Duke Mathematical Journal

Approximation properties for noncommutative Lp-spaces associated with discrete groups

Marius Junge and Zhong-Jin Ruan

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Let $ 1< p < \infty$. It is shown that if $G$ is a discrete group with the approximation property introduced by U. Haagerup and J. Kraus, then the noncommutative $L\sb p(VN(G))$-space has the operator space approximation property. If, in addition, the group von Neumann algebra $VN(G)$ has the quotient weak expectation property (QWEP), that is, is a quotient of a $C\sp \ast$-algebra with Lance's weak expectation property, then $L\sb p(V N(G))$ actually has the completely contractive approximation property and the approximation maps can be chosen to be finite-rank completely contractive multipliers on $L\sb p(V N(G))$. Finally, we show that if $G$ is a countable discrete group having the approximation property and $V N(G)$ has the QWEP, then $L\sb p(V N(G))$ has a very nice local structure; that is, it is a $\mathscr {COL}\sb p$-space and has a completely bounded Schauder basis.

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Duke Math. J., Volume 117, Number 2 (2003), 313-341.

First available in Project Euclid: 26 May 2004

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Zentralblatt MATH identifier

Primary: 46Lxx: Selfadjoint operator algebras ($C^*$-algebras, von Neumann ($W^*$-) algebras, etc.) [See also 22D25, 47Lxx]
Secondary: 22D05: General properties and structure of locally compact groups 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.


Junge, Marius; Ruan, Zhong-Jin. Approximation properties for noncommutative L p -spaces associated with discrete groups. Duke Math. J. 117 (2003), no. 2, 313--341. doi:10.1215/S0012-7094-03-11724-X. https://projecteuclid.org/euclid.dmj/1085598372

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