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.

Completely bounded multiplier, Fourier algebra, Harmonic operator, Locally compact group, Positive definite function
Mathematische Zeitschrift
School of Mathematics and Statistics

Neufang, M, & Runde, V. (Volker). (2007). Harmonic operators: The dual perspective. Mathematische Zeitschrift, 255(3), 669–690. doi:10.1007/s00209-006-0039-6