Compact sets in the free topology
Subsets of the set of g-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, the hull of a single point.
Szegő and widom theorems for the neil algebra
Versions of well-known function theoretic operator theory results of Szegő and Widom are established for the Neil algebra. The Neil algebra is the subalgebra of the algebra of bounded analytic functions on the unit disc consisting of those functions whose derivative vanishes at the origin.
Bilinear matrix inequalities and polynomials in several freely noncommuting variables
Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.