TITLE: Infinite dimensional central simple algebras
QUESTION [8 upvotes]: When constructing the Brauer group of a field, only the finite-dimensional central simple algebras are considered (because of Artin-Wedderburn's characterization).
But what happens to the infinite-dimensional ones? (I.e., to simple algebras which are infinite-dimensional over their centers).

Is there an analogue for the Brauer group?
Is there some structure theorem like Artin-Wedderburn's?
What can be said about cross products of these algebras?

REPLY [7 votes]: Perhaps one approach to thinking about this question would be Dixmier-Douady theory (as described in Brylinski's book "Loop Spaces, Characteristic Classes and Geometric Quantization"). This involves considering certain Banach algebras over the complex numbers as infinite dimensional analogs of central simple algebras, and leads to some useful notions generalizing bundles of central simple algebras over topological spaces.
The point of view here starts with the correspondence between bundles of central simple $\mathbb C$-algebras over a topological space $X$ and torsion elements of the second cohomology $H^2(X, \underline{\mathbb C}^*)$. This comes from the identification of isomorphism classes degree $n$ algebra bundles with $H^1(X, \underline{PGL_n})$ and the nonabelian short exact sequence:
$$ 1 \to \underline{\mathbb C}^* \to \underline{GL_n} \to \underline{PGL_n} \to 1$$
(here I'm using the underline to denote the corresponding sheaves of continuous functions).
Now, when you do this, you find that those classes which come from $H^1(X, \underline{PGL_n})$ (i.e. from degree $n$ central simple algebra bundles) are necessarily $n$-torsion in $H^2(X, \underline{\mathbb C}^*)$. what Dixmier-Doady theory does is identify all the elements of this $H^2$ as arising from infinite dimensional algebras.
A little more precisely, assuming paracompactness of $X$, on fixes some separable Hilbert space $E$, and defines $\mathcal K$ to be the Banach algebra of compact operators on $E$. It then turns out that one can naturally identify the whole of 
$H^2(X, \underline{{\mathbb C}^\ast})$
 with the set of isomorphism classes of locally trivial Banach algebra bundles with fiber isomorphic to $\mathcal K$. What makes this really work is an analog of the above sequence
$$ 1 \to {\mathbb C}^\ast \to {\mathcal L}^\ast \to {\mathcal L}^\ast/{\mathbb C}^\ast \to 1 $$
where $\mathcal L$ is the Banach algebra of bounded (sup-norm) operators on $E$ (which incidentally happens to have center $\mathbb C$), and one uses an identification $\mathcal L^\ast/\mathbb C^\ast \cong Aut(\mathcal K)$.