Schur multipliers and spherical functions on homogeneous trees

TY - JOUR

T1 - Schur multipliers and spherical functions on homogeneous trees

AU - Haagerup, Uffe

AU - Steenstrup, Troels

AU - Szwarc, Ryszard

PY - 2010/10

Y1 - 2010/10

N2 - Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ∥ψ∥S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ∥·∥M0A(G) of the radial functions on the free (non-abelian) group double-struck F signN on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL 2(ℚq) for every prime number q.

AB - Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ∥ψ∥S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ∥·∥M0A(G) of the radial functions on the free (non-abelian) group double-struck F signN on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL 2(ℚq) for every prime number q.

U2 - 10.1142/S0129167X10006537

DO - 10.1142/S0129167X10006537

M3 - Journal article

SN - 0129-167X

VL - 21

JO - International Journal of Mathematics

JF - International Journal of Mathematics

IS - 10

M1 - 1337

ER -