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Sibirsk. Mat. Zh., 2017, Volume 58, Number 2, Pages 243–250 (Mi smj2856)  

This article is cited in 3 scientific papers (total in 3 papers)

Differences of idempotents in $C^*$-algebras

A. M. Bikchentaev

Lobachevskiĭ Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan, Russia

Abstract: Suppose that $P$ and $Q$ are idempotents on a Hilbert space $\mathscr H$, while $Q=Q^*$ and $I$ is the identity operator in $\mathscr H$. If $U=P-Q$ is an isometry then $U=U^*$ is unitary and $Q=I-P$. We establish a double inequality for the infimum and the supremum of $P$ and $Q$ in $\mathscr H$ and $P-Q$. Applications of this inequality are obtained to the characterization of a trace and ideal $F$-pseudonorms on a $W^*$-algebra. Let $\varphi$ be a trace on the unital $C^*$-algebra $\mathscr A$ and let tripotents $P$ and $Q$ belong to $\mathscr A$. If $P-Q$ belongs to the domain of definition of $\varphi$ then $\varphi(P-Q)$ is a real number. The commutativity of some operators is established.

Keywords: Hilbert space, linear operator, idempotent, tripotent, projection, unitary operator, trace class operator, operator inequality, commutativity, $W^*$-algebra, $C^*$-algebra, trace, ideal $F$-norm.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-41-02433
The author was supported by the Russian Foundation for Basic Research and the Government of the Republic of Tatarstan (Grant 15-41-02433).


DOI: https://doi.org/10.17377/smzh.2017.58.201

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English version:
Siberian Mathematical Journal, 2017, 58:2, 183–189

Bibliographic databases:

Document Type: Article
UDC: 517.98
MSC: 35R30
Received: 21.03.2016

Citation: A. M. Bikchentaev, “Differences of idempotents in $C^*$-algebras”, Sibirsk. Mat. Zh., 58:2 (2017), 243–250; Siberian Math. J., 58:2 (2017), 183–189

Citation in format AMSBIB
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\paper Differences of idempotents in $C^*$-algebras
\jour Sibirsk. Mat. Zh.
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\vol 58
\issue 2
\pages 243--250
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\crossref{https://doi.org/10.17377/smzh.2017.58.201}
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\transl
\jour Siberian Math. J.
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\pages 183--189
\crossref{https://doi.org/10.1134/S003744661702001X}
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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, S. A. Abed, “Paranormal elements in normed algebra”, Russian Math. (Iz. VUZ), 62:5 (2018), 10–15  mathnet  crossref  isi
    2. A. M. Bikchentaev, “Differences of idempotents in $C^*$-algebras and the quantum Hall effect”, Theoret. and Math. Phys., 195:1 (2018), 557–562  mathnet  crossref  crossref  adsnasa  isi  elib
    3. A. M. Bikchentaev, “Sled i raznosti idempotentov v $C^*$-algebrakh”, Matem. zametki, 105:5 (2019), 647–655  mathnet  crossref  elib
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