Representations of group algebras in spaces of completely bounded maps Academic Article uri icon

abstract

  • Let G be a locally compact group, π : G → u(H) be a strongly continuous unitary representation, and CBσ (B(H)) the space of normal completely bounded maps on B(H). We study the range of the map Γπ : M(G) → CBσ (B(H)), Γπ(μ) = ∫G π(s) ⊗ π(s)* dμ(s), where we identify CBσ (B(H)) with the extended Haagerup tensor product B(H) ⊗eh B(H). We use the fact that the C*-algebra generated by integrating π to L1(G) is unital exactly when π is norm continuous, to show that Γπ(L 1(G)) ⊂ B(H) ⊗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 characterize positive definite elements of the Varopoulos algebra in terms of completely positive operators. Indiana University Mathematics Journal ©.

author list (cited authors)

  • Smith, R. R., & Spronk, N.

citation count

  • 4

complete list of authors

  • Smith, Roger R||Spronk, Nico

publication date

  • January 2005