Power's Theorem for irreducible representations

Question

Let $A_{\alpha}\subset B(H)$ be a bunch of unital C*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A_{\alpha})$'s. A very nice theorem of Stephen C. Power identifies the character space of $C=C^{\ast}(\cup_{\alpha}A_{\alpha})$ the C*-algebra generated by the union of $A_{\alpha}$'s as a certain subset of the cartesian product of $M(A_{\alpha})$'s. Is there a similar characterisation for the spectrum i.e. the space of all irreducible representations instead of characters?

Answer 1

An important point to remember about Power's result is that if $\varphi$ is a character of $C$ then $\varphi|_{A_\alpha}$ will be a character of $A_\alpha$. This does not happen for irreducible representations.

In particular, consider $C = \overline{\cup M_{2^n}}$, the CAR algebra. Any non-zero irreducible representation $\varphi$ of $C$ will be faithful, since $C$ is simple, and thus $\varphi|_{M_{2^n}}$ is nonzero and so is unitarily equivalent to an infinite direct sum of the identity representation because $M_{2^n}$ has a one point spectrum. However, it is known that the CAR algebra has an uncountable number of non-equivalent irreducible representations (Gårding and Wightman, Representations of the anticommutation relations. Proc. Nat. Acad. Sci. 40, (1954). 617–621.).

Therefore, there cannot be such a characterization.

You may find On representations of finite type by Kadison of some interest as well.