This is an announcement for the paper "Counterexamples for the convexity of certain matricial inequalities" by Marius Junge and Quanhua Xu.

Abstract: In \cite{CL} Carlen and Lieb considered Minkowski type inequalities in the context of operators on a Hilbert space. More precisely, they considered the homogenous expression [ f_{pq}(x_1,...,x_n) \lel \big(tr\big((\sum_{k=1}^n x_k^q)^{p/q}\big)\big)^{1/p} \pl ] defined for positive matrices. The concavity for $q=1$ and $p<1$ yields strong subadditivity for quantum entropy. We discuss the convexity of $f_{pq}$ and show that, contrary to the commutative case, there exists a $q_0>1$ such that $f_{1q}$ is not convex for all $1<q<q_0$. This is achieved by constructing a family of interesting channels on $2\times 2$ matrices.

Archive classification: math.FA math-ph math.MP

Mathematics Subject Classification: 46L25 15A48

The source file(s), cedriv.tex: 58533 bytes, is(are) stored in gzipped form as 0709.0433.gz with size 18kb. The corresponding postcript file has gzipped size 129kb.

Submitted from: junge@math.uiuc.edu

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