Derivations of noncommutative Banach algebras II

Supported in part by a grant from the Ministry of Science of Slovenia.     

Dual equations of convolution type with kernels from different Banach algebras

We study dual integral equations of convolution type with kernels generated by functions from different Banach algebras of the type L₁(-, ) with weights, and defined by an operator equation. We establish theorems on solvability and Fredholmness, representations of solutions and of the resolvent kernel, and formulas for calculating the characteristic and the index of the corresponding operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 803–813, June, 1991.     

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C^∞-functions on the real line. Starting with the convolution algebra of compactly supported C^∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C^∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C^∞-functions of infinitly fast exponential decay at ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the ‘nonselfadjoint analogue’ of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.     

Slowly oscillating functions in semigroup compactifications and convolution algebras

The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications.     

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Let K be an ultrametric complete field and let E be an ultrametric space. Let A be the Banach K-algebra of bounded continuous functions from E to K and let B be the Banach K-algebra of bounded uniformly continuous functions from E to K. Maximal ideals and continuous multiplicative semi-norms on A (resp. on B) are studied by defining relations of stickiness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of A (resp. of B) is in bijection with the set of equivalence classes with respect to stickiness (resp. to contiguousness). Every prime ideal of A or B is included in a unique maximal ideal and every prime closed ideal of A (resp. of B) is a maximal ideal, hence every continuous multiplicative semi-norms on A (resp. on B) has a kernel that is a maximal ideal. If K is locally compact, every maximal ideal of A (resp. of B) is of codimension 1. Every maximal ideal of A or B is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, on A as on B the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of A, Max(A) and the Banaschewski compactification of E which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of A (resp. B) is equal to the whole set of continuous multiplicative semi-norms.     

Spectral synthesis for Banach algebras, II

This paper continues the study of spectral synthesis and the topologies and on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of . Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of . Received: 6 October 1999; in final form: 15 May 2000 / Published online: 25 June 2001     

Zero product preserving maps on Banach algebras of Lipschitz functions

Let (K,d) be a non-empty, compact metric space and α∈]0,1[. Let A be either lipα(K) or Lipα(K) and let B be a commutative unital Banach algebra. We show that every continuous linear map T:A→B with the property that T(f)T(g)=0 whenever f,g∈A are such that fg=0 is of the form T=wΦ for some invertible element w in B and some continuous epimorphism Φ:A→B.     

The greatest regular subalgebra of certain Banach algebras of vector-valued functions

Given an arbitrary commutative complex Banach algebraA, it is shown that, for various classical Banach algebras ofA-valued functions, the greatest regular subalgebra consists precisely of those functions which map into the greatest regular subalgebra ofA. The main result covers the case of continuous and differentiable functions, Lipschitz functions, and Bochner integrable functions on a locally compact abelian group. The principal tools are from the theory of tensor products of Banach algebras.