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 prove that for locally compact, compactly generated self-dual Abelian groups \(G\), there are canonical unitary integral operators on \(L^2(G)\) analogous to the Fourier transform but which have orders 3 and 6. To do this, we establish the existence of a certain projective character on \(G\) whose phase multiplication with the FT gives rise to the Cubic transform (of order 3). (Thus, although the Fourier transform has order 4, one can “make it” have order 3 (or 6) by means of a phase factor!)

Soit \(G\) un groupe localement compact, engendré par un sousensemble compact, et isomorphe à son groupe dual. On construit des operateurs intégrals unitaires canoniques qui sont analogues à la transformée de Fourier, mais qui sont d’ordres trois et six.

Keywords: C*-algebra, Fourier transform, Gaussian sums, Hilbert space, L2 spaces, Locally compact Abelian groups, characters, cyclic groups, integral transforms, projective char- acter, self-dual groups, unitary operators

AMS Subject Classification: Harmonic analysis and almost periodicity, General properties and structure of LCA groups, Compact groups, General properties and structure of locally compact groups, $C^*$-algebras and $W$*-algebras in relation to group representations, General properties and structure of real Lie groups, Integral representations; integral operators; integral equations methods, Integral operators, Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory 11K70, 22B05, 22C05, 22D05, 22D25, 22E15, 31A10, 45P05, 46L35, 46L40, 46L80

PDF(click to download): Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups

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 report on recent results on the existence of Cubic and Hexic integral transforms on self-dual locally compact groups (orders 3 and 6 analogues of the classical Fourier transform) and their application in constructing a canonical continuous section of smooth projections \(\mathcal E(t)\) of the continuous field of rotation C*-algebras \(\{A_t\}_{0 \le t \le 1}\) that is invariant under the noncommutative Hexic transform automorphism. This leads to invariant matrix (point) projections of the irrational noncommutative tori \(A_\theta\). We also present a quick method for computing the (quantized) topological invariants of such projections using techniques from classical Theta function theory.

On décrit des résultats récents sur l’existence d’une transformation intégrale d’ordre trois (ou d’ordre six) sur un groupe localement compact abélien self-dual. On étudie l’application possible à la construction d’un champs continu de projecteurs invariants sous l’automorphisme associé du champs de C*-algèbres de rotation. On calcule certains invariants topologiques de ces projecteurs.

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

AMS Subject Classification: Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory, $K$-theory, String and superstring theories; other extended objects (e.g.; branes), Topological field theories, String and superstring theories 46L35, 46L40, 46L80, 55N15, 81T30, 81T45, 83E30

PDF(click to download): Periodic Integral Transforms and Associated Noncommutative Orbifold Projections

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

S. Walters

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: C*-algebra, Fourier transform, automorphisms, integral transforms, rotation algebras

AMS Subject Classification: General transforms, Special transforms (Legendre; Hilbert; etc.), Automorphisms, K-theory and operator algebras -including cyclic theory 44A05, 44A15, 46L40, 46L80

PDF(click to download): Periodic Integral Transforms and C*-Algebras