Traces, one-parameter flows and K-theory

dc.contributor.authorFrancis, Michael
dc.contributor.supervisorEmerson, Heath
dc.contributor.supervisorLaca, Marcelo
dc.date.accessioned2014-09-02T21:50:55Z
dc.date.available2014-09-02T21:50:55Z
dc.date.copyright2014en_US
dc.date.issued2014-09-02
dc.degree.departmentDepartment of Mathematics and Statisticsen_US
dc.degree.levelMaster of Science M.Sc.en_US
dc.description.abstractGiven 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.proquestcode0280en_US
dc.description.scholarlevelGraduateen_US
dc.identifier.urihttp://hdl.handle.net/1828/5649
dc.languageEnglisheng
dc.language.isoenen_US
dc.rights.tempAvailable to the World Wide Weben_US
dc.rights.urihttp://creativecommons.org/publicdomain/zero/1.0/*
dc.subjectC*-algebrasen_US
dc.subjectK-Theoryen_US
dc.subjectOperator Algebrasen_US
dc.subjectDynamical Systemsen_US
dc.subjectCrossed Productsen_US
dc.subjectHilbert Spacesen_US
dc.subjectTopologyen_US
dc.subjectAlgebraic Topologyen_US
dc.titleTraces, one-parameter flows and K-theoryen_US
dc.typeThesisen_US

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