Topological convolution algebras

Daniel Alpay, Guy Salomon

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


In this paper we introduce a dual of reflexive Fréchet counterpart of Banach algebras of the form p∈NΦp' (where the Φp' are (dual of) Banach spaces with associated norms {dot operator}p), which carry inequalities of the form abp≤Ap,qaqbp and bap≤Ap,qaqbp for p>q+d, where d is preassigned and Ap,q is a constant. We study the functional calculus and the spectrum of the elements of these algebras. We then focus on the particular case Φp'=L2(S,μp), where S is a Borel semi-group in a locally compact group G, and multiplication is convolution. We give a sufficient condition on the measures μp for such inequalities to hold. Finally we present three examples, one is the algebra of germs of holomorphic functions in zero, the second related to Dirichlet series and the third in the setting of non-commutative stochastic distributions.

Original languageEnglish
Pages (from-to)2224-2244
Number of pages21
JournalJournal of Functional Analysis
Issue number9
StatePublished - 1 May 2013


  • Convolution algebra
  • Non-commutative stochastic distributions
  • Topological algebras


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