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Zero Lie product determined Banach algebras

Volume 239 / 2017

J. Alaminos, M. Brešar, J. Extremera, A. R. Villena Studia Mathematica 239 (2017), 189-199 MSC: 43A20, 46H05, 46L05, 47B48. DOI: 10.4064/sm8734-4-2017 Published online: 7 July 2017

Abstract

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $\varphi \colon A\times A\to \mathbb {C}$ with $\varphi (a,b)=0$ whenever $a$ and $b$ commute is of the form $\varphi (a,b)=\tau (ab-ba)$ for some $\tau \in A^*$. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras $L^1(G)$ with $G$ an amenable locally compact group are zero Lie product determined.

Authors

  • J. Alaminos
  • M. Brešar
  • J. Extremera
  • A. R. Villena

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