Khintchine type inequalities for reduced free products and applications

By 'Eric Ricard, Quanhua Xu. 2005

83 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fe15cd73fffab3228694ca3ecf75c8d973e4c2d2

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Weak Amenability of Hyperbolic Groups (2007, background): “rbolic relative to weakly amenable groups is again weakly amenable? There is no serious diﬃculty in extending Theorem to (uniformly) ﬁne hyperbolic graphs in the sense of Bowditch [Bo]. Ricard and Xu [RX] proved that weak amenability with constant one is closed under free products with ﬁnite amalgam”
CCAP for Universal Discrete Quantum Groups (2013, methodology): “Such operators behave well under monoidal equivalence, passing to discrete quantum subgroups and taking free products (the latter by adapting the technique of Ricard and Xu [33]).”
Easy quantum groups and quantum subgroups of a semi-direct product quantum group (2013, background): “In [RX06] it is proved that the free product of groups having CMAP still has CMAP.”
Approximation properties for free orthogonal and free unitary quantum groups (2011, background): “1), and it is also a useful tool for proving other approximation properties, such as the weak (completely) bounded approximation property (see [29] and [30]).”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, background): “However, it is not known whether weak amenability is closed under taking a free product of two groups except in the case that the Cowling–Haagerup constants of both groups are one [60].”
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors (2010, unknown): “Finally (5) is due to [39].”
Bass-Serre rigidity results in von Neumann algebras (2008, background): “It was proven by Ricard & Xu in [36] that such a free product M always has the complete metric approximation property, (denoted c.”