How do I compute the specific map between two isomorphic finite C* algebras? Starting with a finite C* algebra $\mathcal{A} \subset M_{n}\left({\mathbb C}\right)$ (complex $n\times n$ matrices), $\mathcal{A}$ is known to be isomorphic to a canonical algebra of the form $$\mathcal{A} \sim \bigoplus_k {\mathcal{A}_{k}}$$where each C* subalgebra $\mathcal{A}_k \sim M_{n_{k}}\left({\mathbb C}\right)$ is determined by the set of minimal nonzero self-adjoint central projections of $\mathcal{A}$. More specifically for my problem involving quantum noiseless subspaces, $$\mathcal{A} \sim \bigoplus_k {M_{p_{k}}\left({\mathbb C}\right)\otimes {\mathbb 1}_{q_{k}}}$$ where ${\mathbb 1}_{q_{k}}$ is the $q_{k}\times q_{k}$ identity matrix.

I know how to compute the central projections of $\mathcal{A}$ and determine the block structure (i.e. the sets $\left\{ p_k\right\}$ and $\left\{ q_k\right\}$) of this canonical algebra, which I'll denote $\mathcal{A}'$.

By isomorphism, I mean there exists some unitary matrix $U$ such that $\mathcal{A} = U^\dagger \mathcal{A}' U$. My question is: how do I compute $U$, or where can I look to find out how to compute it?

Any assistance or advice would be greatly appreciated.
The most direct way is to decompose the tautological representation of $\mathcal A$ on ${\mathbb C}^n$ to irreducible components. Each of them corresponds to $M_{p_k}$ for certain $k$ and has dimension $p_k$. I guess that projectors to these subspaces are the thing the original poster calls “central projections”, since they belong to the center of $\mathcal A$.

I cannot say how one can compute these things because I do not know how exactly the matrix subalgebra is specified.
The most direct way is to decompose the tautological representation of $\mathcal A$ on ${\mathbb C}^n$ to irreducible components. Each of them corresponds to $M_{p_k}$ for certain $k$ and has dimension $p_k$. I guess that projectors to these subspaces are the thing the original poster calls “central projections”, since they belong to the center of $\mathcal A$.

I cannot say how one can compute these things because I do not know how exactly the matrix subalgebra is specified.