Harmonic operators: The dual perspective

The study of harmonic functions on a locally compact group G has recently been transferred to a "non-commutative" setting in two different directions: Chu and Lau replaced the algebra L ∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L ∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand Jaworski and the first author replaced L ∞(G) by B (L2(G)) to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to B (L2(G)). We study the corresponding space ℋ σ̄ "σ-harmonic operators" i.e. fixed points in B (L2(G)) under the action of σ. We show under mild conditions on either σ or G that is in fact a von Neumann subalgebra of B (L2(G)). Our investigation of ℋσ̄ relies in particular on a notion of support for an arbitrary operator in B (L 2(G)) that extends Eymard's definition for elements of VN(G). Finally we present an approach to ℋσ̄ via ideals in T (L2(G)) where T (L2(G)) denotes the trace class operators on L 2(G) but equipped with a product different from composition as it was pioneered for harmonic functions by Willis