Math. Ann. 313, 39–67 (1999) Mathematische Annalen c  Springer-Verlag 1999 Markov semigroups KMS-symmetric for a weight Stanisl ´ aw Goldstein 1 , J. Martin Lindsay 2 1 Faculty of Mathematics, L ´ ´ od´ z University, ul. Banacha 22, 90-238 L ´ ´ od´ z, Poland (e-mail: goldstei@math.uni.lodz.pl) 2 Department of Mathematics, Nottingham Unversity, Nottingham NG7 2RD, UK (e-mail: jml@maths.nott.ac.uk) Received: 12 December 1997 Mathematics Subject Classification (1991): 46L55, 47D07, 46L50 Introduction Dirichlet forms have become an indispensable tool for the study of Markov processes and the analysis of second order elliptic operators ([FOT], [Da 2], [MRY]). Following Gross’ analysis of a Clifford algebra analogue of the Ornstein-Uhlenbeck semigroup by means of Dirichlet form methods ([Gro]), Albeverio and Høegh-Krohn initiated a general theory of noncom- mutative Dirichlet forms on operator algebras ([AH-K]). In the context of W ∗ -algebras with faithful normal semifinite trace and associated noncom- mutative L p -spaces ([Seg], [Dix]) they established the bijective correspon- dence between tracially symmetric Markov semigroups on the algebra and Dirichlet forms on its L 2 -space. This theory was further developed by Davies and Lindsay who constructed Markov semigroups from unbounded deriva- tions and superderivations on (Z 2 -graded) tracial Hilbert algebras ([DL 1], [DL 2]). Applications of tracial Dirichlet forms have included the construc- tion of the transverse heat semigroup on a Riemannian foliation C ∗ -algebra ([Sau]), and heat kernel bounds on graphs viewed as noncommutative man- ifolds ([Da 3]). A theory of nonsymmetric tracial Dirichlet forms may be found in ([GIS]). The appropriate form of symmetry with respect to a state on a W ∗ - algebra emerges from the ‘symmetric embeddings’ of algebra into predual and standard Hilbert space. The above bijective correspondence is extended to this context in [GL 1] and [Cip]. Whereas we exploited Haagerup’s L p - spaces ([Ha 2]), Cipriani emphasised standard forms and obtained an abstract