This is an announcement for the paper "Khintchine type inequalities for reduced free products and applications" by Eric Ricard and Quanhua Xu.
Abstract: We prove Khintchine type inequalities for words of a fixed length in a reduced free product of $C^*$-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length $d$ is completely bounded with norm depending linearly on $d$. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema's theorem on the stability of exactness under the reduced free product for $C^*$-algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional $C^*$-algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak-$*$ CCAP. In the case of group $C^*$-algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.
Archive classification: Operator Algebras; Functional Analysis
Mathematics Subject Classification: Primary 46L09, 46L54; Secondary 47L07, 47L25
The source file(s), kpl.tex: 94450 bytes, is(are) stored in gzipped form as 0505302.gz with size 30kb. The corresponding postcript file has gzipped size 136kb.
Submitted from: qx@math.univ-fcomte.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.OA/0505302
or
http://arXiv.org/abs/math.OA/0505302
or by email in unzipped form by transmitting an empty message with subject line
uget 0505302
or in gzipped form by using subject line
get 0505302
to: math@arXiv.org.