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Mat. Zametki, 2019, Volume 105, Issue 5, Pages 647–655 (Mi mz11710)  

Trace and Differences of Idempotents in $C^*$-Algebras

A. M. Bikchentaev

Kazan (Volga Region) Federal University

Abstract: Let $\varphi$ be a trace on a unital $C^*$-algebra $\mathcal{A}$, let $\mathfrak{M}_{\varphi}$ be the ideal of definition of the trace $\varphi$, and let $P,Q \in \mathcal{A}$ be idempotents such that $QP=P$. If $Q \in \mathfrak{M}_{\varphi}$, then $P \in \mathfrak{M}_{\varphi}$ and $0 \le \varphi(P) \le \varphi(Q)$. If $Q-P \in \mathfrak{M}_{\varphi}$, then $\varphi(Q-P)\in \mathbb{R}^+$. Let $A,B\in \mathcal{A}$ be tripotents. If $AB=B$ and $A\in \mathfrak{M}_{\varphi}$, then $B \in \mathfrak{M}_{\varphi}$ and $0 \le \varphi (B^2)\le \varphi (A^2)<+\infty$. Let $\mathcal{A}$ be a von Neumann algebra. Then
$$ \varphi(|PQ-QP|)\le \min\{\varphi(P),\varphi(Q),\varphi(|P-Q|)\} $$
for all projections $P,Q \in \mathcal{A}$. The following conditions are equivalent for a positive normal functional $\varphi$ on a von Neumann algebra $\mathcal{A}$:
(i) $\varphi $ is a trace;
(ii) $\varphi(Q-P) \in \mathbb{R}^+$ for all idempotents $P,Q \in \mathcal{A}$ with $QP=P$;
(iii) $ \varphi(|PQ-QP|) \le \min\{\varphi(P),\varphi(Q)\}$ for all projections $P,Q \in \mathcal{A}$;
(iv) $\varphi(PQ+QP) \le \varphi(PQP+QPQ)$ for all projections $P,Q \in \mathcal{A}$.

Keywords: Hilbert space, linear operator, idempotent, tripotent, projection, trace-class operators, commutator, von Neumann algebra, $C^*$-algebra, trace.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 1.9773.2017/8.9
The work was completed at the expense of a subsidy allocated to Kazan Federal University to fulfill the state task in the field of scientific activity (1.9773.2017/8.9).


DOI: https://doi.org/10.4213/mzm11710

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Document Type: Article
UDC: 517.98
Received: 01.06.2017

Citation: A. M. Bikchentaev, “Trace and Differences of Idempotents in $C^*$-Algebras”, Mat. Zametki, 105:5 (2019), 647–655

Citation in format AMSBIB
\Bibitem{Bik19}
\by A.~M.~Bikchentaev
\paper Trace and Differences of Idempotents in $C^*$-Algebras
\jour Mat. Zametki
\yr 2019
\vol 105
\issue 5
\pages 647--655
\mathnet{http://mi.mathnet.ru/mz11710}
\crossref{https://doi.org/10.4213/mzm11710}
\elib{http://elibrary.ru/item.asp?id=37424219}


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