Banach algebras on groups and semigroups
Creators: name: White, Jared
Publisher: name: Lancaster University
Year: 2018

description: This thesis concerns the theory of Banach algebras, particularly those
      coming from abstract harmonic analysis. The focus for much of the thesis
      is the theory of the ideals of these algebras. In the final chapter we use
      semigroup algebras to solve an open probelm in the theory of C*-algebras.
      Throughout the thesis we are interested in the interplay between abstract
      algebra and analysis. Chapters 2, 4, and 5 are closely based upon the
      articles [88], [89], and [56], respectively.
      
      In Chapter 2 we study (algebraic) finite-generation of closed left ideals
      in Banach algebras. Let G be a locally compact group. We prove that the
      augmentation ideal in L 1 pGq is finitely-generated as a left ideal if and
      only if G is finite. We then investigate weighted versions of this result,
      as well as a version for semigroup algebras. Weighted measure algebras are
      also considered. We are motivated by a recent conjecture of Dales and
      Żelazko, which states that a unital Banach algebra in which every maximal
      left ideal is finitely-generated is necessarily finite-dimensional. We
      prove that this conjecture holds for many of the algebras considered.
      Finally, we use the theory that we have developed to construct some
      examples of commutative Banach algebras that relate to a theorem of
      Gleason.
      
      In Chapter 3 we turn our attention to topological finite-generation of
      closed left ideals in Banach algebras. We define a Banach algebra to be
      topologically left Noetherian if every closed left ideal is topologically
      finitely-generated, and we seek infinitedimensional examples of such
      algebras. We show that, given a compact group G, the group algebra L 1 pGq
      is topologically left Noetherian if and only if G is metrisable. For a
      Banach space E satisying a certain condition we show that the Banach
      algebra of approximable operators ApEq is topologically left Noetherian if
      and only if E 1 is separable, whereas it is topologically right Noetherian
      if and only if E is separable. We also define what it means for a dual
      Banach algebra to be weak*-topologically left Noetherian, and give
      examples which satisfy and fail this condition. Along the way, we give
      classifications of the weak*-closed left ideals in MpGq, for G a compact
      group, and in BpEq, for E a reflexive Banach space with AP.
      
      Chapter 4 looks at the Jacobson radical of the bidual of a Banach algebra.
      We prove that the bidual of a Beurling algebra on Z, considered as a
      Banach algebra with the first Arens product, can never be semisimple. We
      then show that rad p` 1 p‘8 i“1Zq 2 q contains nilpotent elements of every
      index. Each of these results settles a question of Dales and Lau. Finally
      we show that there exists a weight ω on Z such that the bidual of ` 1 pZ,
      ωq contains a radical element which is not nilpotent.
      
      In Chapter 5 we move away from the theory of ideals and consider a
      question about the notion of finiteness in C*-alegebras. We construct a
      unital pre-C*-algebra A0 which is stably finite, in the sense that every
      left invertible square matrix over A0 is right invertible, while the
      C*-completion of A0 contains a non-unitary isometry, and so it is
      infinite. This answers a question of Choi. The construction is based on
      semigroup algebras.