2002/03/19 by Volker Runde
Mathematics · #math.FA #msc:46B10 #msc:46B20 #msc:46H20 #msc:46H25 #msc:46M18
published as Arch. Math. (Basel) 77 (2001), 265-272 · 10 pages
arxiv created 2002/03/19 · arxiv updated 2009/11/30
It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If A is a reflexive, amenable Banach algebra such that for each maximal left ideal L of A (i) the quotient A / L has the approximation property and (ii) the canonical map from A \check⊗ L^⊥ to (A / L) \wtensor L^⊥ is open, then A is finite-dimensional. As an application, we show that, if A is an a menable Banach algebra whose underlying Banach space is an \cal Lp-space with p ∈ (1,∞) such that for each maximal left ideal L the quotient A / L has the approximation property, then A is finite-dimensional.