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A short computation of the norms of free convolution operators. (English) Zbl 0599.43007

Author’s abstract: ”Akemann and Ostrand in 1976 gave a formula for the norms of free convolution operators on the \(L^ 2\)-space of a discrete group. Using random walk techniques and generating functions, a short and elementary computation of this formula is given.”
Reviewer: M.Leinert

MSC:

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
60G50 Sums of independent random variables; random walks
20E05 Free nonabelian groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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References:

[1] Charles A. Akemann and Phillip A. Ostrand, Computing norms in group \?*-algebras, Amer. J. Math. 98 (1976), no. 4, 1015 – 1047. · Zbl 0342.22008 · doi:10.2307/2374039
[2] Christian Berg and Jens Peter Reus Christensen, Sur la norme des opérateurs de convolution, Invent. Math. 23 (1974), 173 – 178 (French). · Zbl 0261.22009 · doi:10.1007/BF01405169
[3] Peter Gerl and Wolfgang Woess, Local limits and harmonic functions for nonisotropic random walks on free groups, Probab. Theory Relat. Fields 71 (1986), no. 3, 341 – 355. · Zbl 0562.60011 · doi:10.1007/BF01000210
[4] Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336 – 354. · Zbl 0092.33503
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