Markov semigroups on \(C^*\)-bundles
There is, for \(L^ 2\)-spaces, a one-to-one correspondence between symmetric (sub-)Markov semigroups of operators [cf. \textit{M. Silverstein}, Springer Lect. Notes Math. 516 (1976; Zbl 0331.60046) for the corresponding Markov processes the symmetry appears for the transition functions] and Dirichlet forms [\textit{A. Beurling} and \textit{J. Deny}, Proc. Nat. Acad. Sci. 45, 208-215 (1959; Zbl 0089.08201)]. The semigroups can be extended to operate on \(L^ p\)-spaces; probability theory concerns essentially the \(L^ 1\)-case. Let M denote a compact Riemannian manifold and A a finite-dimensional real \(C^*\)-algebra. The authors are concerned with a ``non- commutative'' theory of symmetric Markov semigroups and Dirichlet forms on vector bundles \({\mathcal A}\) over M with fibres isomorphic to A, and with structure group Aut A. Vector bundles \(L^ p({\mathcal A})\) are constructed using a normalised trace on A invariant for Aut A; \(L^ 2({\mathcal A})\) is indeed the Hilbert space tensor product of \(L^ 2(M)\) and A (with inner product provided by the trace). For a section f define \(f^*(x)=(f(x))^*\). A selfadjoint non-negative operator H on \(L^ 2({\mathcal A})\) such that \(Hf^*=(Hf)^*\) is called (or rather corresponds to) a Dirichlet form, and \((e^{-Ht})_{t\in {\mathbb{R}}^+}\) is called a symmetric Markov semigroup, when, for all t, \(e^{-Ht}\) is positivity preserving on selfadjoint elements of \({\mathcal A}\) and \(e^{-Ht_ 1}\leq 1\). (This is why the semigroup should be called sub-Markov; the term (strictly) Markov usually refers to the restriction \(e^{-Ht_ 1}=1.)\) Equivalent Beurling-Deny conditions are given, as described in \textit{M. Reed} and \textit{B. Simon} [Methods of modern mathematical physics. IV: Analysis of operators (1978; Zbl 0401.47001)]. The set of H-harmonic elements, viz. \(\{f\in L^{\infty}({\mathcal A}):\) \(Hf=0\}\) is denoted by \({\mathcal B}\). It is shown that if H is a 2-positive operator on \(L^ 2({\mathcal A})\) then H is a contraction for \(L^{\infty}({\mathcal A})\) and \({\mathcal B}\) is a \(C^*\)-subalgebra of \(L^{\infty}({\mathcal A})\). The authors consider Clifford bundles \({\mathcal C}(V)\) for a smooth Euclidean vector bundle V on M with connection respecting the Euclidean structure, the fibres being Clifford algebras \(C^+\) or \(C^-\) corresponding to the dimension of quadratic forms on the fibre of V [see \textit{M. F. Atiyah}, \textit{R. Bott} and \textit{A. Shapiro}, Topology 3 suppl. 1, 3-38 (1964; Zbl 0146.19001)]. The Clifford algebras are given a \(C^*\)- structure using the operator norm for the regular representation. It would have been more natural to use, as fibres, Jordan ``spin'' algebras [\textit{N. Jacobson}, Structure and representations of Jordan algebras (1968; Zbl 0218.17010)] instead of artificially constructing \(C^*\)- algebras to fit the first half of the article. Choosing B to be the Bochner Laplacian on C(V) associated with the connection, and defining H to be -B, the H-harmonic elements are the absolutely parallel, or covariant constant, sections and the linear algebra of H-harmonic elements has dimension at most the dimension of \(C^{\pm}\). Following the argument of E. Reed and B. Simon (loc. Cit.) \((e^{-Ht})_ t\) is shown to be a symmetric Markov semigroup, and the authors prove that it is complete-positivity preserving. Complete positivity, rather than positivity, is appropriate because of the ``non-commutativity''.
non-commutative theory of symmetric Markov semigroups and Dirichlet forms on vector bundles
symmetric (sub-)Markov semigroups of operators
scientific article; zbMATH DE number 4136788
scientific article