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Separable algebras in multitensor C$^*$-categories are unitarizable

Luca Giorgetti, Wei Yuan, XuRui Zhao

Abstract

Recently, S. Carpi et al. (Comm. Math. Phys., 402:169-212, 2023) proved that every connected (i.e. haploid) Frobenius algebra in a tensor C$^*$-category is unitarizable (i.e. isomorphic to a special C$^*$-Frobenius algebra). Building on this result, we extend it to the non-connected case by showing that an algebra in a multitensor C$^*$-category is unitarizable if and only if it is separable.

Separable algebras in multitensor C$^*$-categories are unitarizable

Abstract

Recently, S. Carpi et al. (Comm. Math. Phys., 402:169-212, 2023) proved that every connected (i.e. haploid) Frobenius algebra in a tensor C-category is unitarizable (i.e. isomorphic to a special C-Frobenius algebra). Building on this result, we extend it to the non-connected case by showing that an algebra in a multitensor C-category is unitarizable if and only if it is separable.
Paper Structure (4 sections, 12 theorems, 24 equations)

This paper contains 4 sections, 12 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebras and modules in multitensor C$^*$-categories
  4. Separable algebras are unitarizable

Key Result

Proposition 3.5

Let $(A, m, \iota)$ be an algebra in $\EuScript{C}$.

Theorems & Definitions (32)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • Example 3.6
  • Definition 3.7
  • Definition 3.8
  • Proposition 3.9
  • ...and 22 more