This thesis is split up into two parts: the first concerns certain applications of the de la Harpe-Skandalis determinant to K-theory of appropriately regular C*-algebras. The second is concerned with (unital) inclusions of C*-algebras which satisfy a strong tensorial absorption condition. The first chapter following the preliminary section is joint work with Aaron Tikuisis [ST23], while the following chapters are independent. The penultimate chapter is [Sar23b] and the last chapter is essentially [Sar23a].
In the first chapter following the preliminaries, we examine the interplay between the algebraic K₁-group and the unitary algebraic K₁-group of a unital C*-algebra. We prove that for an abundance of unital C*-algebras, the algebraic K₁-group splits naturally as a direct sum of the unitary algebraic K₁-group and the space of continuous real-valued affine functions on the trace simplex. We further prove that if one considers Hausdorffized variants, then for any unital C*-algebra, there is a natural splitting of the Hausdorffized algebraic K₁-group in terms of the Hausdorffized unitary algebraic K₁-group and the space of continuous real-valued affine functions on the trace simplex. Moreover, this a splitting of topological groups.
The following chapter studies how certain group homomorphisms between unitary groups of C*-algebras induce maps on the trace simplex. In particular, we show that a contractive group homomorphism between unital C*-algebras which sends the circle to the circle, induces a map between their trace simplices. Under mild regularity conditions these further induce maps between Elliott invariants. As a consequence we show that certain inclusions of C*-algebras are in a correspondence with certain inclusions of unitary groups.
Finally we investigate what we call "D-stable inclusions" of C*-algebras, where D is strongly self-absorbing. We give a systematic study and prove that such inclusions between unital, separable, D-stable C*-algebras exist, are abundant, and are non-trivial.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/45463 |
| Date | 25 September 2023 |
| Creators | Sarkowicz, Pawel |
| Contributors | Tikuisis, Aaron, Giordano, Thierry |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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