Vol. 137, No. 2, 1989

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Coordinates for triangular operator algebras. II

Paul Scott Muhly, Kichi-Suke Saito and Baruch Solel

Vol. 137 (1989), No. 2, 335–369

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.

Mathematical Subject Classification 2000
Primary: 46L10
Secondary: 47D25
Received: 2 February 1988
Published: 1 April 1989
Paul Scott Muhly
Kichi-Suke Saito
Baruch Solel