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 [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser.22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. 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.

Morita equivalent unital locally matrix algebras

We consider the Hamming space of periodic (0, 1)-sequences and a continual family of its subspaces defined as direct limits of finite Hamming spaces. These subspaces form a complete lattice under inclusion which is isomorphic to the lattice of supernatural numbers. We explicitly describe the isometry groups of these spaces. This involves certain constructions similar to the hyperoctahedral groups but accounting for additional structures on the underlying sets.