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Passing $C^*$-correspondence Relations to the Cuntz-Pimsner algebras

Menevşe Eryüzlü

Abstract

We construct a functor that maps $C^*$-correspondences to their Cuntz-Pimsner algebras. Applications include a generalization of the well-known result of Muhly and Solel: Morita equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras; as well as the result of Muhly, Pask, and Tomforde: regular strong shift equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras.

Passing $C^*$-correspondence Relations to the Cuntz-Pimsner algebras

Abstract

We construct a functor that maps -correspondences to their Cuntz-Pimsner algebras. Applications include a generalization of the well-known result of Muhly and Solel: Morita equivalent -correspondences have Morita equivalent Cuntz-Pimsner algebras; as well as the result of Muhly, Pask, and Tomforde: regular strong shift equivalent -correspondences have Morita equivalent Cuntz-Pimsner algebras.

Paper Structure

This paper contains 10 sections, 25 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Prelimineries
  3. Categories
  4. A Covariant Representation
  5. The Functor
  6. Applications
  7. Muhly & Solel Theorem
  8. Cuntz-Pimsner Algebras of Shift Equivalent $C^*$-correspondences
  9. Pimsner Dilations
  10. Final Notes

Key Result

Proposition 2.1

enchilada Let $X_0$ be an $A_0-B_0$ pre-correspondence, and let $Z$ be an $A-B$ correspondence. If there is a map $\Phi : X_0 \rightarrow Z$ satisfying for all $a\in A_0$ and $x,y\in X_0$, then $\Phi$ extends uniquely to an injective $A-B$ correspondence homomorphism $\tilde{\Phi}: X \rightarrow Z .$

Figures (1)

  • Figure :

Theorems & Definitions (50)

  • Proposition 2.1
  • Lemma 2.2: fowler
  • Lemma 2.3
  • Lemma 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • ...and 40 more