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
Ψ : ∑ g ∈ G μ g λ g ↦ ∑ g ∈ G ψ ( g ) μ g λ g . {\displaystyle \Psi :~\sum \limits _{g\in G}\mu _{g}\lambda _{g}\mapsto \sum \limits _{g\in G}\psi (g)\mu _{g}\lambda _{g}.}
See also Figà-Talamanca–Herz algebra Fourier algebra
References
Pisier, Gilles (1995), "Multipliers and lacunary sets in non-amenable groups", American Journal of Mathematics, 117 (2), The Johns Hopkins University Press: 337–376, arXiv:math/9212207, doi:10.2307/2374918, ISSN 0002-9327, JSTOR 2374918, MR 1323679, S2CID 2958712 Figà-Talamanca, Alessandro; Picardello, Massimo A. (1983), Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, New York: Marcel Dekker Inc., ISBN 978-0-8247-7042-6, MR 0710827 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.