Representations of Group Algebras in Spaces of Completely Bounded Maps
نویسنده
چکیده
Let G be a locally compact group, π : G → U(H) be a strongly continuous unitary representation, and CB(B(H)) the space of normal completely bounded maps on B(H). We study the range of the map Γπ : M(G)→ CB(B(H)), Γπ(μ) = Z G π(s)⊗ π(s)dμ(s) where we identify CB(B(H)) with the extended Haagerup tensor product B(H)⊗ B(H). We use the fact that the C*-algebra generated by integrating π to L(G) is unital exactly when π is norm continuous, to show that Γπ(L (G)) ⊂ B(H) ⊗ B(H) exactly when π is norm continuous. For the case that G is abelian, we study Γπ(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. 2000 Mathematics Subject Classification: Primary 46L07, 22D20; Secondary 22D10, 22D25, 22B05.
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