# The inner structure of finite-dimensional $C^*$-algebras

This is just to complete the questions about the inner structure of finite-dimensional $$C^*$$-algebras. Please correct the statements and help me with the appropriate links if it is not very difficult. Maybe I am wrong somewhere.

Let $$H$$ be a Hilbert space with a finite or countable basis. Let $$\mathscr{A}$$ be a finite-dimensional unital $$C^*$$-algebra of operators acting on $$H$$. One of the diamonds, the Wedderburn theorem states that $$\mathscr{A}$$ is a direct sum of simple matrix algebras $$\mathscr{A}\cong\bigoplus_{n=1}^N\mathbb{C}^{m_n\times m_n}.$$ But we need a bit more, namely the characterization of its inner structure in terms of unitary transformations of Hilbert spaces. Let $$\mathcal{E}^n_{jj}$$ be the orthogonal projectors corresponding to the diagonal matrix units in $$\mathbb{C}^{m_n\times m_n}$$. I think that the following statements are true $$i)\ \ \ \ \ {\rm rank}(\mathcal{E}^n_{11})=...={\rm rank}(\mathcal{E}^n_{m_nm_n}),\ \ n=1,...,N;$$ $$ii)\ \ \ \ \ \sum_{n=1}^Nm_n{\rm rank}(\mathcal{E}^n_{11})=\dim H.$$ The ranks can be finite or infinite. Now, I think that the set of pairs of numbers $${\rm struc}(\mathscr{A})=\{(m_n,{\rm rank}(\mathcal{E}^n_{11}))\}_{n=1}^N$$ is the complete characteristic of the inner structure of $$\mathscr{A}$$. Namely, a unital $$C^*$$-alebra $$\mathscr{A}_1$$ of operators acting on a Hilbert space $$H_1$$ has the same $${\rm struc}(\mathscr{A})={\rm struc}(\mathscr{A}_1)$$ if and only if there is a unitary $$\mathcal{U}:H\to H_1$$ such that $$\mathscr{A}_1=\mathcal{U}\mathscr{A}\mathcal{U}^{-1}$$. Moreover, for any finite set of pairs $$s\subset{\mathbb{N}}\times\overline{\mathbb{N}}$$ there is a finite-dimensional unital $$C^*$$-algebra $$\mathscr{A}$$ of operators acting on some Hilbert space such that $${\rm struc}(\mathscr{A})=s$$.

I think the idea of the Proof of this statement is the same as in the comment Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspaces or can be extracted from the Proof of Wedderburn theorem directly. I am sorry for the multiple more or less basic questions about the structure of $$C^*$$-algebras, but I know the people for whom this information can be useful.