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Frobenius-Rieffel norms on matrix algebras (Konrad Aguilar, Pomona)
September 22, 2022 @ 4:00 pm - 5:00 pm
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).