Hilbert space — 2 articles found.

The Surprising Power of Averaging over Groups

C. R. Math. Rep. Acad. Sci. Canada Vol. 42 (3) 2020, pp. 38-41
Vol.42 (3) 2020
James Hogan; Samuel Li Details
(Received: 2020-09-20 )
(Received: 2020-09-20 )

James Hogan Department of Mathematics, University of Toronto, 40 St George St, Toronto, ON M5S 2E4
e-mail: james.hogan@mail.utoronto.ca

Samuel Li Department of Mathematics, University of Toronto, 40 St George St, Toronto, ON M5S 2E4
e-mail: samuelj.li@mail.utoronto.ca

Abstract/Résumé:

We highlight the surprising power of averaging via a few illuminating examples. Two of these problems involve characterizations of Hilbert space, and the third is a fundamental result in noncommutative geometry.

Nous soulignons le pouvoir surprenant de la moyenne par quelques exemples éclairants. Deux de ces problèmes concernent la caractérisation de l’espace de Hilbert, et le troisième est un résultat fondamental en géométrie non commutative.

Keywords: Averaging, Hilbert space, Irrational rotation algebra, noncommutative geometry

AMS Subject Classification: Instructional exposition (textbooks; tutorial papers; etc.), Characterizations of Hilbert spaces, Noncommutative geometry (__ la Connes) 00-01, 46C15, 58B34

PDF(click to download): The Surprising Power of Averaging over Groups

Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups

C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (4) 2015, pp. 121-130
Vol.37 (4) 2015
Sam Walters Details
(Received: 2014-10-10 , Revised: 2014-10-10 )
(Received: 2014-10-10 , Revised: 2014-10-10 )

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