In this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

A group is said to be factorizable if it has a finite number of abelian subgroups, H1, H2, … Hn, such that G = H1H2 … Hn. It is shown that, if G is a factorizable or connected locally compact group, then every derivation from L1 (G) to an arbitrary L1 (G)-bimodule X is continuous.

Let F ⊂ G be closed and A(F) = A(G)/IF. If F is a Helson set then A(F)** is an amenable (semisimple) Banach algebra. Our main result implies the following theorem: Let G be a locally compact group, F ⊂ G closed, a ∈ G. Assume either (a) For some non-discrete closed subgroup H, the interior of F ∩ aH in aH is non-empty, or (b) R ⊂ G, S ⊂ R is a symmetric set and aS ⊂ F. Then A(F)** is a non-amenable non-semisimple Banach algebra. This raises the question: How ‘thin’ can F be for A(F)** to remain a non-amenable Banach algebra?

In this paper, we study the structure of certain conditional expectation on crossed product C*-algebra. In particular, we prove that the index of a conditional expectation E: B → A is finite if and only if the index of the induced expectation from B ⋊ G onto A ⋊ G is finite where G is a discrete group acting on B.

The bidual of a unital infrabarrelled l.m.c. C* algebra E, equipped with the bidual topology and the regualr Arens product, is always an l.m.c. C*-algebra. On the other hand, a unital l.m.c. *-algebra E has the C*-property if and only if every self-adjoint element x of E is strongly hermitian (x has real numerical range), or the sets of normalized states and normalized continuous positive linear forms of E coincide. Finally, every unital cpmplete l.m.c. C* algebra satisfying, locally, the property ‘the extreme points are dense in that set of continuous positive linear forms” (antiliminal algebra) has the complexes as its only normal elements.

Many Banach algebras A have the property that, although there are discontinuous homomorphisms from A into other Banach algebras, every homomorphism from A into another Banach algebra is automatically continuous on a dense subspace—preferably, a subalgebra—of A. Examples of such algebras are C*-algebras and the group algebras L1(G), where G is a locally compact, abelian group. In this paper, we prove analogous results for , where E is a Banach space, and . An important rôle is played by the second Hochschild cohomology group of and , respectively, with coefficients in the one-dimensional annihilator module. It vanishes in the first case and has linear dimension one in the second one.

The paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

Gelfand-type duality results can be obtained for locally convex algebras using a quasitopological structure on the spectrum of an algebra (as opposed to the topologies traditionally considered). In this way, the duality between (commutative, with identity) C*-algebras and compact spaces can be extended to pro-C*-algebras and separated quasitopologies. The extension is provided by a functional representation of an algebra A as the algebra of all continuous numerical functions on a quasitopological space. The first half of the paper deals with uniform spaces and quasitopologies, and has independent interest.

We develop several iterative methods for computing generalized inverses using both first and second order optimization methods in C*-algebras. Known steepest descent iterative methods are generalized in C*-algebras. We introduce second order methods based on the minimization of the norms ‖Ax − b‖2 and ‖x‖2 by means of the known second order unconstrained minimization methods. We give several examples which illustrate our theory.

Let A be a semisimple modular annihilator Banach algebra and let LA be the left regular representation of A. We show how the strong radical of A is related to the strong radical of LA.

The connection between Clifford analysis and the Weyl functional calculus for a d-tuple of bounded selfadjoint operators is used to prove a geometric condition due to J. Bazer and D. H. Y. Yen for a point to be in the support of the Weyl functional calculus for a pair of hermitian matrices. Examples are exhibited in which the support has gaps.

The quotient bounded and the universally bounded elements in a calibrated locally convex algebra are defined and studied. In the case of a generalized B*-algebra A, they are shown to form respectively b* and B*-algebras, both dense in A. An internal spatial characterization of generalized B*-algebras is obtained. The concepts are illustrated with the help of examples of algebras of measurable functions and of continuous linear operators on a locally convex space.

Let G be a locally compact group G (which may be non-abelian) and Ap(G) the p-Fourier algebra of Herz (1971). This paper is concerned with the Fourier algebra Al, p(G) = Ap(G) ∩ L1(G) and various relations that exist between Al, p(G), Ap(G) and G.

We present a symmetric version of a normed algebra of quotients for each ultraprime normed algebra. In addition, a C*-a1gebra of quotients of an arbitrary C*-a1gebra is introduced.

Topological algebra bundles whose fibre (-algebras) admit functional representations constitute a category, antiequivalent with that of (topological) fibre bundles having completely regular bundle spaces and locally compact fibres.

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25–39) and of Cline and Greville (Linear Algebra Appl. 29(1980), 53–62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakočević and Wei is solved.

We deal with the dual Banach algebras for a locally compact group G. We investigate compact left multipliers on , and prove that the existence of a compact left multiplier on is equivalent to compactness of G. We also describe some classes of left completely continuous elements in .

In this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions, and then investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.

The question, whether a given element of a C*-algebra has an image of rank one in some faithful representation, was studied in [3]. Such elements were characterised there by the property of being “single” (as defined below). As was pointed out in [3], Section 5, this criterion fails for general Banach algebras and the purpose of this paper is to provide a stronger condition giving the required representation property for any semi-simple Banach algebra.