Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC, V2N 4Z9, Canada; e-mail: walters@unbc.ca

Abstract/Résumé:

We demonstrate, in a rather quantitative manner, the existence of topological obstructions to approximating the irrational rotation C*-algebra \(A_\theta\) by Fourier invariant unital C*-subalgebras of either of the forms \[M \oplus B \oplus \sigma(B), \qquad M \oplus N \oplus D \oplus \sigma(D) \oplus \sigma^2(D) \oplus \sigma^3(D),\] where \(M, N\) are Fourier invariant matrix algebras (over \(\mathbb C\)), \(B\) is a C*-subalgebra whose unit projection is flip invariant and orthogonal to its Fourier transform, and \(D\) is a C*-subalgebra whose unit projection is orthogonal to its orbit under the Fourier transform. Here, \(\sigma\) is the noncommutative Fourier transform automorphism of \(A_\theta\) defined by \(\sigma(U) = V^{-1},\ \sigma(V)=U\) on the canonical unitary generators \(U,V\) obeying the unitary Heisenberg commutation relation \(VU = e^{2\pi i\theta}UV\).

On montre l’existence d’obstructions topologiques à l’approximation du tore non-commutatif par sous-algèbres de certains types qui sont invariantes sous l’automorphisme de Fourier.

Keywords: C*-algebra, Fourier transform, automorphisms, inductive limits, noncommutative tori, rotation algebra, topological invariants, topological obstructions, unbounded traces

AMS Subject Classification: Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory 46L35, 46L40, 46L80

PDF(click to download): Topological Obstruction to Approximating the Irrational Rotation C*-algebra by Certain Fourier Invariant C*-subalgebras