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 infinite-dimensional Hilbert space. We study the problem of description of all subspaces of the vector space of all infinite-dimensional \(n\times n\)-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 infinite-dimensional matrices. If we can describe all maximal involutive algebras of infinite-dimensional 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 infinite-dimensional 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

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