Noncommutative Boundaries Arising in Dynamics and Representations of the Cuntz Relations. (3rd April 2020) - Jorgensen, Palle; Tian, James
Abstract: In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of noncommutative operators; noncommutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these noncommutative algebras are the Cuntz algebras; denoted O N, N for the number of isometry generators. N is at least 2. Our focus is on the representations of O N . We aim to develop new noncommutative tools, involving both representation theory and stochastic processes. They serve to connect these parallel developments.In outline, boundaries, Poisson, or Martin, are certain measure spaces (often associated to random walk models), designed to encode the asymptotic behavior, e.g., how trajectories diverge when the number of steps goes to infinity. We stress that our present boundaries (commutative or noncommutative) are purely measure-theoretical objects. Although, as we show, in some cases our boundaries may be compared with more familiar topological boundaries.