Representations of Group Algebras in Spaces of Completely Bounded Maps

TL;DR

This paper investigates the structure of representations of group algebras within spaces of completely bounded maps, establishing conditions for their inclusion and characterizing positive definite elements in the context of locally compact groups.

Contribution

It provides a new characterization of the range of the map Gamma_pi for group algebra representations, linking norm continuity to inclusion in the extended Haagerup tensor product.

Findings

Gamma_pi(L^1(G)) is in B(H)⊗^{eh}B(H) iff pi is norm continuous

For abelian G, Gamma_pi(M(G)) is studied within the Varopoulos algebra

Positive definite elements of the Varopoulos algebra are characterized via completely positive operators

Abstract

Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi(L^1(G))\subset B(H)\otimes^{eh}B(H) exactly when pi is norm continuous. For the case that G is abelian, we study Gamma_pi(M(G)) as a subset of the Varopoulos algebra. We also characterise positive definite elements of the Varopoulos algebra in terms of completely positive operators.