Derivations and automorphisms of locally matrix algebras (Q2662194)

In the paper under review, the author describes derivations and automorphisms of countable dimensional locally matrix algebras. Let \(A\) be a locally matrix algebra, \(\mathrm{Der}(A)\) the Lie algebra of all derivations of the algebra \(A\), \(\mathrm{Inder}(A)\) the ideal of all inner derivations, \(\mathrm{Outder}(A)\) the algebra of outer derivations of A, \(\mathrm{Map}(A, A)\) the set of all mappings \(A \to A\), \(\mathrm{Aut}(A)\) and \(\mathrm{Inn}(A)\) the group of automorphisms and the group of inner automorphisms of the algebra \(A\), respectively. In the first part of the paper, the author studies the Tychonoff topology on the set \(\mathrm{Map}(A, A)\) and prove that the ideal \(\mathrm{Inder}(A)\) is dense in \(\mathrm{Der}(A)\). Moreover, if \(A\) is unital, the completion of \(\mathrm{Inn}(A)\) in \(\mathrm{Map}(A, A)\) in the Tychonoff topology is the semigroup \(P(A)\) of injective endomorphisms (embeddings) of \(A\). In particular, \(\mathrm{Inn}(A)\) is dense in \(\mathrm{Aut}(A)\). Then derivations of an infinite tensor product of matrix algebras are described. More precisely the following result is proved: if \(A\) is a countable dimensional locally matrix algebra. Then the Lie algebra \(\mathrm{Outder}(A)\) is not locally finite dimensional. In the second part, the author also proves that: \begin{itemize} \item If \(A=\otimes_{i\in I} A_i\), where \(I\) is an infinite set and each algebra \(A_i\) is a matrix algebra over a field \(\mathbb{F}\) of the dimension \(> 1\), then \[\dim_{\mathbb{F}} \mathrm{Der}(A) = \dim_{\mathbb{F}} \mathrm{Outder}(A) = | \mathbb{F} |^{|I|}.\] \item If \(A\) is a countable dimensional locally matrix algebra over a field \(\mathbb{F}\), then \[\dim_{\mathbb{F}} \mathrm{Der}(A) = \dim_{\mathbb{F}}\mathrm{Outder}(A) = | \mathbb{F} |^{\aleph_0}\] where \(\aleph_0\) stands for the countable cardinality. \end{itemize} The final result of the paper is devoted to the study of automorphisms. Specifically it is proved that if \(A\) is a countable dimensional locally matrix algebra over a field \(\mathbb{F}\), then \[| \mathrm{Aut}(A) | = | \mathrm{Out}(A) | = | \mathbb{F} |^{\aleph_0}.\]