On Minimal and Maximal p-operator Space Structures

We show that L∞(µ), in its capacity as multiplication operators on Lp(µ), is minimal as a p-operator space for a decomposable measure μ. We conclude that L1(μ) has a certain maximal type p-operator space structure that facilitates computations with L1(μ) and the projective tensor product.

Representing Multipliers of the Fourier Algebra on Non-Commutative Lp Spaces

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 Lp 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 Lp 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 Lp spaces, say . It is shown that is isometric to L1(G), generalising the abelian situation.

Order Structure of the Figà-Talamanca-Herz Algebra

We study the interplay between the order structure and the -operator space structure of Figà-Talamanca-Herz algebra of a locally compact group . We show that for amenable groups, an order and algebra isomorphism of Figà-Talamanca-Herz-algebras yields an isomorphism or anti-isomorphism of the underlying groups. We also give a bound for the norm of a -completely positive linear map from Figà-Talamanca-Herz algebra to its dual space.