Analysis Seminar. Konrad Aguilar. Quantum metrics on the tensor product of a commutative C*-algebra and an AF C*-algebra
Speaker: Konrad Aguilar (Arizona State University)
Title: Quantum metrics on the tensor product of a commutative C*-algebra and an AF C*-algebra
Abstarct: Quantum metric spaces were introduced in 1998 by M. A. Rieffel as noncommutative analogues to metric spaces to study the metric aspects of Noncommutative Geometry of A. Connes. In this talk, given a compact metric space X and a unital AF (approximately finite-dimensional) C*-algebra A equipped with a faithful tracial state, we place quantum metrics on the tensor product of C(X) and A given established quantum metrics on C(X) and A from work with Bice and Latremoliere. We prove the inductive limit of C(X) tensor A given by A is a metric limit in the Gromov-Hausdorff propinquity, which is a noncommutative analogue to the Gromov-Hausdorff distance on the class of compact metric spaces. We show that our quantum metric is compatible with the tensor product by producing a Leibniz rule on elementary tensors and showing the diameter of our quantum metric on the tensor product is bounded above the diameter of the Cartesian product of the quantum metric spaces. Finally, we provide continuous families in the Gromov-Hausdorff propinquity of C(X) tensor A, which extends our previous results with Latremoliere on UHF algebras. We will define many of the terms used in this talk including quantum metric spaces and C*-algebras.
Tuesday, October 22 at 3:30pm to 4:20pm