KMS-symmetric Markov semigroups
Positivity preserving contraction semigroups on a \(W^*\)-algebra which are symmetric with respect to a state on the algebra naturally extend to the Haagerup \(L^p\)-spaces over the algebra. The self-adjoint contraction semigroups on \(L^2\) that arise in this way are characterised by Beurling-Deny type criteria for the corresponding quadratic form generator. Conversely, a closed Dirichlet form on \(L^2\) uniquely determines a KMS-symmetric, positivity preserving contraction semigroup on the algebra. Moreover complete positivity for the semigroup corresponds to the quadratic form being completely Dirichlet in a natural sense. The notion of symmetry employed here is, in general, different from detailed balance as usually defined, and involves the symmetric embedding, rather than the GNS embedding, of the algebra into \(L^2\). The present theory extends earlier results of Albeverio and Høegh- Krohn, and recent work of Davies and Lindsay, on semigroups which are symmetric with respect to a trace.
positivity preserving contraction semigroups on a \(W^*\)-algebra
scientific article; zbMATH DE number 795324