I was recently reading the following paper: Structure of Leavitt Path Algebras of Polynomial Growth by Alahmedi et al. and I have a question about one of the results.
Fix a basis $\mathcal{B} = \{e_i : i \in \mathbb{Z}^+\}$ for a vector space $V$ over a field $K$ and let $E = \text{End}_K(V)$ be the $K$-algebra of endomorphisms of $V$; assume the endomorphisms act on the left. It is well known that any element of $E$ may be written as an infinite matrix where every column has only finitely many nonzero entries (any individual column may have an arbitrarily large number of entries however); because of this we will identify $E$ with these column finite matrices. There exists a non-unital subalgebra $M_\infty(K)$ of $E$ which consists of those infinite matrices which have only finitely many nonzero entries.
Theorem 12 of the paper states:
For an arbitrary automorphism $\varphi$ of $M_\infty(K)$ there exists an invertible element $T \in E$ such that $\varphi(a) = T^{-1} a T$ for all $a \in M_\infty(K)$.
The theorem cites Nathan Jacobson's The Structure of Rings as the location of the proof of this theorem, but no specific location within the book is given.
I have looked through the book and cannot find a result which matches the statement. The closest I could find is Section 10, Theorem 3
Let $A$ be a ring with an identity which possesses two sets of matrix units $\{e_{ij} : i,j = 1, \ldots, s\}$ and $\{f_{kl} : k,l = 1, \ldots, t\}$ such that the corresponding coefficient rings $B$ and $C$ are completely primary. Than $s = t$ and there exists a unit $u$ such that $f_{ij} = u^{-1}e_{ij}u$ for $i,j = 1, \ldots, s$ and $C = u^{-1} B u$.
This result from Jacobson only assumes the existence of a finite set of matrix units while $M_\infty(K)$ is the span of an infinite set of matrix units. This seems to be a severe obstruction to extending this result.
Did I miss another result in the book, or can anybody provide a reference to another resource which contains this result?