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TMF, 2018, Volume 195, Number 1, Pages 75–80 (Mi tmf9351)  

This article is cited in 1 scientific paper (total in 1 paper)

Differences of idempotents in $C^*$-algebras and the quantum Hall effect

A. M. Bikchentaev

Kazan (Volga Region) Federal University, Kazan, Russia

Abstract: Let $\varphi$ be a trace on the unital $C^*$-algebra $\mathcal{A}$ and $\mathfrak{M}_{\varphi}$ be the ideal of the definition of the trace $\varphi$. We obtain a $C^*$ analogue of the quantum Hall effect: if $P,Q\in\mathcal{A}$ are idempotents and $P-Q\in\mathfrak{M}_{\varphi}$, then $\varphi((P-Q)^{2n+1})=\varphi (P-Q)\in \mathbb{R}$ for all $n\in\mathbb{N}$. Let the isometries $U\in\mathcal{A}$ and $A=A^*\in\mathcal{A}$ be such that $I+A$ is invertible and $U-A\in\mathfrak{M}_{\varphi}$ with $\varphi (U-A)\in \mathbb{R}$. Then $I-A, I-U \in\mathfrak{M}_{\varphi}$ and $\varphi (I-U)\in \mathbb{R}$. Let $n\in\mathbb{N}$, $\dim \mathcal{H}=2n+1$, the symmetry operators $U,V\in\mathcal{B}(\mathcal{H})$, and $W=U-V$. Then the operator $W$ is not a symmetry, and if $V=V^*$, then the operator $W$ is nonunitary.

Keywords: Hilbert space, linear operator, idempotent, symmetry, projection, unitary operator, trace-class operator, $C^*$-algebra, trace, quantum Hall effect.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-41-02433
Ministry of Education and Science of the Russian Federation 1.1515.2017/4.6
1.9773.2017/8.9
This research is supported by the Russian Foundation for Basic Research, the government of the Republic of Tatarstan (Grant No. 15-41-02433), and subsidies given to Kazan Federal University for performing the governmental target program in the field of science (1.1515.2017/4.6 and 1.9773.2017/8.9).


DOI: https://doi.org/10.4213/tmf9351

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English version:
Theoretical and Mathematical Physics, 2018, 195:1, 557–562

Bibliographic databases:

Document Type: Article
Received: 09.02.2017

Citation: A. M. Bikchentaev, “Differences of idempotents in $C^*$-algebras and the quantum Hall effect”, TMF, 195:1 (2018), 75–80; Theoret. and Math. Phys., 195:1 (2018), 557–562

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Bikchentaev, “Sled i raznosti idempotentov v $C^*$-algebrakh”, Matem. zametki, 105:5 (2019), 647–655  mathnet  crossref  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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