Given a representation θ: A → B(H) of a Banach algebra A on a Hilbert space H, H is said to have the reduction property as an A—module if every closed invariant subspace of H is complemented by a closed invariant subspace; A has the total reduction property if for every representation θ: A → B(H), H has the reduction property.

We show that a C*—algebra has the total reduction property if and only if all its representations are similar to *—representations. The question of whether all C*-algebras have this property is the famous ‘similarity problem’ of Kadison.

We conjecture that non-self-adjoint operator algebras with the total reduction property are always isomorphic to C*-algebras, and prove this result for operator algebras consisting of compact operators.

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.

Valuated Butler groups of finite rank are investigated. The valuated B2-groups are both epic images and pure subgroups of completely decomposable valuated groups of finite rank (Theorem 3.1). However, there are fundamental changes in the theory of Butler groups when valuations are involved. We introduce valuated B1-groups and show that they are valuated B2-groups. Surprisingly, valuated B2-groups of rank greater than 1 need not be valuated B1 -groups, unless they carry a special kind valuation, see Theorem 7.1. Theorem 6.5 gives a full characterization of valuated B1 -groups of finite rank, generalizing Bican's characterization of Butler groups.

In this article, a Blackburn group refers to a finite non-Dedekind group for which the intersection of all nonnormal subgroups is not the trivial subgroup. By completing the arguments of M. Hertweck, we show that all conjugacy class preserving automorphisms of Blackburn groups are inner automorphisms.

A module M is said to satisfy the condition (℘*) if M is a direct sum of a projective module and a quasi-continuous module. In an earlier paper, we described the structure of rings over which every (countably generated) right module satisfies (℘*), and it was shown that such a ring is right artinian. In this note some additional properties of these rings are obtained. Among other results, we show that a ring over which all right modules satisfy (℘*) is also left artinian, but the property (℘*) is not left-right symmetric.

For α > 0 let α denote the set of functions which can be expressed where μ is a complex-valued Borel measure on the unit circle. We show that if f is an analytic function in Δ = {z ∈ : |z| < 1} and there are two nonparallel rays in /f(Δ) which do not meet, then f ∈ α where απ denotes the largest of the two angles determined by the rays. Also if the range of a function analytic in Δ is contained in an angular wedge of opening απ and 1 < α < 2, then f ∈ α.

We prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

Let aπ denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We characterize elements b such that 1 − (bπ − aπ)2 is invertible. We also apply this result in rings with involution to obtain a characterization of the perturbation of EP elements. In Banach algebras we obtain a characterization in terms of matrix representations and derive error bounds for the perturbation of the Drazin Inverse. This work extends recent results for matrices given by the same authors to the setting of rings and Banach algebras. Finally, we characterize generalized Drazin invertible operators A, B ∈ (X) such that pr(Bπ) = pr(Aπ + S), where pr is the natural homomorphism of (X) onto the Calkin algebra and S ∈(X) is given.

2000 Mathematics subject classification: primary 16A32, 16A28, 15A09.

For 0 < p < ∞, we let pp−1 denote the space of those functions f that are analytic in the unit disc Δ = {z ∈ C: |z| < 1} and satisfy ∫Δ(1 – |z|)p−1|f′(z)|pdx dy < ∞ The spaces pp−1 are closely related to Hardy spaces. We have, p−1p ⊂ Hp, if 0 < p ≦ 2, and Hp ⊂ pp−1, if 2 ≦ p < ∞. In this paper we obtain a number of results about the Taylor coefficients of pp-1 -functions and sharp estimates on the growth of the integral means and the radial growth of these functions as well as information on their zero sets.

2000 Mathematics subject classification: primary 30D35, 30D55, 46E15.

Free algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.

2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.