Let M be a von Neumann
algebra and let A be a maximal abelian self-adjoint subalgebra (masa) of M. A
subalgebra T of M is called triangular (with respect to A) if T ∩ T∗= A, where T∗
denotes the collection of adjoints of the elements in T. If T is not contained in any
larger triangular subalgebra of M, then T is called maximal triangular. If A is a
Cartan subalgebra, then M may be realized as an algebra of matrices indexed
by an equivalence relation on a standard Borel space and if T is σ-weakly
closed and maximal triangular, then T may be realized as the collection of
matrices supported on the graph of a partial order that totally orders each
equivalence class. In this paper we will be concerned with the relation between the
structure of these algebras and the theory of analytic operator algebras. It
turns out that this relation is complex: it involves the cohomology of the
equivalence relation, the order type of the partial order and the type of
M.
