A C*-cover lattice dichotomy

TL;DR

This paper characterizes the structure of the lattice of C*-covers for non-selfadjoint operator algebras, showing it is either trivial or uncountably infinite, and explores conditions affecting this lattice.

Contribution

It proves the dichotomy of the C*-cover lattice size and constructs examples with a one-point lattice using novel methods.

Findings

The lattice of C*-covers is either a singleton or uncountably infinite.

Explicit examples of algebras with a one-point lattice are provided.

The C*-envelope may lack an immediate successor in the lattice.

Abstract

In this paper, we show that the lattice of C*-covers of a non-selfadjoint operator algebra is either one point or uncountable. We prove that there are non-selfadjoint operator algebras with a one-point lattice in two ways: as an explicit subalgebra of the C*-algebra of a universal contraction, and via a direct limit construction inspired by the work of Kirchberg and Wassermann for operator systems. We also establish that the C*-envelope need not have an immediate successor C*-cover in the lattice, and that a semi-Dirichlet non-selfadjoint operator algebra never has a one-point lattice.