Schur Multipliers and Spherical Functions on Homogeneous Trees

2009/08/31 by Uffe Haagerup, Haagerup, Uffe, Troels Steenstrup +3 · 1 citation
Mathematics · #20E08 (Primary) 22E50 #43A90 #46L07 (Secondary) #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #advanced mathematical theories #math.GR #math.OA #msc:20E08 #msc:22E50 #msc:43A90 #msc:46L07

paper · pdf · doi:10.48550/arxiv.0908.4424

51 pages

arxiv created 2009/08/31 · openalex publication_date 2009/08/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X times X. Moreover, we find a closed expression for the Schur norm of f. As applications, we obtain a closed expression for the completely bounded Fourier multiplier norm of the radial functions on the free (non-abelian) group on N generators (for N between 2 and infinity) and of the spherical functions on the p-adic group PGL₂(Q_q) for every prime number q.

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