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Regular representations of Dirichlet spaces

Masatoshi Fukushima

1971
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Transactions of the American Mathematical Society
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We construct a regular and a strongly regular Dirichlet space which are equivalent to a given Dirichlet space in the sense that their associated function algebras are isomorphic and isometric. There is an appropriate strong Markov process called a Ray process on the underlying space of each strongly regular Dirichlet space. Introduction. A. Beurling and J. Deny [1] introduced the notion of Dirichlet spaces and developed the general theory of kernel-free potentials. Recently the author [6]
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... he author [6] adopted Dirichlet spaces relative to L2-spaces (we will call them L2-Dirichlet spaces or D-spaces as an abbreviation) to describe boundary conditions for multidimensional Brownian motions. A .D-space is a certain space of functions that are defined on an underlying measure space (X, m). When (X, m) is fixed, there is a one-to-one correspondence between the set of all symmetric sub-Markov resolvent operators on L2(X; m) and the set of all £>-spaces. In particular, any sub-Markov resolvent kernel on X which is symmetric with respect to m generates a £>-space. The present paper and the subsequent one [9] concern the problem of whether conversely any Z)-space guarantees the existence of a suitable strong Markov process or not. The present paper aims at constructing a regular and a strongly regular Z)-space which are equivalent to a given D-space. A D-space is called regular if it densely contains sufficiently many continuous functions vanishing at infinity on its underlying space. There corresponds a potential theory of a type of Beurling-Deny to each regular D-space. A strongly regular Z)-space is a regular one which is generated by a Ray resolvent kernel. According to D. Ray [15] , there is a right continuous strong Markov process on the underlying space of each strongly regular D-space. Suppose that we are given a D-space with underlying space (X, m). Theorem 2 in §5 states that there exists then a regular D-space with some modified underlying space (A", m') in such a way that these two D-spaces are equivalent to each other as function spaces. The latter D-space will be called a regular representation of the given one. The regular representation will be carried out depending on a subalgebra L of L°°(X; m) satisfying a certain condition denoted by (C). Actually we will take as X' the space of all regular maximal ideals of L.

doi:10.1090/s0002-9947-1971-0281256-1
fatcat:5b5cxgah2jc6beo2xmnvrel4oa