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Contents of PMS, Vol. 33, Fasc. 2,
pages 213 - 223
 

HERZ–SCHUR MULTIPLIERS AND NON-UNIFORMLY BOUNDED REPRESENTATIONS OF LOCALLY COMPACT GROUPS

Troels Steenstrup

Abstract: Let G be a second countable, locally compact group and let φ be a continuous Herz–Schur multiplier on G . Our main result gives the existence of a (not necessarily uniformly bounded) strongly continuous representation π of G on a Hilbert space H , together with vectors ξ,η ∈ H , such that φ (y- 1x ) = ⟨π(x )ξ,π(y-1)*η⟩ for x,y ∈ G and sup ∥π(x)ξ∥⋅sup ∥π(y-1)*η∥ = ∥φ ∥ x∈G y∈G M0A(G ) . Moreover, we obtain control over the growth of the representation in the sense that ∥π(g)∥ ≤ exp (cd(g,e)) 2 for g ∈ G , where e ∈ G is the identity element, c is a constant, and d is a metric on G .

2000 AMS Mathematics Subject Classification: Primary: 46L07; Secondary: 22D12.

Keywords and phrases: Herz–Schur multipliers, non-uniformly bounded representations, free groups.

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