Abstract view
Representing Multipliers of the Fourier Algebra on NonCommutative $L^p$ Spaces


Published:20110325
Printed: Aug 2011
Matthew Daws,
School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
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 noncommutative $L^p$ 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 noncommutative $L^p$ 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 FigaTalamancaHerz
algebra built out of these noncommutative $L^p$ spaces, say
$A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to
$L^1(G)$, generalising the abelian situation.
MSC Classifications: 
43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 show english descriptions
Homomorphisms and multipliers of function spaces on groups, semigroups, etc. Fourier and FourierStieltjes 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 FourierStieltjes 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
