Theory for finite-dimensional representations of C*-algebras

Question

As the title suggests, I am interested if there is any well-developed theory of finite-dimensional representations of $C^*$-algebras (the algebra itself may not necessarily be finite-dimensional). For example, character theory provides a powerful tool for studying representations of finite groups while highest weights are used for studying representations of Lie algebras. Both of these theories provide (at least up to some extent) useful characterisations of irreducible representations and identification of representations up to equivalence classes.

Is there anything analogous for finite-dimensional representations of $C^*$-algebras (or particular type of $C^*$-algebras if no general theory exists)? Would be much appreciated if you could point me to any relevant references to further study the area.

Answer 1

First some notation & definitions: Let $A$ be a $C^*$-algebra, $\hat{A}$ be its structure space (the set of equivalence classes of irreducible representations) with the hull-kernel topology. Let $F_n(\hat{A})\subseteq\hat{A}$ be the subset comprised of $n$-dimensional representations and $\displaystyle F(\hat{A})=\bigcup_{n\in\mathbb{N}} F_n(\hat{A})$. $A$ is said to be residually finite dimensional (RFD) if $\hat{A}=\overline{F(\hat{A})}$; equivalently, if finite dimensional representations separate the points of $A$. $A$ is $n$-homogeneous if $\hat{A}=F_n(\hat{A})$, and $A$ is $n$-subhomogeneous if $\hat{A}=\bigcup_{k=1}^n F_k(\hat{A})$. I suggest you to search the literature with these keywords.

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If $A$ is a $C^*$-algebra, then $I=\ker{F(\hat{A})} =\{x\in A: \pi(x)=0\hspace{3mm}\forall\pi\in F(\hat{A})\}$ is a closed *-ideal of $A$, $I$ has no finite dimensional representations, and $A/I$ is a RFD $C^*$-algebra. Thus, the study of fin. dim. representations reduces to the study of RFD algebras, e.g., see Archbold1995 for a list of properties.

If not only do we have $\hat{A}=\overline{F(\hat{A})}$, but $\hat{A}=F(\hat{A})$ (equivalently, all irreducible representations are finite dimensional), then we can say more. The following are equivalent for a $C^*$-algebra $A$ (Thm 1, Hamana1977)

1. $\hat{A}=F(\hat{A})$.
2. $A$ has the Dunford-Pettis property.
3. $A^{**}$ is a finite type I $W^*$-algebra.
4. $A^{**}$ is a finite $W^*$-algebra.

These four conditions are also equivalent to

5. The multiplication of $A$ is jointly weakly sequentially continuous.