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I have been thinking about what can we say about decomposing any representation as a direct sum of irreducible representations. I know that every cyclic representation corresponds to a representation coming from a state, which is the essence of the GNS construction. Moreover such a representation is irreducible $\textbf{iff}$ the state is a pure state.

I also know that every representation can be decomposed into a direct sum of cyclic representations. Now my question is whether we can decompose any representation of a C* algebra $A$ into direct sum of irreducible representations. It seems to me that not every representation can be represenation can be decomposed in this way. Is there a description/classification of those representations which can be decomposed in this way.

I don't know much representation theory but I think such results are true for representations of some other class of objects.

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It can be shown that if $A$ is separable and $\pi:A\to B(H)$ is a representation with $H$ separable, then $\pi$ is approximately unitarily equivalent to a direct sum of irreducible representations.

But the "approximate" part is essential. For instance let $A=C_r^*(\mathbb F_2)$, the reduced C$^*$-algebra of the free group in two generators, and consider the identity representation $\pi:A\to B(\ell^2(\mathbb F_2))$ (where $A$ is the C$^*$-algebra generated by the image of the left regular representation of $\mathbb F_2$). If $\pi_1$ is irreducible and $\pi=\pi_1\oplus\pi_2$, then $p=\pi_1(I_A)$ is a projection in the commutant $C_r^*(\mathbb F_2)'=R(\mathbb F_2)$, which is known to be a II$_1$-factor. Because $A$ is simple, $\pi_1$ is faithful, and so $\pi_1(A)=pA$; and $pA$ cannot be dense in $B(pH)$, because $p$ has nontrivial subprojections in $R(\mathbb F_2)$, so $pA$ has nontrivial commutant in $B(pH)$, contradicting the irreducibility of $\pi_1$.

With similar ideas as above, one can show that any representation of a II$_1$-factor into a separable Hilbert space cannot be a sum of irreducible representations, because any irreducible representation of a II$_1$-factor is uncountably-dimensional.

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  • $\begingroup$ Is this "approximate" a technical term? I have never come across such aterm. If yes can you suggest a reference for it. $\endgroup$ – Parish Mar 28 '17 at 16:19
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    $\begingroup$ Approximate unitary equivalence is a fairly standard notion. Two representations $\pi$ and $\rho$ are approximately unitarily equivalent if there exists a sequence $\{u_n\}$ of unitaries such that $$\rho(a)=\lim_{n\to\infty}u_n\pi(a)u_n^*,\ \ \ \ \ a\in A.$$ You can find it defined for instance in [Davidson, II.4] or [Brown-Ozawa, 1.7]. $\endgroup$ – Martin Argerami Mar 28 '17 at 16:32

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