Bulletin of the Belgian Mathematical Society - Simon Stevin

Approximate Connes-amenability of dual Banach algebras

G. H. Esslamzadeh, B. Shojaee, and A. Mahmoodi

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We introduce the notions of approximate Connes-amenability and approximate strong Connes-amenability for dual Banach algebras. Then we characterize these two types of algebras in terms of approximate normal virtual diagonals and approximate $\sigma WC-$virtual diagonals. We investigate these properties for von Neumann algebras, measure algebra and the algebra of $p$-pseudomeasures on locally compact groups. In particular we show that a von Neumann algebra is approximately Connes-amenable if and only if it has an approximate normal virtual diagonal. This is the ``approximate'' analog of the main result of Effros in [10]. We show that in general the concepts of approximate Connes-amenability and Connes-amenability are distinct, but for measure algebras these two concepts coincide. Moreover cases where approximate Connes-amenability of $\mathcal{A}^{**}$ implies approximate Connes-amenability or approximate amenability of $\mathcal{A}$ are also discussed.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 2 (2012), 193-213.

First available in Project Euclid: 24 May 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46H20: Structure, classification of topological algebras
Secondary: 46H35: Topological algebras of operators [See mainly 47Lxx]

Approximately inner derivation Approximately Connes amenable Approximately strongly Connes amenable Approximate normal virtual diagonal Approximate $\sigma WC-$virtual diagonal


Esslamzadeh, G. H.; Shojaee, B.; Mahmoodi, A. Approximate Connes-amenability of dual Banach algebras. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 2, 193--213. doi:10.36045/bbms/1337864267. https://projecteuclid.org/euclid.bbms/1337864267

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