Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A C*-cover lattice dichotomy

Adam Humeniuk, Christopher Ramsey, Marcel Scherer

TL;DR

The paper analyzes the lattice of C*-covers for non-selfadjoint operator algebras and proves a dichotomy: the lattice is either a singleton or has continuum cardinality. It develops a robust lattice framework using boundary/Shilov ideals and direct-limit techniques, and constructs two explicit one-point examples. It then shows that for separable algebras the lattice size must be 1 or continuum, and proves a lattice-isomorphism between an algebra and its Shilov extension A+I, while characterizing the maximal covers and envelopes in this setting. The work further demonstrates uncountability phenomena, including the semi-Dirichlet case, and provides an example with no immediate successor to the envelope, thereby clarifying the landscape of C*-cover lattices for non-selfadjoint algebras and offering concrete tools to engineer algebras with prescribed lattice structures.

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.

A C*-cover lattice dichotomy

TL;DR

The paper analyzes the lattice of C*-covers for non-selfadjoint operator algebras and proves a dichotomy: the lattice is either a singleton or has continuum cardinality. It develops a robust lattice framework using boundary/Shilov ideals and direct-limit techniques, and constructs two explicit one-point examples. It then shows that for separable algebras the lattice size must be 1 or continuum, and proves a lattice-isomorphism between an algebra and its Shilov extension A+I, while characterizing the maximal covers and envelopes in this setting. The work further demonstrates uncountability phenomena, including the semi-Dirichlet case, and provides an example with no immediate successor to the envelope, thereby clarifying the landscape of C*-cover lattices for non-selfadjoint algebras and offering concrete tools to engineer algebras with prescribed lattice structures.

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.
Paper Structure (11 sections, 37 theorems, 183 equations)

This paper contains 11 sections, 37 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Background
  3. C*-covers
  4. Boundary ideals
  5. Maximal u.c.c. maps
  6. Direct limits
  7. Non-trivial one point lattice example
  8. Extension of an operator algebra by a Shilov ideal
  9. Uncountability of the C*-cover lattice
  10. An example with no immediate successor to the C*-envelope
  11. Semi-Dirichlet C*-covers

Key Result

Proposition 2.1

Let $A$ be an operator algebra. Then $\text{C$^*$-Lat}(A)$ with the ordering $\preceq$ forms a complete lattice.

Theorems & Definitions (75)

  • Proposition 2.1: Hamidi and Thompson
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • Proposition 2.9
  • ...and 65 more