Douglas Farenick, University of Regina
"Low-dimensional operator systems, equivalence, and free convexity"
There are various forms of equivalence that apply to Hilbert space operators, the most operator-theoretical faithful being unitary equivalence and, perhaps, approximate unitary equivalence. Much weaker forms of equivalence arise through unital isometries and unital complete isometries defined on the smallest operator system \mathcal R_T that contains a given Hilbert space operator T. For example, T and S are operator system equivalent if there exists a unital complete isometry \varphi:\mathcal R_T\rightarrow\mathcal R_S such that \varphi(T)=S. There are natural several-variable formulations of this single-operator case, as in the work of Davidson, Dor-on, Shalit, and Solel (2017).
In this lecture I will consider operator system equivalence for certain irreducible weighted shifts on finite- and infinite-dimensional spaces, as well as for finite upper-triangular Toeplitz matrices. The analysis makes connections with C*-envelopes, boundary representations, and free convexity.