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.


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