Journal of Operator Theory

Volume 55, Issue 2, Spring 2006 pp. 295-310.

A finiteness result for commuting squa res of matrix algebras

Authors: Remus Nicoar\u{a}
Author institution: Department of Mathematics, Vanderbilt University, Nashville TN 37240, USA and Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Summary: We consider a condition for non-degenerate commuting squares of matrix algebras (finite dimensional von Neumann algebras) called the \emph{span condition}, which in the case of the $n$-dimensional standard spin models is shown to be satisfied if and only if $n$ is prime. We prove that the commuting squares satisfying the span condition are isolated among all commuting squares (modulo isomorphisms). In particular, they are finitely many for any fixed dimension. Also, we give a conceptual proof of previous constructions of certain one-parameter families of complex Hadamard matrices.

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