1. CJM 2016 (vol 68 pp. 1067)
||On Positive Definiteness over Locally Compact Quantum Groups|
The notion of positive-definite functions over locally compact
groups was recently introduced and studied by Daws and Salmi.
on this work, we generalize various well-known results about
functions over groups to the quantum framework. Among these are
on "square roots" of positive-definite functions, comparison
various topologies, positive-definite measures and characterizations
of amenability, and the separation property with respect to compact
Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation property
Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89
2. 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
3. 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