An [Formula: see text]-algebra [Formula: see text] with unit [Formula: see text] is said to be a locally matrix algebra if an arbitrary finite collection of elements [Formula: see text] from [Formula: see text] lies in a subalgebra [Formula: see text] with [Formula: see text] of the algebra [Formula: see text], that is isomorphic to a matrix algebra [Formula: see text], [Formula: see text]. To an arbitrary unital locally matrix algebra [Formula: see text], we assign a Steinitz number [Formula: see text] and study a relationship between [Formula: see text] and [Formula: see text].

We study an abstract class of Hamming spaces (known also as measure algebras) that generalizes standard Hamming spaces [Formula: see text]. We classify countable locally standard Hamming spaces and show that each of them can be realized as the Boolean algebra of idempotents of a Cartan subalgebra of a locally matrix algebra.

We construct a unital locally matrix algebra of uncountable dimension that(1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [1] and [5]. We also show that for an arbitrary infinite Steinitz number s there exists a unital locally matrix algebra A having the Steinitz number s and not isomorphic to a tensor product of finite dimensional matrix algebras.