Traces, one-parameter flows and K-theory
| dc.contributor.author | Francis, Michael | |
| dc.contributor.supervisor | Emerson, Heath | |
| dc.contributor.supervisor | Laca, Marcelo | |
| dc.date.accessioned | 2014-09-02T21:50:55Z | |
| dc.date.available | 2014-09-02T21:50:55Z | |
| dc.date.copyright | 2014 | en_US |
| dc.date.issued | 2014-09-02 | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Master of Science M.Sc. | en_US |
| dc.description.abstract | Given a C*-algebra $A$ endowed with an action $\alpha$ of $\R$ and an $\alpha$-invariant trace $\tau$, there is a canonical dual trace $\widehat \tau$ on the crossed product $A \rtimes_\alpha \R$. This dual trace induces (as would any suitable trace) a real-valued homomorphism $\widehat \tau_* : K_0(A \rtimes_\alpha \R) \to \R$ on the even $K$-theory group. Recall there is a natural isomorphism $\phi_\alpha^i : K_i(A) \to K_{i+1}(A \rtimes_\alpha \R)$, the Connes-Thom isomorphism. The attraction of describing $\widehat \tau_* \circ \phi_\alpha^1$ directly in terms of the generators of $K_1(A)$ is clear. Indeed, the paper where the isomorphisms $\{\phi_\alpha^0,\phi_\alpha^1\}$ first appear sees Connes show that $\widehat \tau_* \phi_\alpha^1[u] = \frac{1}{2 \pi i} \tau(\delta(u) u^*)$, where $\delta = \frac{d}{dt} \big|_{t=0} \alpha_t(\cdot)$ and $u$ is any appropriate unitary. A careful proof of the aforementioned result occupies a central place in this thesis. To place the result in its proper context, the right-hand side is first considered in its own right, i.e., in isolation from mention of the crossed-product. A study of 1-parameter dynamical systems and exterior equivalence is undertaken, with several useful technical results being proven. A connection is drawn between a lemma of Connes on exterior equivalence and projections, and a quantum-mechanical theorem of Bargmann-Wigner. An introduction to the Connes-Thom isomorphism is supplied and, in the course of this introduction, a refined version of suspension isomorphism $K_1(A) \to K_0(\susp A)$ is formulated and proven. Finally, we embark on a survey of unbounded traces on C*-algebras; when traces are allowed to be unbounded, there is inevitably a certain amount of hard, technical work needed to resolve various domain issues and justify various manipulations. | en_US |
| dc.description.proquestcode | 0280 | en_US |
| dc.description.scholarlevel | Graduate | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/5649 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights.temp | Available to the World Wide Web | en_US |
| dc.rights.uri | http://creativecommons.org/publicdomain/zero/1.0/ | * |
| dc.subject | C*-algebras | en_US |
| dc.subject | K-Theory | en_US |
| dc.subject | Operator Algebras | en_US |
| dc.subject | Dynamical Systems | en_US |
| dc.subject | Crossed Products | en_US |
| dc.subject | Hilbert Spaces | en_US |
| dc.subject | Topology | en_US |
| dc.subject | Algebraic Topology | en_US |
| dc.title | Traces, one-parameter flows and K-theory | en_US |
| dc.type | Thesis | en_US |