Derivations and automorphisms of locally matrix algebras

TL;DR

This paper characterizes derivations and automorphisms of infinite tensor products of matrix algebras, revealing that the dimensions of their outer derivations and automorphism groups are tied to the cardinality of the base field.

Contribution

It provides a detailed description of derivations and automorphisms for locally matrix algebras, establishing their structure and cardinality relations.

Findings

Outer derivations form a Lie algebra of dimension |F|^{aleph_0}

Outer automorphisms form a group of order |F|^{aleph_0}

Results depend on the cardinality of the base field F

Abstract

We describe derivations and automorphisms of infinite tensor products of matrix algebras. Using this description we show that for a countable--dimensional locally matrix algebra $A$ over a field $\mathbb{F}$ the dimension of the Lie algebra of outer derivations of $A$ and the order of the group of outer automorphisms of $A$ are both equal to $|\mathbb{F}|^{\aleph_0},$ where $|\mathbb{F}|$ is the cardinality of the field $\mathbb{F}.$