Markov semigroups on \(C^*\)-bundles
type: Item
instance of: scholarly article
title: Markov semigroups on \(C^*\)-bundles
zbMATH Open document ID: 0694.46049
DOI: 10.1016/0022-1236(89)90037-2
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 57R25
Mathematics Subject Classification ID: 17C10
zbMATH DE Number: 4136788
zbMATH Keywords: non-commutative theory of symmetric Markov semigroups and Dirichlet forms on vector bundles
zbMATH Keywords: completely positive vector bundle
zbMATH Keywords: trace
zbMATH Keywords: Jordan spin algebra
zbMATH Keywords: symmetric (sub-)Markov semigroups of operators
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: finite-dimensional real \(C^*\)-algebra
zbMATH Keywords: Hilbert space tensor product
zbMATH Keywords: H-harmonic elements
zbMATH Keywords: Clifford bundles
zbMATH Keywords: connection
zbMATH Keywords: Clifford algebras
zbMATH Keywords: Bochner Laplacian
zbMATH Keywords: complete-positivity preserving
author: E. Brian Davies
author: Oscar S. Rothaus
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90037-2
OpenAlex ID: W2033715261
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description: scientific article; zbMATH DE number 4136788