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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

transactions of the american mathematical society

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