In this talk, which is based upon joint work with Erin Griesenauer and Baruch Solel, I will discuss advances in the problem of identifying Arveson’s boundary representations and -envelopes of homogeneous -algebras built from algebras of generic matrices. These algebras, in turn, may be viewed as cross sections of certain holomorphic matrix bundles that arise naturally when one tries to view the free algebra on some number of generators in terms of matrix-valued functions on its space of finite-dimensional representations.
Holomorphic Matrix Bundles: A Natural Setting for Free Theory Sponsored by the Meyer Fund