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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Norm of convolution by operator-valued
functions on free groups


Author: Artur Buchholz
Journal: Proc. Amer. Math. Soc. 127 (1999), 1671-1682
MSC (1991): Primary 43A30; Secondary 43A65
Published electronically: February 4, 1999
MathSciNet review: 1476122
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Abstract: We present a connection between the Leinert sets and the non-crossing two-partitions and we use this connection to give a simple proof of the free Khintchine inequality in discrete non-commutative $L_p$-spaces. Moreover we extend the inequality of Haagerup-Pisier,

\begin{displaymath}\left\| \sum _{g\in S} \lambda(g)\otimes a_g\right\|_{C_\lambda^*(F_n)\otimes A} \le 2\max\left\{\left\| \sum _{g\in S} a_g^*a_g\right\|^{\frac 12}, \left\|\sum _{g\in S} a_g a_g^*\right\|^{\frac 12}\right\}, \end{displaymath}

where $\lambda$ is the left regular representation of the group $F_n$, $a_g$ are elements of the $C^*$-algebra $A$, and $S$ is the set of the words with length one, to the set $S$ of the words with arbitrary fixed length.


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Additional Information

Artur Buchholz
Affiliation: Institute of Mathematics, University of Wroclaw, Wroclaw pl. Grunwaldzki 2/4, Poland
Email: buchholz@math.uni.wroc.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04660-2
PII: S 0002-9939(99)04660-2
Keywords: Convolution operator, free group, Leinert's set, Khintchine inequality
Received by editor(s): September 23, 1996
Received by editor(s) in revised form: September 3, 1997
Published electronically: February 4, 1999
Additional Notes: This paper is part of the author’s Master Thesis under Prof.\ M.\ Bożejko, supported by KBN grant 2P03A05108
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society