Automorphisms of Infinite Dimensional Algebras–Part 1

I’m going to be using this, and most likely the next post or two to organize my thoughts about a fun little problem I’ve been toying around with.

Recently, I’ve become interested in directly infinite algebras. Gien a field of characteristic zero K, these are K-algebras A which have elements x and y such that xy = 1 but yx \neq 1. Whenever an algebra is directly infinite, one can also say that A has a sub-algebra which is isomorphic to the non-unital algebra of infinite matrices with finitely many nonzero entries. One can see this by considering the algebra generated by

M = \{y^{i-1}(1-yx)x^{j-1} : i,j \in \mathbb{Z}\}.

This is certainly a sub-algebra of A and calculation shows that M is a countably infinite set of matrix units, let’s denote the individual matrix units by E_{ij} = y^{i-1}(1-yx)x^{j-1}.

The rest of this post will be devoted to investigating some of the properties of a particularly nice (or not nice if you are a glass half-empty person) example of a directly infinite algebra. This algebra, this algebra, called the “Toeplitz-Jacobson” algebra, was first identified by Jacobson in the 50’s here. In that paper, he investigated properties of elements which are only “one-sided invertible.” Particularly interesting to me is his construction of infinite matrix representations of the generators.

x \mapsto Z := \begin{pmatrix}0 &1 &0 &0 &\cdots \\ 0 &0 &1 &0 &\cdots \\ 0 &0 &0 &1 &\cdots \\ 0 &0 &0 &0 &\cdots \\ \vdots &\vdots &\vdots &\vdots &\ddots\end{pmatrix} and y \mapsto Y := \begin{pmatrix}0 &0 &0 &0 &\cdots \\ 1 &0 &0 &0 &\cdots \\ 0 &1 &0 &0 &\cdots \\ 0 &0 &1 &0 &\cdots \\ \vdots &\vdots &\vdots &\vdots &\ddots\end{pmatrix}

Under these mappings, A may be embedded in the row and column finite matrix algebra, via the mapping A = \langle x,y | xy = 1\rangle \hookrightarrow \langle X, Y \rangle \subseteq B(K) where B(K) denotes the K-algebra of infinite matrices indexed by \mathbb Z^+ \times \mathbb Z^+ where each row and column has finitely many nonzero entries. Note that, if one were to v isualize these matrices as operators on an infinite dimensional vector space V with ordered basis \{b_1, b_2, \ldots\}, x and y can be visualized as the shift operators on V. It’s due to this property that the Toeplitz-Jacobson algebra gains the former of its names. The Toeplitz algebra from functional analysis is merely the completion (in the appropriate norm) of the algebra A (where K is taken to be \mathbb C.

The set of matrix units M then takes on a very important role in this embedding. The map sends the set of matrix units of A, E_{ij}, to e_{ij}, the set of matrix units of B(K). Moreover, \text{Span}_K\{E_{ij}\; | \; i,j \in \mathbb Z^+\} is a minimal ideal of A whose embedding in B(K) consists of the set of infinite matrices with only finitely many nonzero entries, which will be denoted by M_\infty(K) (we’ll abuse notation a bit and also denote the subalgebra generated by the E_{ij}‘s in A by M_\infty(K) also.

This ability to create an infinite set of matrix units is one of the charms of directly infinite algebras. Since \{E_{ii} \; |\; i \in \mathbb Z^+\} is an infinite set of orthogonal idempotents, the Toeplitz-Jacobson algebra fails many of the finiteness properties. Because one can create a family of ideals

A(\sum_{i = 1}^n E_{ii})

which increases without bound, A (as well as any directly infinite algebra) fails to be Noetherian, and thus also fails to be Artinian.

The Toeplitz-Jacobson algebra occupies an interesting space in the realm of module decompositions. Because it has a quotient isomorphic to a commutative ring (A/M_\infty(K) \simeq K[x,x^{-1}], the algebra of Laurent polynomials), it has invariant basis number, namely if A^n \simeq A^m then $n = m$. Thus it cannot be broken up into some number of copies of itself. However, it may be decomposed into a copy of itself and another A-module. Note that since xy = 1, there is a left A-module isomorphism between A_A and Ax. Then a simple argument shows that A \simeq Ayx \oplus A(1-yx) \simeq Ax \oplus \text{ann}_l(y), where \text{ann}_l(a) denotes the left annihilator of the element a \in A. The isomorophism between $A$ and $Ax$ then implies that

A \simeq A \oplus M

for some nonzero left A-module M. As with previous results, this is a consequence of the failure of A to be directly finite rather than a consequence of any specific property of the Toeplitz-Jacobson algebra. This decomposition property is the reason that the Toeplitz-Jacobson algebra seems to spin out of control of many particularly nice properties. It can “bud off” these submodules M which gives it just enough room to run away.

This exposition is by no means a complete narrative of all the properties of the Toeplitz-Jacobson algebra. For example, I left out much of the rich representation theory investigated by Iovanov and Sistko in their paper (as well as many others), and it also fails to mention its importance in the theory of Leavitt path algebras. However, these properties are the ones that come first to mind mind when I think of directly infinite algebras generally and the Toeplitz-Jacobson algebra specifically in a research context.

In the next post we’ll talk about automorphisms of this algebra.