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 Mat. Sb., 2008, Volume 199, Number 4, Pages 3–20 (Mi msb3851)

On the representation of elements of a von Neumann algebra in the form of finite sums of products of projections. III. Commutators in $C^*$-algebras

A. M. Bikchentaev

N. G. Chebotarev Research Institute of Mathematics and Mechanics, Kazan State University

Abstract: It is proved that every skew-Hermitian element of any properly infinite von Neumann algebra can be represented in the form of a finite sum of commutators of projections in this algebra. A new commutation condition for projections in terms of their upper (lower) bound in the lattice of all projections of the algebra is obtained. For the full matrix algebra the set of operators with canonical trace zero is described in terms of finite sums of commutators of projections and the domain in which the trace is positive is described in terms of finite sums of pairwise products of projections. Applications to $AF$-algebras are obtained.
Bibliography: 33 titles.

DOI: https://doi.org/10.4213/sm3851

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English version:
Sbornik: Mathematics, 2008, 199:4, 477–493

Bibliographic databases:

UDC: 517.983+517.987
MSC: Primary 46L10; Secondary 47C15
Received: 12.03.2007 and 24.10.2007

Citation: A. M. Bikchentaev, “On the representation of elements of a von Neumann algebra in the form of finite sums of products of projections. III. Commutators in $C^*$-algebras”, Mat. Sb., 199:4 (2008), 3–20; Sb. Math., 199:4 (2008), 477–493

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb3851
• https://doi.org/10.4213/sm3851
• http://mi.mathnet.ru/eng/msb/v199/i4/p3

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This publication is cited in the following articles:
1. A. M. Bikchentaev, “Commutativity of projectors and trace characterization on von Neumann algebras. I”, Russian Math. (Iz. VUZ), 53:12 (2009), 68–71
2. A. M. Bikchentaev, “Commutativity of projections and characterization of traces on von Neumann algebras”, Siberian Math. J., 51:6 (2010), 971–977
3. A. M. Bikchentaev, “Commutation of Projections and Trace Characterization on von Neumann Algebras. II”, Math. Notes, 89:4 (2011), 461–471
4. A. M. Bikchentaev, “Trace and Differences of Idempotents in $C^*$-Algebras”, Math. Notes, 105:5 (2019), 641–648
5. A. M. Bikchentaev, “Idealnye $F$-normy na $C^*$-algebrakh. II”, Izv. vuzov. Matem., 2019, no. 3, 90–96
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