Geometric properties of Dirichlet forms under order isomorphisms

We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with arbitrary (irreducible) Dirichlet forms and show that any intertwining order isomorphism is necessarily unitary (up to a constant). In the topological setting we deal with quasi-regular forms and show that any intertwining order isomorphism induces a quasi-homeomorphism between the underlying spaces. In the geometric setting we deal with both regular Dirichlet forms as well as resistance forms and essentially show that the geometry defined by these forms is preserved by intertwining order isomorphisms. In particular, we prove in the strongly local regular case that intertwining order isomorphisms induce isometries with respect to the intrinsic metrics between the underlying spaces under fairly mild assumptions. This applies to a wide variety of metric measure spaces including $\mathrm{RCD}(K,N)$-spaces, complete weighted Riemannian manifolds and complete quantum graphs. In the non-local regular case our results cover in particular graphs as well as fractional Laplacians as arising in the treatment of $\alpha$-stable L\'evy processes. For resistance forms we show that intertwining order isomorphisms are isometries with respect to the resistance metrics. Our results can can be understood as saying that diffusion always determines the Hilbert space, and -- under natural compatibility assumptions -- the topology and the geometry respectively. As special instances they cover earlier results for manifolds and graphs.