# Number of non-isomorphic C$^*$-algebras of given dimension

There is a neat result classifying all finite-dimensional C$^*$-algebras:

Every non-zero finite-dimensional C$^*$-algebra is $*$-isomorphic to $M_{n_1}(\mathbb C)\oplus\cdots\oplus M_{n_k}(\mathbb C)$ for some $n_1,\ldots,n_k\in\mathbb N$.

This led me to the wonder how many non-isomorphic C$^*$-algebras there are of a given dimension. That is, given some $n\in\mathbb N$, what is the number of C$^*$-algebras $A$ such that $\dim(A)=n$ (up to $*$-isomorphism)? Clearly the answer is the number of elements of the set $$\left\{(n_1,n_2,\ldots)\in(\mathbb N\cup\{0\})^\mathbb N: n_1\geq n_2\geq\cdots, n=\sum_{k=1}^\infty n_k^2\right\},$$ but I don't have the machinery necessary to calculate this. So I figured I would ask the MSE community.

If $f:\mathbb N\to\mathbb N$ is defined by $f(n)=|G_n|$, where $G_n$ is the set defined above, is there any closed-formula for computing $f(n)$ for any given $n\in\mathbb N$?

• This is probably quite complicated. Here are formulas for how many partitions into 2, 3, 4 squares a number admits. – s.harp Mar 25 '18 at 2:05