TWISTED COMMUTING SQUARES

We consider generalizations of commuting squares, called twisted commuting squares, obtained by having the commuting square orthogonality condition hold with respect to the inner product given by a faithful state on a finite-dimensional *-algebra. We present various examples of twisted commuting squares, most of which are computationally easy to work with, and we prove an isolation result. We also show how parametric families of (not necessarily) twisted commuting squares yield associative deformations of the matrix multiplication.

Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

A NOTE ON COMMUTING SQUARES ARISING FROM AUTOMORPHISMS ON SUBFACTORS

Using an idea due to Popa, we can associate a commuting square of factors to any given finite set of automorphisms acting on an inclusion of factors of finite index. We use this setting to obtain a simple proof of Popa's classification theorem of strongly outer actions of finitely generated discrete strongly amenable groups on a strongly amenable inclusion of type II 1 factors. We also obtain a new complete outer conjugacy invariant for arbitrary automorphisms, which contains the higher obstruction of Kawahigashi and the standard invariant as a special case.