2023/04/13 by Brian Forrest, Forrest, Brian, John Sawatzky +3
Mathematics · #Advanced Topics in Algebra #Advanced Operator Algebra Research #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.2304.06195
Let G be a locally compact group. In this paper, we study various invariant subspaces of the duals of the algebras AM(G) and Acb(G) obtained by taking the closure of the Fourier algebra A(G) in the multiplier algebra MA(G) and completely bounded multiplier algebra McbA(G) respectively. In particular, we will focus on various functorial properties and containment relationships between these various invariant subspaces including the space of uniformly continuous functionals and the almost periodic and weakly almost periodic functionals. Amongst other results, we show that if A(G) is either AM(G) or Acb(G), then UCB(A(G))⊆ WAP(G) if and only if G is discrete. We also show that if UCB(A(G))=A(G)^*, then every amenable closed subgroup of G is compact. Let i:A(G)→ A(G) be the natural injection. We show that if X is any closed topologically introverted subspace of A(G)^* that contains L1(G), then i^*(X) is closed in A(G) if and only if G is amenable.