A finiteness result for commuting squares of matrix algebras

TL;DR

This paper investigates a specific condition for commuting squares of matrix algebras, showing it is satisfied only for prime dimensions in certain models, and proves these squares are isolated and finite in number for fixed dimensions.

Contribution

It introduces the span condition for commuting squares, proves their finiteness and isolation in the space of all such squares, and provides a conceptual proof for existing biunitary families.

Findings

Span condition holds iff dimension n is prime in standard spin models.

Commuting squares satisfying the span condition are isolated.

There are finitely many such squares for each fixed dimension.

Abstract

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 finiteley many for any fixed dimension. Also, we give a conceptual proof of previous constructions of certain one-parameter families of biunitaries.