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Crossed products and C*-covers of semi-Dirichlet operator algebras

Adam Humeniuk, Elias G. Katsoulis, Christopher Ramsey

TL;DR

The paper develops the theory of semi-Dirichlet operator algebras by studying their C$^*$-covers and crossed products. It proves that semi-Dirichlet C$^*$-covers form a complete lattice with a maximal element $C^*_{sD-max}(\mathcal{A})$, and that the full crossed product of a semi-Dirichlet dynamical system is isomorphic to the relative full crossed product with respect to this maximal cover. It also shows that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product and that the crossed product remains semi-Dirichlet, with Takai duality clarifying the abelian case. The results unify dilation theory, representation theory, and crossed-product constructions for semi-Dirichlet algebras, connecting to tensor-algebra realizations and Shilov representations.

Abstract

In this paper, we show that the semi-Dirichlet C*-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet C*-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product.

Crossed products and C*-covers of semi-Dirichlet operator algebras

TL;DR

The paper develops the theory of semi-Dirichlet operator algebras by studying their C-covers and crossed products. It proves that semi-Dirichlet C-covers form a complete lattice with a maximal element , and that the full crossed product of a semi-Dirichlet dynamical system is isomorphic to the relative full crossed product with respect to this maximal cover. It also shows that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product and that the crossed product remains semi-Dirichlet, with Takai duality clarifying the abelian case. The results unify dilation theory, representation theory, and crossed-product constructions for semi-Dirichlet algebras, connecting to tensor-algebra realizations and Shilov representations.

Abstract

In this paper, we show that the semi-Dirichlet C*-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet C*-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product.
Paper Structure (4 sections, 20 theorems, 98 equations)

This paper contains 4 sections, 20 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Semi-Dirichlet operator algebras
  3. Semi-Dirichlet C*-covers
  4. Coextending semi-Dirichlet dynamics

Key Result

Lemma 2.2

Suppose ${\mathcal{A}}$ and ${\mathcal{B}}$ are completely isometrically isomorphic operator algebras. Then ${\mathcal{A}}$ is semi-Dirichlet if and only if ${\mathcal{B}}$ is semi-Dirichlet.

Theorems & Definitions (48)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4: Proposition 4.2.i & ii DKDoc
  • Proposition 2.5: Lemma 5.2 KatRamMem
  • Definition 2.6
  • Proposition 2.7: Muhly-Solel MS1
  • Theorem 2.8: Katsoulis-Ramsey, Theorem 7.5 KatRamMem
  • Corollary 2.9
  • ...and 38 more