# On a Poisson Summation Formula for Noncommutative Tori

On a Poisson Summation Formula for Noncommutative Tori Complex Anal. Oper. Theory Complex Analysis https://doi.org/10.1007/s11785-018-0812-7 and Operator Theory On a Poisson Summation Formula for Noncommutative Tori Igor Nikolaev Received: 8 January 2018 / Accepted: 31 May 2018 © Springer International Publishing AG, part of Springer Nature 2018 Abstract It is proved that a maximal abelian subalgebra of the noncommutative torus commutes with the Laplace operator on a complex torus. As a corollary, one gets an analog of the Poisson summation formula for noncommutative tori. Keywords Selberg trace formula · Noncommutative torus Mathematics Subject Classiﬁcation 11F72 (Selberg trace formula) · 46L85 (noncommutative topology) 1 Introduction The Poisson summation formula is an elementary and fundamental fact of harmonic analysis and representation theory. The simplest case of such a formula says that for every function f ∈ C (R) it holds f (n) = f (n), (1) n∈Z n∈Z ∞ ∞ where C (R) is the set of C -smooth complex-valued functions with a compact −2πi νx support on the real line R and f (ν) := f (x )e dx is the Fourier transform −∞ Communicated by Daniel Aron Alpay. B Igor Nikolaev igor.v.nikolaev@gmail.com Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, New York, NY http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Complex Analysis and Operator Theory Springer Journals

# On a Poisson Summation Formula for Noncommutative Tori

, Volume OnlineFirst – Jun 5, 2018
9 pages

/lp/springer_journal/on-a-poisson-summation-formula-for-noncommutative-tori-dBVjuEkyDs
Publisher
Springer Journals
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general; Operator Theory; Analysis
ISSN
1661-8254
eISSN
1661-8262
D.O.I.
10.1007/s11785-018-0812-7
Publisher site
See Article on Publisher Site

### Abstract

Complex Anal. Oper. Theory Complex Analysis https://doi.org/10.1007/s11785-018-0812-7 and Operator Theory On a Poisson Summation Formula for Noncommutative Tori Igor Nikolaev Received: 8 January 2018 / Accepted: 31 May 2018 © Springer International Publishing AG, part of Springer Nature 2018 Abstract It is proved that a maximal abelian subalgebra of the noncommutative torus commutes with the Laplace operator on a complex torus. As a corollary, one gets an analog of the Poisson summation formula for noncommutative tori. Keywords Selberg trace formula · Noncommutative torus Mathematics Subject Classiﬁcation 11F72 (Selberg trace formula) · 46L85 (noncommutative topology) 1 Introduction The Poisson summation formula is an elementary and fundamental fact of harmonic analysis and representation theory. The simplest case of such a formula says that for every function f ∈ C (R) it holds f (n) = f (n), (1) n∈Z n∈Z ∞ ∞ where C (R) is the set of C -smooth complex-valued functions with a compact −2πi νx support on the real line R and f (ν) := f (x )e dx is the Fourier transform −∞ Communicated by Daniel Aron Alpay. B Igor Nikolaev igor.v.nikolaev@gmail.com Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, New York, NY

### Journal

Complex Analysis and Operator TheorySpringer Journals

Published: Jun 5, 2018

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