Frobenius-Rieffel norms on finite-dimensional C*-algebras

TL;DR

This paper explores Frobenius-Rieffel norms on finite-dimensional C*-algebras, showing they generalize the Frobenius norm, providing explicit formulas, and comparing them to operator norms with continuous equivalence constants.

Contribution

It introduces a generalization of Frobenius norms via Rieffel's construction, with explicit formulas and norm comparisons for finite-dimensional C*-subalgebras.

Findings

Frobenius-Rieffel norms generalize the Frobenius norm.

Explicit formulas for conditional expectations onto subalgebras.

Equivalence constants vary continuously with irrational parameters.

Abstract

In 2014, Rieffel introduced norms on certain unital C*-algebras built from conditional expectations onto unital C*-subalgebras. We begin by showing that these norms generalize the Frobenius norm, and we provide explicit formulas for certain conditional expectations onto unital C*-subalgebras of finite-dimensional C*-algebras. This allows us compare these norms to the operator norm by finding explicit equivalence constants. In particular, we find equivalence constants for the standard finite-dimensional C*-subalgebras of the Effros-Shen algebras that vary continuously with respect to their given irrational parameters.