Graduate Seminar

Location:  RI 208 and Livestream

Speaker: Patrick Melanson

PhD Student supervised by Remus Floricel

Title:  Irrational Rotation Algebras and Non-Commutative Solenoids

Zoom: https://uregina-ca.zoom.us/j/92053635111?pwd=U25NcEp6amlGV1YrRDdnUm5qNlhaUT09

Abstract:

In the 1980s, Rieffel introduced the idea of irrational rotation algebras, also called the non-commutative torus. Let \(\theta\) be a real number (thought of as an angle), and U,V being bounded unitaries from \(L^2(R/Z)\) with the property that \(VU = e^{i\pi\theta} UV\). Then \(A_{\theta}\) is the C*-algebra generated by U,V. We are interested in the case that \(\theta\) is an irrational number, since we then find that \(A_{\theta}\) is a simple algebra.

Given a sequence \(\theta = \{ \theta_i \}\) and N > 1 a natural number satisfying certain properties, we can construct a directed system whose direct limit is the N non-commutative solenoid \(A^S_{\theta}\). The non-commutative solenoid can also be realized as a twisted group C*-algebra, defined by a specific cocycle. We can call the above the non-commutative 2-torus, and the non-commutative 2-N-solenoid and then generalize to higher dimensions, with analogous characterizations of simplicity.