Algebras of operators on Banach spaces, and homomorphisms thereof
Creators: name: Horvath, Bence
Publisher: name: Lancaster University
Year: 2019

description: This thesis concerns the theory of Banach algebras of operators on Banach
      spaces. The emphasis for most of the thesis is on the homomorphisms and
      perturbations of homomorphisms of such algebras. Chapter 2 of this thesis
      is devoted to the study of perturbations of homomorphisms between Banach
      algebras. We say that a bounded linear map ϕ : A → B between Banach
      algebras A and B is δ-multiplicative, if δ &gt; 0 and ∥ϕ(ab)−ϕ(a)ϕ(b)∥
      ≤ δ∥a∥∥b∥ for every a, b ∈ A. Intuitively, if ϕ is δ-multiplicative for a
      small δ then ϕ may be considered to be almost homomorphic. Another way of
      measuring how perturbed a bounded linear map is to see how near it is to
      the closed set of multiplicative bounded linear maps between A and B. One
      would like to explore, of course, the connection between these two
      notions. If a bounded linear map is ϵ-close to a genuine homomorphism in
      norm, then it is δ-multiplicative for some δ depending on ϵ and the norm
      of the map. More precisely: Let A, B be Banach algebras and let ϵ, K
      &gt; 0. It is easy to check (see [38, Proposition 1.1]) that there is
      a δ &gt; 0 such that for any bounded linear map ϕ : A → B with ∥ϕ∥ ≤
      K, if there is a bounded linear multiplicative map ψ : A → B with ∥ϕ − ψ∥
      &lt; δ, then ϕ is ϵ-multiplicative. The converse direction is far from
      being true, in general. Pairs of Banach algebras (A, B) which satisfy the
      converse direction, are said to have the AMNM property, see Definition
      2.1.1 for a precise formulation. Although this topic has its roots in the
      classical Hyers–Ulam stability theory, it was really K. Jarosz who
      initiated the study of this field in [36]. Later, B. E. Johnson in [37]
      studied the AMNM property of Banach algebras of the form (A, K), where A
      is a commutative Banach algebra over some scalar field K. He substantially
      extended the scope of his study in his seminal paper [38], where he now
      turned his attention to general pairs of Banach algebras. Arguably, the
      most important result of his paper is [38, Theorem 3.1], which states that
      if A is an amenable Banach algebra and B is a dual Banach algebra then (A,
      B) has the AMNM property. By carefully studying the proof of Johnson’s
      theorem, we extend this result in Theorem 2.3.3. This allows us to prove
      the main result (Theorem 2.1.2) of Chapter 2, namely, that (B(E), B(F))
      has the AMNM property whenever E ∈ {ℓ1, Lp[0, 1] : 1 ≤ p &lt; ∞} or E
      is a Banach space with a subsymmetric, shrinking Schauder basis, and F is
      a separable, reflexive Banach space. (Here B(E) denotes the Banach algebra
      of bounded linear operators on E.) This extends another result of Johnson;
      it was shown in [38, Proposition 6.3] that (B(H), B(H)) has the AMNM
      property, where H is a separable Hilbert space. We would like to draw the
      reader’s attention to the fact that rather non-trivial methods are needed
      to obtain this latter result. The reason behind this is twofold: Firstly,
      B(E) is very rarely amenable for infinite-dimensional Banach spaces E, see
      [69]. Secondly, when considering bounded linear maps between B(E) and
      B(F), they are not assumed to be weak∗ - to - weak∗ continuous, even if
      both E and F are reflexive. In the penultimate section of Chapter 2 we
      study δ-multiplicative maps φ : A(E) → B(F), where E and F are Banach
      spaces and A(E) denotes the Banach algebra of approximable operators on E.
      Inspired by a result of M. D. P. Daws ([17, Lemma 3.3.14]), we show that
      if E has a Schauder basis, φ is a δ-multiplicative, norm one linear map
      from A(E) to B(F) for a sufficiently small δ &gt; 0, then F has a
      closed subspace isomorphic to E. In the last section of Chapter 2 we
      establish a connection between the second bounded Hochschild cohomology
      group of a Banach algebra A with coefficients in a Banach A-bimodule X,
      and the AMNM property of the pair (A, A ⋉ X), where A ⋉ X denotes the
      semi-direct product of A with X. In Chapter 3 we study representations of
      Banach algebras of the form B(E). More precisely, we are interested in the
      following question: Suppose E is a Banach space. If F is a non-zero Banach
      space and ψ : B(E) → B(F) is a surjective algebra homomorphism, is ψ
      necessarily injective? Positive answer to this question would imply — by a
      classical result of Eidelheit, see e.g. [15, Theorem 2.5.7] — that E and F
      are isomorphic. If E satisfies the property above for every non-zero
      Banach space F and every surjective algebra homomorphism ψ : B(E) → B(F),
      then we say that E has the SHAI property (Definition 3.1.1). As we shall
      see in Chapter 3, many of the classical Banach spaces and Hilbert spaces
      of arbitrary density character (that is, of arbitrarily large dimension)
      satisfy this property (see Example 3.3.8 and Theorem 3.4.8), as well as
      some more exotic ones (Corollary 3.3.12 and Theorem 3.4.6). We demonstrate
      in Proposition 3.5.3 that the SHAI property is stable under taking finite
      sums. We also give many examples for Banach spaces which fail to have this
      property, see Examples 3.2.2. In the last section of Chapter 3, we study
      the question in the “opposite direction”: Can we find infinite-dimensional
      Banach spaces E and F and a surjective algebra homomorphism ψ : B(E) →
      B(F) which is not injective? We shall see in Theorem 3.6.13 that for every
      separable, reflexive Banach space F it is possible to find a Banach space
      E and a surjective algebra homomorphism ψ : B(E) → B(F) which is not
      injective. In Chapter 4 we study certain ring-theoretic properties of
      B(E), characterised by their idempotents. In the classical
      (non-commutative) ring-theoretic context the notions of
      Dedekind-finiteness, Dedekind-infiniteness and proper infiniteness are
      very well know. In a Banach algebraic context, there is a stronger notion
      than Dedekindfiniteness; this is having stable rank one, in the sense of
      Rieffel, see Definition 4.1.3 and Lemma 4.2.1. The study of these
      ring-theoretic properties of B(E) was laid out by N. J. Laustsen in [47].
      In particular, it was shown in the aforementioned paper that B(E) is
      Dedekind-finite whenever E is a complex hereditarily indecomposable Banach
      space. We shall see in Corollary 4.2.9 that in fact more is true, for such
      a Banach space E, its algebra of operators B(E) has stable rank one. To
      complement this, we show in Theorems 4.2.16 and 4.3.5 that there are
      examples for both complex and real Banach spaces such that their algebras
      of operators are Dedekind-finite but they do not have stable rank one. The
      complex example is provided by the complexification of Tarbard’s
      indecomposable but not hereditarily indecomposable space X∞, and the real
      example is C(K, R), where K is a connected Koszmider space. In the last
      section of Chapter 4 we explore the connection between the existence of a
      certain unique maximal ideal in a Banach algebra and Dedekind-finiteness.