What is the smallest dimension a non-commutative C-star-algebra can have? Let $d$ denote this dimension. Clearly, $d\leq 4$ as $M_{2}(\mathbb{C})$ is a $4$-dimensional non-commutative C-star-algebra. Also, $\{0\}$ and $\mathbb{C}$ are the only $0$- and $1$-dimensional C*-algebras, respectively, and they are both commutative, thus $d\geq2$. But are there $2$- or $3$-dimensional non-commutative C-star-algebras?
I can't find any examples, so I think $d=4$.
See Wikipedia. A finite-dimensional $C^*$ algebra is isomorphic to a direct sum of full matrix algebras. The smallest noncommutative full matrix algebra is $M(2,\mathbb C)$ which is $4$-dimensional.