We consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤm, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤm, 2) and of the homotopy classes of maps from M(ℤm, 2) to M(ℤn, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤm, 2) to M(ℤn, 2) for each co-H-structure on M(ℤm, 2) and on M(ℤn, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤn, 2) on the set of co-H-structures of M(ℤm, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤm, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤm, 2) are associative and non-commutative.

This paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine

Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.

Let G be an almost simple complex algebraic group defined over R, and let G(R) be the group of real points of G. We enumerate the G(R)-conjugacy classes of maximal R-tori of G. Each of these conjugacy classes is also a single G(R)˚-conjugacy class, where G(R)˚ is the identity component of G(R), viewed as a real Lie group. As a consequence we also obtain a new and short proof of the Kostant-Sugiura's theorem on conjugacy classes of Cartan subalgebras in simple real Lie algebras.

A connected real Lie group P is said to be weakly exponential (w.e.) if the image of its exponential map is dense in P. This concept was introduced in [HM] where also the question of identifying all w.e. almost simple real Lie groups was raised. By using a theorem of A. Borel and our classification of maximal R-tori we answer the above question when P is of the form G(R)˚.

Let M2(K, *) be the algebra of 2 × 2 matrices with involution over a field K of characteristic 0. We obtain the exact values of the cocharacters, codimensions and Hilbert series of the *-T-ideal of the polynomial identities for M2(K, *).

Let A be an algebra of continuous real functions on a topological space X. We study when every nonzero algebra homomorphism φ: A → R is given by evaluation at some point of X. In the case that A is the algebra of rational functions (or real-analytic functions, or Cm-functions) on a Banach space, we provide a positive answer for a wide class of spaces, including separable spaces and super-reflexive spaces (with nonmeasurable cardinal).

We consider a class of real flag manifolds which occur as Fürstenberg boundaries of ordered symmetric spaces and study the image of associated momentum maps. The presence of the order structure is responsible for much stronger convexity properties than in the general case.

We consider minimal left ideals L of the universal semigroup compactification of a topological semigroup S. We show that the enveloping semigroup of L is homeomorphically isomorphic to if and only if given q ≠ r in , there is some p in the smallest ideal of with qp ≠ rp. We derive several conditions, some involving minimal flows, which are equivalent to the ability to separate q and r in this fashion, and then specialize to the case that S = , and the compactification is . Included is the statement that some set A whose characteristic function is uniformly recurrent has .

We introduce a new method to calculate compositions of Brown-Peterson operations. We derive a formula for rn1 and a formula for commutators.

We give sufficient conditions for the simplicity of reduced amalgamated products of C*-algebras. We show that in some situations a minimal projection in a unital C*-algebra A is minimal in a free product A *-cB. We show that in certain situations if a minimal projection in A were minimal in a particular reduced free product of A and B then the reduced free product would be a simple C*-algebra which has finite and infinite projections.

Let with . If G is a nondiscrete locally compact group which is amenable as a discrete group and m ∈ LIM(CB(G)), then we can embed into the set of all extensions of m to left invariant means on L∞(G) which are mutually singular to every element of TLIM(L∞(G)), where LIM(S) and TLIM(S) are the sets of left invariant means and topologically left invariant means on S with S = CB(G) or L∞(G). It follows that the cardinalities of LIM(L∞(G)) ̴ TLIM(L∞(G)) and LIM(L∞(G)) are equal. Note that which contains is a very big set. We also embed into the set of all left invariant means on CB(G) which are mutually singular to every element of TLIM(CB(G)) for G = G1 ⨯ G2, where G1 is nondiscrete, non–compact, σ–compact and amenable as a discrete group and G2 is any amenable locally compact group. The extension of any left invariant mean on UCB(G) to CB(G) is discussed. We also provide an answer to a problem raised by Rosenblatt.

We prove the following result. Let A be a direct limit of direct sums of C*-algebras of the form C(X) ⊗ Mn, with the spaces X being compact metric. Suppose that there is a finite upper bound on the dimensions of the spaces involved, and suppose that A is simple. Then the C* exponential rank of A is at most 1 + ε, that is, every element of the identity component of the unitary group of A is a limit of exponentials. This is true regardless of whether the real rank of A is 0 or 1.

In recent years there has been a growing interest in problems of factorization for bounded linear operators. We first show that many of these problems properly belong to the category of C*-algebras. With this interpretation, it becomes evident that the problem is fundamental both to the structure of operator algebras and the elements therein. In this paper we consider the direct integral algebra with separable and infinite dimensional. We generalize a theorem of Wu (1988) and characterize those decomposable operators which are products of non-negative decomposable operators. We do this by first showing that various results on operator ranges may be generalized to “measurable fields of operator ranges”.

Let L be an orthomodular poset. A positive measure ξ on L is said to be weakly purely finitely additive if the zero measure is the only completely additive measure majorized by ξ. It was shown in [15] that, in an arbitrary orthomodular poset L, every positive measures μ is the sum v + ξ of a positive completely additive measure v and a weakly purely finitely additive measure ξ. We give sufficient conditions for this Yosida-Hewitt-type decomposition to be unique.

A positive measure λ on L is said to be filtering if every non-zero element p in L majorizes a non-zero element q on which λ vanishes. A filtering measure is weakly purely finitely additive. Filtering measures play a mediator role throughout these investigations since some of the aforementioned conditions are given in terms of these.

The results obtained here are then viewed in the context of Boolean lattices and applied to lattices of idempotents of non-associative JBW-algebras.

In this article we examine conditions for the appearance or nonappearance of the two extra-special 2-groups of order 32 as Galois groups over a field F of characteristic not 2. The groups in question are the central products DD of two dihedral groups of order 8, and DQ of a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups over F implies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability of DD and DQ. We conclude by providing an explicit construction of field extensions with Galois group DD.