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

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 is an element of 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 BAT if there is an idempotent u is an element of A** such that I = I-u and the subalgebra Au is norm closed in A**. b) For any closed ideal I of A with a BAT, 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 VN(G) of an amenable group G is naturally complemented in VN(G) if the spectrum of X belongs to the closed coset ring R-c(G(d)) of G(d), the discrete version of G. This paper contains several applications of these results