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: C.-H. Chu and A. T.-M. Lau replaced the algebra \(L^\infty(G)\) by the group von Neumann algebra \(VN(G)\) and the convolution action of a probability measure \(\mu\) on \(L^\infty(G)\) by the canonical action of a positive definite function \(\sigma\) on \(\VN(G)\); on the other hand, W. Jaworski and the first-named author replaced \(L^\infty(G)\) by \(B(L^2(G))\) to which the convolution action by \(\mu\) can be extended in a natural way. We establish a link between both approaches. The action of \(\sigma\) on \(VN(G)\) can be extended to \(B (L^2(G))\). We study the corresponding space \(\tilde{H}_\sigma\) of ``\(\sigma\)-harmonic operators'', i.e., fixed points in \(B(L^2(G))\) under the action of \(\sigma\). We show, under mild conditions on either \(\sigma\) or \(G\), that \(\tilde{H}_\sigma\) is in fact a von Neumann subalgebra of \(B (L^2(G))\). Our investigation of \(\tilde{H}_\sigma\) 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 \(\tilde{H}_\sigma\) via ideals in \(T (L^2(G))\) - where \(T(L^2(G))\) 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 G. A. Willis.