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The Lattice of C*-covers of an operator algebra

Adam Humeniuk, Christopher Ramsey

TL;DR

This work analyzes the lattice of C*-covers $C^*$-Lat({\mathcal A}) of a non-selfadjoint operator algebra ${\mathcal A}$ and shows it can fail to distinguish algebras up to complete isometric isomorphism. It introduces four notions of 'sameness' for lattices, proves they are all distinct and explores their behavior under direct sums and tensor products, including cases where the lattice is preserved when tensoring with nuclear or simple C*-algebras. The paper also demonstrates pathologies and limitations: there exist operator algebras generating exactly $n$ C*-algebras up to $*$-isomorphism for any $n$, and it constructs a simple operator algebra not CB-isomorphic to any C*-algebra. These results illuminate the nuanced relationship between non-selfadjoint operator algebras and their C*-covers, with implications for invariants and classifications in this setting.

Abstract

In this paper it is shown that the lattice of C*-covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C*-covers are developed and proven to be distinct. The lattice of C*-covers of direct sums and tensor products are studied. Along the way key examples are found of an operator algebra that generates exactly n C*-algebras up to *-isomorphism and a simple operator algebra that is not similar to a C*-algebra.

The Lattice of C*-covers of an operator algebra

TL;DR

This work analyzes the lattice of C*-covers -Lat({\mathcal A}) of a non-selfadjoint operator algebra and shows it can fail to distinguish algebras up to complete isometric isomorphism. It introduces four notions of 'sameness' for lattices, proves they are all distinct and explores their behavior under direct sums and tensor products, including cases where the lattice is preserved when tensoring with nuclear or simple C*-algebras. The paper also demonstrates pathologies and limitations: there exist operator algebras generating exactly C*-algebras up to -isomorphism for any , and it constructs a simple operator algebra not CB-isomorphic to any C*-algebra. These results illuminate the nuanced relationship between non-selfadjoint operator algebras and their C*-covers, with implications for invariants and classifications in this setting.

Abstract

In this paper it is shown that the lattice of C*-covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C*-covers are developed and proven to be distinct. The lattice of C*-covers of direct sums and tensor products are studied. Along the way key examples are found of an operator algebra that generates exactly n C*-algebras up to *-isomorphism and a simple operator algebra that is not similar to a C*-algebra.
Paper Structure (8 sections, 29 theorems, 101 equations)

This paper contains 8 sections, 29 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Equivalences of the lattice of C*-covers
  3. A one point lattice of C*-covers?
  4. Tensor Products
  5. Properties preserved (or not) by C*-lattice equivalences
  6. Residually finite-dimensional
  7. Action-admissible
  8. Simplicity

Key Result

Proposition 2.4

If ${\mathcal{A}}$ is an operator algebra, then the ordering $\preceq$ on $\text{C$^*$-Lat}({\mathcal{A}})$ makes $\text{C$^*$-Lat}({\mathcal{A}})$ into a complete lattice. Given an arbitrary family $({\mathcal{B}}_\lambda,\iota_\lambda)$ of C*-covers for ${\mathcal{A}}$, their join is To describe their meet, it is easiest to use the maximal C*-algebra $(C^\ast_\text{max}({\mathcal{A}}),\mu)$. Le

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Hamidi and Thompson
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 54 more