On tensorial absorption of the Jiang-Su algebra

The Jiang-Su algebra Z and the notion of Z-stability (i.e. tensorial absorption of the Jiang-Su algebra) are now widely acknowledged to be of particular importance in the classification and structure theory of separable nuclear C*-algebras. The key results in this thesis are early attempts to explore Z-stability outside the constraints of unital and of nuclear C*-algebras. Standard unitisations of a separable Z-stable C*-algebra are not Z-stable and we therefore explore possible unitisations that preserve Z-stability. We construct the minimal Z-stable unitisation of a separable Z-stable C*-algebra and show that it satisfies an appropriate universal property. An interesting area in which to exploit Z-stability outside of the context of nuclear C*-algebras is the so-called Kadison’s similarity problem. We show that the tensor product of two separable unital C*-algebras has Kadison’s similarity property if one of them is nuclear and admits a unital *-homomorphism from (the building blocks of) the Jiang-Su algebra. An immediate consequence of this is that any separable unital Z-stable C*-algebra also has this property