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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.
- Source :
-
Transactions of the American Mathematical Society . Aug2014, Vol. 366 Issue 8, p4151-4171. 21p. - Publication Year :
- 2014
-
Abstract
- 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 Iu = {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 = Iu and the subalgebra 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 VN(G) of an amenable group G is naturally complemented in VN(G) iff the spectrum of X belongs to the closed coset ring Rc(Gd) of Gd, the discrete version of G. This paper contains several applications of these results. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MATHEMATICS
*BANACH algebras
*SUBSPACES (Mathematics)
*HILBERT space
*BANACH spaces
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 366
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 96101992
- Full Text :
- https://doi.org/10.1090/S0002-9947-2014-06336-8