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DOI: 10.23671/VNC.2018.2.14714 Maximal Commutative Involutive Algebras on a Hilbert Space
Arzikulov F. N.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2.
Abstract:
This paper is devoted to involutive algebras of bounded linear operators on an infinitedimensional Hilbert space. We study the problem of description of all subspaces of the vector space of all infinitedimensional \(n\times n\)matrices over the field of complex numbers for an infinite cardinal number \(n\) that are involutive algebras. There are many different classes of operator algebras on a Hilbert space, including classes of associative algebras of unbounded operators on a Hilbert space. Most involutive algebras of unbounded operators, for example, \(\sharp\)algebras, \(EC^\sharp\)algebras and \(EW^\sharp\)algebras, involutive algebras of measurable operators affiliated with a finite (or semifinite) von Neumann algebra, we can represent as algebras of infinitedimensional matrices. If we can describe all maximal involutive algebras of infinitedimensional matrices, then a number of problems of operator algebras, including involutive algebras of unbounded operators, can be reduced to problems of maximal involutive algebras of infinitedimensional matrices. In this work we give a description of maximal commutative involutive subalgebras of the algebra of bounded linear operators in a Hilbert space as the algebras of infinite matrices.
Keywords: involutive algebra, algebra of operators, Hilbert space, infinite matrix, von Neumann algebra
Language: Russian
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For citation: Arzikulov F. N. Maximal Commutative Involutive Algebras on a Hilbert Space. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol.
20, no. 2, pp.1622. DOI 10.23671/VNC.2018.2.14714 ← Contents of issue 
 

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