Wigner coefficients or Clebsch-Gordan coefficients [1; 25; 27].

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.

In this paper we construct modular forms from combinatorial designs, and codes over finite fields. We construct codes from designs, and lattices from codes. Then we use the combinatorial properties of the designs and the weight (or shape) structures of the codes to study the theta functions of the associated lattices. These theta functions are shown to be modular forms for the modular group or for various congruence subgroups. The levels of the forms we examine are determined by the dimensions of the codes and the characteristics of the fields. Using the Lee polynomial of the codes we can write the theta functions as homogeneous polynomials in modified Jacobi theta functions. By extending the underlying combinatorial structure, a modular form of half-integral weight is associated with a modular form of integral weight.

There are several notions of positive definiteness for functions on topological groups, the two of which are: Bochner type positive definite functions and integrally positive definite functions. The class P(F) of positive definite functions for the class F can be defined more generally and it is interesting to observe that a change in F produces a different class P(F) of positive definite functions. The purpose of this paper is to study the functions in P(LP(G)) which are positive definite for the class LP(G) (1 ≦ p < ∞), where G is a compact or locally compact group. The relevant information about the class P(F) can be found in [1; 2; 3 and 8].

A quasigroup is an ordered pair (Q, •), where Q is a set and (•) is a binary operation on Q such that the equations ax — b and ya — b are uniquely solvable for every pair of elements a,b in Q. It is well-known (see, for example, [11]) that the multiplication table of a quasigroup defines a Latinsquare, that is, a Latin square can be viewed as the multiplication table of a quasigroup with the headline and sideline removed. We are concerned mainly with finite quasigroups in this paper. A quasigroup (Q, •) is called idempotent if the identity x2 = x holds for all x in Q.

In two preceding papers [2; 3] the author has studied the algebras of the irreducible representations λ and the classes Ci of a finite group G. Integral representations {λ} and {Ci} of these algebras are derivable from the appropriate multiplication tables [4]. It should be emphasized, however, that the symmetry properties of the two sets of structure constants are not the same, and this leads to somewhat greater complexity in the formulae relating to classes as compared to representations.

In the theory of Jordan algebras one encounters several definitions of the trace, and it is sometimes unclear whether the different notions are equivalent or not. If we restrict attention to the so–called JB–algebras studied in [2] and their weakly closed analogues JBW–algebras [8], we shall in the present note show that the different concepts are all equivalent for JBW–algebras, and that the conditions not involving projections are equivalent for JB–algebras. Among the seven equivalent conditions we shall consider, the second (ii) was used by Alfsen and Shultz [1] to show that if the JBW–algebra, is the self–adjoint part of a von Neumann algebra, then the condition characterizes traces on the enveloping von Neumann algebra. Condition (iii) appears in Robertson's paper [7] together with the implication (ii) ⇒ (iii). The inequality (iv) is a Jordan analogue of Gardner's inequality | ϕ(x) ≦ ϕ(|x|)|, [3], characterizing traces on Cast;–algebras.