suppose we are given a vector space basis of a unital *-algebra $\mathcal{A} \subset \mathcal{M}_d(\mathbb{C})$. I found a proof showing that one can find a unitary $U$ such that $$U\mathcal{A}U^* = \bigoplus_{k} \mathcal{M}_{d_k} \otimes \mathbb{1}_{m_k} $$ where $\mathbb{1}_{m_k}$ is the identity in $\mathcal{M}_{m_k} $ and the dimensions on the right hand side add up to d.
Is it possible to calculate $U$ explicitly? The proof I found involves minimal projections in the center of $\mathcal{A}$ but I don't even know if it's possible to calculate the center of the algebra.
Thanks
You cannot expect to obtain information about an algebra from a vector space basis . For instance, your algebra could be $\mathbb C\oplus\mathbb C \subset M_2 (\mathbb C)$, $d=2$, and your basis $$\begin {bmatrix}57&0\\0&58\end {bmatrix},\ \ \ \begin {bmatrix}-\sqrt2&0\\0&\pi\end {bmatrix}. $$ Then $U $ could be the identity, or $$U=\begin {bmatrix}-1&0\\0&-i\end {bmatrix} , $$ or $$U=\begin {bmatrix}0&\frac {1+\sqrt3}2\\ i&0\end {bmatrix}, $$ or another of an infinity of choices.