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

2014
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Transactions of the American Mathematical Society
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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 ∈ 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

doi:10.1090/s0002-9947-2014-06336-8
fatcat:zlfcas34ejeovid7tpozvmd73a