On $C^*$-algebras associated with $C^*$-correspondences

TL;DR

This paper investigates $C^*$-algebras derived from $C^*$-correspondences, establishing key theorems and conditions for properties like nuclearity, exactness, and $K$-theory sequences.

Contribution

It proves the gauge-invariant uniqueness theorem and provides criteria for nuclearity, exactness, and the UCT for these $C^*$-algebras.

Findings

Proved the gauge-invariant uniqueness theorem.

Derived conditions for nuclearity and exactness.

Established a 6-term $K$-theory exact sequence.

Abstract

We study $C^*$-algebras arising from $C^*$-correspondences, which was introduced by the author. We prove the gauge-invariant uniqueness theorem, and obtain conditions for our $C^*$-algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6-term exact sequence of $K$-groups involving the $K$-groups of our $C^*$-algebras.