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?



Power's Theorem for irreducible representations
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.](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528005/)). 


Therefore, there cannot be such a characterization.


You may find [*On representations of finite type*](http://www.pnas.org/content/95/23/13392.full.pdf) by Kadison of some interest as well.


