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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


Authors: Anthony To-Ming Lau and Ali Ülger
Journal: Trans. Amer. Math. Soc. 366 (2014), 4151-4171
MSC (2010): Primary 46H20, 43A25, 43A46; Secondary 43A22
DOI: https://doi.org/10.1090/S0002-9947-2014-06336-8
Published electronically: April 7, 2014
MathSciNet review: 3206455
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Abstract: Let $ A$ be a commutative Banach algebra with a BAI (=bounded approximate identity). We equip $ A^{\ast \ast }$ with the (first) Arens multiplication. To each idempotent element $ u$ of $ A^{\ast \ast }$ we associate the closed ideal $ I_{u}=\{a\in 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^{\ast \ast }$. The main results are: a) A closed ideal $ I$ of $ A$ has a BAI iff there is an idempotent $ u\in A^{\ast \ast }$ such that $ I=I_{u}$ and the subalgebra $ Au$ is norm closed in $ A^{\ast \ast }$. b) For any closed ideal $ I$ of $ A$ with a BAI, the quotient algebra $ A/I$ is isomorphic to a subalgebra of $ A^{\ast \ast }$. We also show that a weak$ ^{\ast }$ 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 $ \Re _{c}(G_{d})$ of $ G_{d}$, the discrete version of $ G$. This paper contains several applications of these results.


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Additional Information

Anthony To-Ming Lau
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: tlau@math.ualberta.ca

Ali Ülger
Affiliation: Department of Mathematics, Koc University, 34450 Sariyer, Istanbul, Turkey
Email: aulger@ku.edu.tr

DOI: https://doi.org/10.1090/S0002-9947-2014-06336-8
Keywords: Commutative Banach algebras, invariant complementation, idempotent, Fourier algebra, Fourier-Stieltjes algebra, group von Neumann algebra, closed ideal, coset ring, bounded approximate identity, multiplier, Arens product
Received by editor(s): August 17, 2012
Published electronically: April 7, 2014
Additional Notes: The first author was supported by NSERC grant MS 100
Article copyright: © Copyright 2014 American Mathematical Society