1. CJM 2011 (vol 63 pp. 798)
|Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ 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 $L^p$ 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
Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52
2. CJM 2009 (vol 61 pp. 241)
|Operator Integrals, Spectral Shift, and Spectral Flow |
We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general \vNa s. For semifinite \vNa s we give applications to the Fr\'echet differentiation of operator functions that sharpen existing results, and establish the Birman--Solomyak representation of the spectral shift function of M.\,G.\,Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.
Categories:47A56, 47B49, 47A55, 46L51