CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces

  Published:2011-03-25
 Printed: Aug 2011
  • Matthew Daws,
    School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF  

Abstract

We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation.
Keywords: multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation
MSC Classifications: 43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 show english descriptions Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
Noncommutative measure and integration
$C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]
Multipliers
Operator spaces and completely bounded maps [See also 47L25]
Noncommutative function spaces
43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
46L51 - Noncommutative measure and integration
22D25 - $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]
42B15 - Multipliers
46L07 - Operator spaces and completely bounded maps [See also 47L25]
46L52 - Noncommutative function spaces
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/