Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ induced by $\mathcal{A}$ and the connections between them (if any). For example if $\mathcal{A}$ is a multiplicative abelian group of self-adjoint operators then $\mathcal{H}$ has the following decomposition over the characters of $\mathcal{A}$:
\begin{equation}
\mathcal{H} = \bigoplus\_{\lambda\in\hat{A}} \mathcal{H}\_\lambda
\end{equation}
where $\mathcal{H}\_\lambda = \{ v\in\mathcal{H}: Tv = \lambda(T)v \,,\forall T\in \mathcal{A}\}$.
For a different angle I came across the following theorem in *Theory of Quantum Error Correction for General Noise* by Knill, Laflamme and Viola, Phy. Rev. Lett. Vol 84, No. 11, (2000) (Theorem 5). Which states (using their notation)


**Theorem**: Let $\mathcal{A}$ be a $\dagger$-closed algebra of operators on Hilbert space $\mathcal{S}$, including the identity. Then $\mathcal{S}$ is isomorphic to a direct sum,
\begin{equation}
\mathcal{S} \simeq \bigoplus\_i \mathcal{C}\_i\otimes\mathcal{Z}\_i
\end{equation}
in such a way that in the representation on the right-hand side,
\begin{equation}
\mathcal{A} = \bigoplus\_i \operatorname{Mat}(C\_i)\otimes I^{(Z\_i)}
\end{equation}
and the commutant of $\mathcal{A}$ is given by
\begin{equation}
Z(\mathcal{A}) = \bigoplus\_i I^{(C\_i)}\otimes \operatorname{Mat}{(\mathcal{Z}\_i)}.
\end{equation}
The paper only references this result to C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, 1962 but I cannot find this result in this book or anywhere. A related paper says to consider "the decomposition of $\mathcal{S}$ into irreducible representations of a $\*$-subalgebra $\mathcal{A}$ …" for which I am trying to understand what this actually means.


So my questions are:


1. Looking for a proof, or reference to a proof, or idea of a proof, for the above theorem and to understand the statement "the decomposition of $\mathcal{S}$ into irreducible representations of a $\*$-subalgebra $\mathcal{A}$".
2. What, if any, is the connection between the two decompositions, especially when $\mathcal{A}$ is the group that I first stated and secondly when $\mathcal{A}$ is extended to become a $\*$-subalgebra?


So for example take $\mathcal{A}$ to be an abelian subgroup of the $n$-qubit Pauli group $\mathcal{P}^{\otimes n}$ where $\mathcal{P}=\langle iI,X,Z\rangle$ which is the situation that I am interested in to understand tensor product structures.



Decomposition of Hilbert spaces via groups and algebras representations
This is basic C${}^\*$-algebra theory and should be found in most introductory texts --- I feel sure it's in *C${}^\*$-algebras by Example* by Davidson, for instance.


