This talk will attempt to introduce the definition and some examples of the so-called Commuting Squares, whose importance has recently been realised in the study of von Neumann algebras. The notion of a commuting square first made its apearance (in the manner defined here) in the paper [PP], and the 'classifica- tion of commuting squares' is now recognised as being one of the central problems of the theory of subfactors. Further, the commuting square condition is intimately connected with what are known as 'Reidemeister moves of type II' in knot theory.
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