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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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
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by Anthony To-Ming Lau and Ali Ülger PDF
Trans. Amer. Math. Soc. 366 (2014), 4151-4171 Request permission

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
  • MR Author ID: 110640
  • Email: tlau@math.ualberta.ca
  • Ali Ülger
  • Affiliation: Department of Mathematics, Koc University, 34450 Sariyer, Istanbul, Turkey
  • Email: aulger@ku.edu.tr
  • Received by editor(s): August 17, 2012
  • Published electronically: April 7, 2014
  • Additional Notes: The first author was supported by NSERC grant MS 100
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3206455