Bezushchak O. Structural theory and asymptotic constructions of locally matrix algebras

The thesis is devoted to asymptotic constructions and structure theory of locally matrix algebras and their applications to groups and algebras of infinite matrices and Hamming spaces. We introduced new examples of locally matrix algebras of arbitrary dimensions and defined their Steinitz invariants. It is shown that this invariant does not determine a locally matrix algebra of an uncountable dimension up to an isomorphism, however it determines an algebra up to a universal elementary equivalence. We have also characterized Morita equivalence of countable-dimensional unital locally matrix algebras in terms of their Steinitz invariants. The thesis includes analysis of decompositions into tensor products of matrix algebras and primary locally matrix algebras. In particular, we constructed examples of uncountable-dimensional unital locally matrix algebras that do not decompose into a tensor product of primary algebras, which gives a negative answer to the question of Kurochkin. We introduced a new Steinitz invariant of a not necessarily unital locally matrix algebra: its spectrum that determines a countable-dimensional locally matrix algebra up to an isomorphism. We give a complete classification of saturated sets of Steinitz numbers that appear as spectrums of locally matrix algebras. It is proved that an countable unital locally standard Hamming space decomposes as a tensor product of standard Hamming spaces. These spaces are related to Cartan subalgebras of locally matrix algebras and are determined by their Steinitz invariants. For a not necessarily unital locally standard Hamming space we defined its spectrum which is a saturated set of Steinitz numbers and proved an analog of Dixmier’s theorem. The thesis includes study of automorphisms and derivations of locally matrix algebras, groups of infinite periodic matrices and derivations of associative and Lie algebras of infinite matrices. It is proved that the ideal of inner derivations of a locally matrix algebra is dense in the Lie algebra of all derivations in Tykhonoff topology and the subgroup of inner automorphisms of a unital locally matrix algebra is dense in the group of all autmorphisms and the semigroup of injective endomorphisms in Tykhonoff topology. We describe derivations and injective endomorphisms of infinite tensor products of matrix algebras as converging infinite sums of inner derivations or converging infinite products of inner automorphisms of a special type. It is proved that for a countable-dimensional locally matrix algebra the dimension of the Lie algebra of outer derivations and the order of the group of outer automorphisms are equal to |F|אּ0. It is proved also that the Lie algebra of outer derivations is not locally finite dimensional (an analog of the theorem of Strade). We used density of the algebra of inner derivations of a locally matrix algebra to show that derivations of the associative algebra of infinite matrices M∞(I,F) and special linear algebras sl∞(I,F), so∞(I,F), sp∞(I,F) are adjoint operators of elements from Mrcf(I,F) and glrcf(I,F) respectively, and derivations of the algebras Mrcf(I,F), glrcf(I,F) and the algebra of Jacobi matrices and its adjoint algebra are inner.