Frobenius-Rieffel norms on matrix algebras (Konrad Aguilar, Pomona)

Noncommutative metric geometry is the study of certain noncommuative algebras in the context of metric geometry. For instance, the Lipschitz constant (which measures the maximum slope obtained by a real-valued continuous function on a metric space (allowed to be infinite)) is a vital tool in metric geometry, and a main feature of noncommutative metric geometry is the introduction of a noncommutative notion of the Lipschitz constant, called an L-seminorm, due to M.A. Rieffel. The purpose of our work is to introduce suitable L-seminorms on matrix algebras. To accomplish this, we used norms introduced by Rieffel on certain unital C*-algebras built from conditional expectations onto unital C*-subalgebras. We begin by showing that these norms generalize the Frobenius norm on matrix algebras, and we provide explicit formulas for certain conditional expectations onto unital C*-subalgebras of finite-dimensional C*-algebras. This allows us to compare these norms to the unique C*-norm (the operator 2-norm), by finding explicit equivalence constants. (This is joint work with Stephan R. Garcia and Elena Kim (’21), arxiv: 2112.13164).