A ring is artinian if it has no infinite descending chains of ideals. Of course finite-dimensional algebras are artinian. I'm wondering if it's possible to have an artinian C*-algebra (or Banach algebra) which is infinite-dimensional. I'm not sure if it suffices to assume the descending chain condition on closed one-sided ideals, or if you need it on all ideals.
Of course if artinian implies finite-dimensional then $A$ must be unital after all.
The reason I ask is because I'm wondering if there is a C*-version of the Artin--Wedderburn Theorem:
If $A$ is a (semiprimitive) artinian C*-algebra then it is $*$-isomorphic direct sum of full matrix algebras over division algebras: $A\simeq \bigoplus M_{d_i}(\Delta_i)$.
Note that semiprimitive (Jacobson radical = 0) is free for C*-algebras, and the only C*-division algebra is $\mathbf{C}$. So my hope is that this would yield that artinian is a direct sum of full matrix algebras over $\mathbf{C}$, hence finite-dimensional.
Some problems I see with this are: how would we know that $\Delta_i$'s are closed in the topology on $A$? If we knew that, we could make the above argument that $\Delta_i\simeq \mathbf{C}$. If we apply the classical Artin--Wedderburn Theorem we get an algebraic isomorphism with a direct sum of matrix algebras over division algebras, but it's not clear to me that these division algebras would be finite-dimensional if all we assume is that $A$ is artinian.
Thoughts?
If a Banach algebra $A$ does not possess an infinite increasing (resp. decreasing) chain of closed left ideals, then $A$ is finite dimensional, see Theorem 3 and Remark 5 in SinclairTullo1974. Thus, every Noetherian (resp. Artinian) Banach algebra is finite dimensional.