Let $X$ be any finite dimensional $C^*$ algebra over $\mathbb{C}$. Let $\{\alpha_1,\cdots,\alpha_n\}$ be a basis for $X$. Now we have $\alpha_i \alpha_j \in X$. Write it as a linear combination interms of the above basis. So you will be getting a vector in $\mathbb{C}^n$ say denoted by $a_{ij}$. So if you take the multiplication table of the above basis, we will be getting a matrix $A=[a_{ij}] \in M_n({\mathbb{C}}^n)$.
My question is as follows. Is there any special basis for $X$ so that each column and each row of $A$ is an orthonormal basis for $\mathbb{C}^n$?
A finite dimensional $C^*$-algebra is isomorphic to $$ M_{n_1}(\mathbb C) \oplus \cdots \oplus M_{n_r}(\mathbb C), $$ where $n_1,n_2,\cdots,n_r \in \mathbb N$. In particular they have a canonical basis $$ \{e_{ij}^{(1)} \}_{i,j=1}^{n_1} \cup \cdots \cup \{e_{ij}^{(r)}\}_{i,j=1}^{n_r}. $$ The matrix you are looking for is the identity matrix in $M_l\mathbb C)$, where $l = n_1+n_2+\cdots+n_r$.