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Archiv der Mathematik

, Volume 77, Issue 3, pp 265–272 | Cite as

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

  • V. Runde
Article

Abstract.

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If \( {\frak A} \) is a reflexive, amenable Banach algebra such that for each maximal left ideal L of \( {\frak A} \) (i) the quotient \( {\frak A}/L \) has the approximation property and (ii) the canonical map from \( {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp \) is open, then \( {\frak A} \) is finite-dimensional. As an application, we show that, if \({\frak A}\) is an amenable Banach algebra whose underlying Banach space is an Lp-space with \( p\in (1,\infty) \) such that for each maximal left ideal L the quotient \( {\frak A}/L \) has the approximation property, then \( {\frak A} \) is finite-dimensional.

Keywords

Banach Space Open Problem Space Property Banach Algebra Approximation Property 

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • V. Runde
    • 1
  1. 1.Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1Canada

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