Algebras of Vector Functions over Normed Fields

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

On constructing multiplications on Banach spaces

Duality and operator algebras II: operator algebras as Banach algebras

Let A be a Banach algebra. For f ∈ A*, we inspect the weak sequential properties of the well-known map Tf : A → A*, Tf (a) = fa, where fa ∈ A* is defined by fa(x) = f (ax) for all x ∈ A. We provide equivalent conditions for when Tf is completely continuous for every f ∈ A*, and for when Tf maps weakly precompact sets onto L-sets for every f ∈ A*. Our results have applications to the algebra of compact operators K(X) on a Banach space X.

We now turn to the definitions and fundamental properties of (in general, non commutative) Banach algebras and some more general topological algebras. Thus we are moving from the foundations given in Chapter 1, and shall now examine the implications of the additional assumption that the algebras that we are considering also have a topology; in each case, the topology is connected to the algebraic structure by the simple requirement that the algebraic operations be continuous.

We define spatial [Formula: see text] AF algebras for [Formula: see text], and prove the following analog of the Elliott AF algebra classification theorem. If [Formula: see text] and [Formula: see text] are spatial [Formula: see text] AF algebras, then the following are equivalent: [Formula: see text] and [Formula: see text] have isomorphic scaled preordered [Formula: see text]-groups. [Formula: see text] as rings. [Formula: see text] (not necessarily isometrically) as Banach algebras. [Formula: see text] is isometrically isomorphic to [Formula: see text] as Banach algebras. [Formula: see text] is completely isometrically isomorphic to [Formula: see text] as matricial [Formula: see text] operator algebras. As background, we develop the theory of matricial [Formula: see text] operator algebras, and show that there is a unique way to make a spatial [Formula: see text] AF algebra into a matricial [Formula: see text] operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered [Formula: see text]-group of a spatial [Formula: see text] AF algebra.

Since the appearance of Banach algebra theory, the interaction between the theories ofBanach algebras with involution and that of bounded linear operators on a Hilbert space hasbeen extensively developed. The connections of Banach algebras with the theory ofbounded linear operators on a Hilbert space have also evolved, and Calkin Algebras andAlgebras of Operators on Banach Spaces provides an introduction to this set of ideas.The book begins with a treatment of the classical Riesz-Schauder theory which takesadvantage of the most recent developments-some of this material appears here for the firsttime. Although the reader should be familiar with the basics of functional analysis, anintroductory chapter on Banach algebras has been included. Other topics dealt with includeFredholm operators, semi-Fredholm operators, Riesz operators. and Calkin algebras.This volume will be of direct interest to both graduate students and research mathematicians.

On Banach algebras of band-dominated operators and their order structure

Quantum Harmonic Analysis and Geometric Invariants

We show that a real Banach algebra $A$ such that $\Vert a^{2}\Vert =\Vert a\Vert^{2}$ for $a\in A$ is a subalgebra of the algebra $C_{\mathbb{H}}( X) $ of continuous quaternion-valued functions on a compact set $X$.

This work has arisen from lecture courses given by the authors on important topics within functional analysis. The authors, who are all leading researchers, give introductions to their subjects at a level ideal for beginning graduate students, and others interested in the subject. The collection has been carefully edited so as to form a coherent and accessible introduction to current research topics. The first chapter by Professor Dales introduces the general theory of Banach algebras, which serves as a background to the remaining material. Dr Willis then studies a centrally important Banach algebra, the group algebra of a locally compact group. The remaining chapters are devoted to Banach algebras of operators on Banach spaces: Professor Eschmeier gives all the background for the exciting topic of invariant subspaces of operators, and discusses some key open problems; Dr Laursen and Professor Aiena discuss local spectral theory for operators, leading into Fredholm theory.

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

We investigate two systematic constructions of inverse-closed subalgebras of a given Banach algebra or operator algebra $\ensuremath {\mathcal {A}}$ , both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on $\ensuremath {\mathcal {A}}$ and yields a family of smooth inverse-closed subalgebras of $\ensuremath {\mathcal {A}}$ that resemble the usual Hölder–Zygmund spaces. The second construction starts with a graded sequence of subspaces of $\ensuremath{\mathcal{A}}$ and yields a class of inverse-closed subalgebras that resemble the classical approximation spaces. We prove a theorem of Jackson–Bernstein type to show that in certain cases both constructions are equivalent. These results about abstract Banach algebras are applied to algebras of infinite matrices with off-diagonal decay. In particular, we obtain new and unexpected conditions of off-diagonal decay that are preserved under matrix inversion.

Let G be a locally compact abelian group and let \({\bf T}{\text{ = }}{\left\{ {T{\left( g \right)}} \right\}}_{{g \in G}} \) be a representation of G by means of isometries on a Banach space. We define WT as the closure with respect to the weak operator topology of the set \({\left\{ {\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)}:f \in L^{1} {\left( G \right)}} \right\}}, \) where \(\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)} = {\int\limits_G {f{\left( g \right)}T{\left( g \right)}dg} } \) is the Fourier transform of f ∈L1(G) with respect to the group T. Then WT is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that if the Arveson spectrum sp(T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra WT is semisimple.

Many Banach algebrasAhave the property that, although there are discontinuous homomorphisms fromAinto other Banach algebras, every homomorphism fromAinto another Banach algebra is automatically continuous on a dense subspace—preferably, a subalgebra—ofA. Examples of such algebras areC*-algebras and the group algebrasL1(G), whereGis a locally compact, abelian group. In this paper, we prove analogous results for, whereEis a Banach space, and. An important rôle is played by the second Hochschild cohomology group ofand, respectively, with coefficients in the one-dimensional annihilator module. It vanishes in the first case and has linear dimension one in the second one.