Dirichlet forms as Banach algebras and applications
Let \((\mathcal E,\mathcal F)\) be a regular Dirichlet form on \(L^2(X,m)\) where \((X,m)\) is a locally compact, metrizable space with countable base such that \(m\) is a Radon measure with full support. Let \((\mathcal E,\mathcal F_e)\) be the extended Dirichlet space. It is shown that \(\mathcal B_e := C_0(X)\cap\mathcal F_e\) (\(C_0\) are the continuous functions vanishing at infinity) and \(\mathcal A_e := L^\infty(X,m)\cap\mathcal F_e\) are complex Banach algebras when endowed with the norm \(\| \bullet\| _\infty + [\mathcal E(\bullet)]^{1/2}\) which can be made into involutive complex Banach algebras. \(\mathcal A_e\) is unital iff \(X\) is compact and \((\mathcal E,\mathcal F)\) is recurrent. Moreover, \(\mathcal A_e, \mathcal B_e\) have a unique Banach algebra topology. The Gelfand spectrum of \(\mathcal B_e\) is homeomorphic to \(X\), the Gelfand transform conicides with the natural embedding of \(\mathcal B_e\) into \(C_0(X)\). The above machinery can be applied to strongly local Dirichlet forms showing that \((\mathcal E_1,\mathcal F_1)\) and \((\mathcal E_2,\mathcal F_2)\) are quasi-equivalent (i.e.\ form-comparable) if and only if \(\mathcal F_1=\mathcal F_2\). Finally, the ideal structure of \(\mathcal B_e\) is described, showing that the algebraic \(K\)-theory of \(\mathcal B_e\) is isomorphic to the topological \(K\)-theory. This allows to construct Dirichlet structures on (sections of) finite-dimensional, locally trivial vector bundles over \(X\).
scientific article; zbMATH DE number 5129579