Markov semigroups KMS-symmetric for a weight
Let \(A\) be a von Neumann algebra with faithful normal semifinite weight \(\varphi\), \(M\) be the crossed product of \(A\) with the modular automorphism group \(\sigma_t(x)= h^{it}xh^{-it}\) of \(\varphi\), and let \(S\) be an operator on \(L^p(A)\) with domain \(\text{Dom}(S)\), where \(p\in [1,\infty]\). \(S\) is called Markov on \(L^p(A)\), \(h^{(1/p)}\), or Markov with respect to \(h^{1/p}\), if (i) \([0,h^{1/p}]_{L^p}\subset \text{Dom}(S)\), (ii) \(S([0, h^{1/p}]_{L^p})\subset [0, h^{1/p}]_{L^p}\); \(S\) is Markov on \((A,\varphi)\) if \((\text{i}')\) \([0,1]_m\subset \text{Dom}(S)\), where \(m=\{x\in A:\varphi(|x|)<\infty\}\). \((\text{ii}')\) \(S([0, 1]\cap \text{Dom}(S))\subset [0,1]_A\), An operator \(T\) on \(A\) (with domain \(\text{Dom}(T)\)) is called KMS-symmetric (with respect to \(\varphi\)) if \(m\subset\text{Dom}(T)\) and \(\text{tr}(Ta: i(b))= \text{tr }i(a)\), for all \(a,b\in m\). In the paper, the authors consider Markov semigroups which are KMS-symmetric with respect to a faithful normal semifinite weight, and show that these correspond to selfadjoint contraction semigroups on its standard Hilbert space whose form generator is Dirichlet. It is reminded that a nonnegative quadratic form \(Q\) in \(L^2(A)\) is called Dirichlet (with respect to \(\varphi\)) if \(Q\cdot \pi_I\leq Q\), where \(\pi_I\) is the nearest point projection onto the closed convex set \([0,h^{1/2}]_{L^2}\).
scientific article; zbMATH DE number 1336971