Herz–Schur multiplier

In the mathematical field of representation theory, a Herz–Schur multiplier (named after Carl S. Herz and Issai Schur) is a special kind of mapping from a group to the field of complex numbers.

Definition

Let Ψ be a mapping of a group G to the complex numbers. It is a Herz–Schur multiplier if the induced map Ψ: N(G) → N(G) is a completely positive map, where N(G) is the closure of the span M of the image of λ in B(ℓ ²(G)) with respect to the weak topology, λ is the left regular representation of G and Ψ is on M defined as

[math]\displaystyle{ \Psi:~\sum\limits_{g\in G}\mu_g\lambda_g\mapsto\sum\limits_{g\in G}\psi(g)\mu_g\lambda_g. }[/math]

See also

• Figà-Talamanca–Herz algebra
• Fourier algebra

References

• Pisier, Gilles (1995), "Multipliers and lacunary sets in non-amenable groups", American Journal of Mathematics (The Johns Hopkins University Press) 117 (2): 337–376, doi:10.2307/2374918, ISSN 0002-9327 
• Figà-Talamanca, Alessandro; Picardello, Massimo A. (1983), Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, 87, New York: Marcel Dekker Inc., ISBN 978-0-8247-7042-6 
• Carl S. Herz. Une généralisation de la notion de transformée de Fourier-Stieltjes. Annales de l'Institut Fourier, tome 24, no 3 (1974), p. 145-157.