Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces
Printed: Aug 2011
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.
multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation
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