Harmonic operators: the dual perspective
| dc.creator | Neufang, Mathias | |
| dc.creator | Runde, Volker | |
| dc.date | 2005-08-16 | |
| dc.date | 2006-05-09 | |
| dc.date.accessioned | 2026-07-07T07:44:46Z | |
| dc.date.available | 2026-07-07T07:44:46Z | |
| dc.description | 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 $μ$ on $L^\infty(G)$ by the canonical action of a positive definite function $σ$ 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 $μ$ 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 $\tilde{H}_σ$ of ``$σ$-harmonic operators'', i.e., fixed points in $B(L^2(G))$ under the action of $σ$. We show, under mild conditions on either $σ$ or $G$, that $\tilde{H}_σ$ is in fact a von Neumann subalgebra of $B (L^2(G))$. Our investigation of $\tilde{H}_σ$ 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 $\tilde{H}_σ$ 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. | |
| dc.description | 26 pages; LaTeX2e; more revisions & references updated | |
| dc.identifier | https://arxiv.org/abs/math/0508301 | |
| dc.identifier | http://arxiv.org/abs/math/0508301 | |
| dc.identifier | Math. Z. 255 (2007), 669-690 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123315 | |
| dc.subject | Functional Analysis | |
| dc.subject | Operator Algebras | |
| dc.subject | Primary 22D99; Secondary 22D20, 22D25, 22D35, 43A35, 46L07, 46L10, 47L50 | |
| dc.title | Harmonic operators: the dual perspective | |
| dc.type | text |