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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.

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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.

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    $\begingroup$ Apparently I cannot read ... Theorem I.10.8 of Ken's book in a slightly different form for identifying the multiplicities.Thanks $\endgroup$ – vand Mar 18 at 19:41
  • $\begingroup$ No problem, you are welcome! $\endgroup$ – Nik Weaver Mar 18 at 19:52
  • $\begingroup$ @ConfusedSpider, if this answer was useful to you, then don't forget to upvote it in addition to accepting it. $\endgroup$ – LSpice Mar 18 at 20:26
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    $\begingroup$ Thanks. I did not know that I could do that - on my own question that is. Done $\endgroup$ – vand Mar 18 at 21:42

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