Characterization of closed ideals with bounded approximate identities in commutative Banach algebras, complemented subspaces of the group von Neumann algebras and applications

Anthony To-Ming Lau, Ali Ülger
2014 Transactions of the American Mathematical Society  
Let A be a commutative Banach algebra with a BAI (=bounded approximate identity). We equip A * * with the (first) Arens multiplication. To each idempotent element u of A * * we associate the closed ideal I u = {a ∈ A : au = 0} in A. In this paper we present a characterization of the closed ideals of A with BAI's in terms of idempotent elements of A * * . The main results are: a) A closed ideal I of A has a BAI iff there is an idempotent u ∈ A * * such that I = I u and the subalgebra Au is norm
more » ... algebra Au is norm closed in A * * . b) For any closed ideal I of A with a BAI, the quotient algebra A/I is isomorphic to a subalgebra of A * * . We also show that a weak * closed invariant subspace X of the group von Neumann algebra V N(G) of an amenable group G is naturally complemented in V N(G) iff the spectrum of X belongs to the closed coset ring c (G d ) of G d , the discrete version of G. This paper contains several applications of these results.
doi:10.1090/s0002-9947-2014-06336-8 fatcat:zlfcas34ejeovid7tpozvmd73a