Let $A$ be a C\*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional. 



> 
> Q. Is $A$ finite dimensional? 
> 
> 
> 



finite dimensional C*-algebras
The answer is yes. To prove it, assume first that $A$ is separable. In
this case, by Theorem 7.10 in John B. Conway's book "A course in
Operator Theory", we may choose a faithful representation $\pi$ of $A$
on a separable Hilbert space $H$. Choose an orthonormal basis
$\{e\_n\}\_{n\in{\mathbb N}}$ for $H$ and define
 $$
 \varphi(a) = \sum\_{n\in{\mathbb N}}{1\over
2^n}\langle\pi(a)e\_n,e\_n\rangle, \quad \forall a\in A,
 $$
 so that $\varphi$ is a state on $A$. Observe that $\varphi$ is
faithful because if $\varphi(a^\*a)=0$, then $\pi(a)e\_n=0$, for every
$n$, which implies that $\pi(a)=0$, and hence also that $a=0$, by virtue
of the faithfulness of $\pi$.


Notice that we have just proven that every separable C\*-algebra has a
faithful state.


It is then an easy exercise to prove that the GNS representation of $A$
given by $\varphi$ is faithfull. It is obviously also cyclic, hence
finite dimensional by hypothesis, so $A$ is finite dimensional.


The general case follows from Matthew Daws' reduction to the separable
case: Let us prove that every separable closed \*-subalgebra $B\subseteq
A$ satisfies the condition in the question, that is, every cyclic
representation of $B$ is finite dimensional. Indeed, given a cyclic
representation $\pi$ of $B$, let $\xi$ be a cyclic vector with unit
norm, and consider the associated state
 $$
 \varphi(b) = \langle\pi(b)\xi,\xi\rangle, \quad \forall b\in B.
 $$
 By Corollary 7.6 in Conway's book, $\varphi$ admits an extension to a
state $\psi$ on $A$. The GNS representation of $A$ given by
$\psi$, which I am going to denote by $\rho$, is then
finite dimensional by hypothesis. If $\eta$ is the canonical
cyclic vector for $\rho$, then
 $$
 K:=\overline{\hbox{span}}\{\rho(b)\eta:b\in B\}
 $$
 is an invariant subspace for the restriction $\rho|\_B$, and if
one further restricts $\rho|\_B$ to a representation of $B$ acting
on $K$, which I am going to cal $\tau$, then $\tau$ is cyclic, with
cyclic vector $P(\eta)$, where $P$ is the orthogonal projection
from the space of $\rho$ onto $K$. The state on $B$ given by
 $$
 \omega(b) = \langle\tau(b)P(\eta),P(\eta)\rangle, \quad
\forall b\in B
 $$
 is then equal to $\varphi$ (to see this one should first prove that
$\rho(b)P(\eta)=P\rho(b)\eta = \rho(b)\eta\, $), and then by the
uniquenes of the GNS construction (Theorem 7.7 in Conway's book), one
has that $\tau$ is equivalent to $\pi$.


Since $K$ is a subspace of the space of $\rho$, it
is finite dimensional and hence $\pi$ is also a finite dimensional
representation.


This completes the proof that every cyclic representation of every
separable closed \*-subalgebra of $A$ is finite dimensional.


By the first part of the argument we have that any separable closed
\*-subalgebra of $A$ is finite dimensional, so it follows easily that $A$
is finite dimensional.


