Abstract
The study of harmonic functions on a locally compactgroupGhas 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(L 2 (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(L 2 (G)). We study the corresponding space ˜ Hσ
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