Maximal commutative involutive algebras on a Hilbert space

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.