Regular representations of Dirichlet spaces

Masatoshi Fukushima
1971 Transactions of the American Mathematical Society
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  introduced the notion of Dirichlet spaces and developed the general theory of kernel-free potentials. Recently the author 
more » ... he author  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  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  , 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.