Examples of this $C^*$ Algebra

Question

Does anyone have some concrete examples of $AW^*$ algebra? Wiki gives several characterizations, but I'd like to have some examples. In particular, I'm wondering about $\ell^{\infty}$ in the real and complex case. Wiki claims that a $C^*$ algebra is $AW^*$ if its spectrum is extremally disconnected. In the commutative case, this says that the Gelfand dual is extremally disconnected, which I think is false for complex small l infinity?

Answer 1

Any von Neumann algebra is an AW$^*$-algebra. So in particular $\ell^\infty(\mathbb N)$ is an AW$^*$-algebra. Its spectrum is $\beta\mathbb N$, the Stone-Cech compactification of $\mathbb N$ which is, indeed, extremely disconnected.

To get non-von Neumann examples you need to get C$^*$-algebras with little or no normal states. The protoptypical example is the Dixmier Algebra; this is the algebra obtained by taking the quotient of the bounded Borel functions on $[0,1]$ by the ideal of functions which are zero outside of a meagre set. Tensoring this with von Neumann factors one can get non-von Neumann AW$^*$-algebras of any type.