Representations of Group Algebras in Spaces of Completely Bounded Maps

Details Representations of Group Algebras in Spaces of Completely Bounded Maps Representations of Group Algebras in Spaces of Completely Bounded Maps

2003-10-27

arxiv.org/abs/math/0310421v2

Indiana Univ. Math. J. 54 (3):873-896, 2005.

Mathematics

Functional Analysis

29 pages. Accepted in Indiana Univ. math J

Scientific paper

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.

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