1. CJM Online first
 Runde, Volker; Viselter, Ami

On positive definiteness over locally compact quantum groups
The notion of positivedefinite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various wellknown results about
positivedefinite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positivedefinite functions, comparison
of
various topologies, positivedefinite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.
Keywords:bicrossed product, locally compact quantum group, noncommutative $L^p$space, positivedefinite function, positivedefinite measure, separation property Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89 

2. CJM 2011 (vol 63 pp. 798)
 Daws, Matthew

Representing Multipliers of the Fourier Algebra on NonCommutative $L^p$ 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 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.
Keywords:multiplier, Fourier algebra, noncommutative $L^p$ space, complex interpolation Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 

3. CJM 2004 (vol 56 pp. 983)
 Junge, Marius

Fubini's Theorem for Ultraproducts \\of Noncommutative $L_p$Spaces
Let $(\M_i)_{i\in I}$, $(\N_j)_{j\in J}$ be families of von
Neumann algebras and $\U$, $\U'$ be ultrafilters in $I$, $J$,
respectively. Let $1\le p<\infty$ and $\nen$. Let $x_1$,\dots,$x_n$ in
$\prod L_p(\M_i)$ and $y_1$,\dots,$y_n$ in $\prod L_p(\N_j)$ be
bounded families. We show the following equality
$$
\lim_{i,\U} \lim_{j,\U'} \Big\ \summ_{k=1}^n x_k(i)\otimes
y_k(j)\Big\_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U}
\Big\ \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\_{L_p(\M_i\otimes \N_j)} .
$$
For $p=1$ this Fubini type result is related to the local
reflexivity of duals of $C^*$algebras. This fails for $p=\infty$.
Keywords:noncommutative $L_p$spaces, ultraproducts Categories:46L52, 46B08, 46L07 

4. CJM 2004 (vol 56 pp. 225)
 Blower, Gordon; Ransford, Thomas

Complex Uniform Convexity and Riesz Measure
The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~NeumannSchatten trace ideals. Banach spaces that are $q$uniformly
$\PL$convex in the sense of Davis, Garling and TomczakJaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$uniformly $\PL$convex for
$1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals Categories:46B20, 46L52 

5. CJM 2001 (vol 53 pp. 979)
 Nagisa, Masaru; Osaka, Hiroyuki; Phillips, N. Christopher

Ranks of Algebras of Continuous $C^*$Algebra Valued Functions
We prove a number of results about the stable and particularly the
real ranks of tensor products of \ca s under the assumption that one
of the factors is commutative. In particular, we prove the following:
{\raggedright
\begin{enumerate}[(5)]
\item[(1)] If $X$ is any locally compact $\sm$compact Hausdorff space
and $A$ is any \ca, then\break
$\RR \bigl( C_0 (X) \otimes A \bigr) \leq
\dim (X) + \RR(A)$.
\item[(2)] If $X$ is any locally compact Hausdorff space and $A$ is
any \pisca, then $\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$.
\item[(3)] $\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$ for any
nonzero \ca\ $A$, and $\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$
for any unital \ca\ $A$.
\item[(4)] If $A$ is a unital \ca\ such that $\RR(A) = 0$, $\sr (A) =
1$, and $K_1 (A) = 0$, then\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\item[(5)] There is a simple separable unital nuclear \ca\ $A$ such
that $\RR(A) = 1$ and\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\end{enumerate}}
Categories:46L05, 46L52, 46L80, 19A13, 19B10 
