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Let $A$ be a C*-algebra. It seems that for given an *-representation $\pi$ of $A$, there is unique central projection $z_{\pi}$ in $A^{**}$ such that $\pi$ is just (unitary equivalent to) $\rho_z$ where $$\rho_z:A\to A^{**} : a\to az$$

It means that there is a one to one correspondence between Rep($A$), the equivalence class of *-representations of $A$, and $Z(A^{**})$, the set of all central projectionas in $A^{**}$

ABB
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    What is the question? – Nik Weaver Dec 31 '15 at 13:07
  • I would like to be sure that $\pi\simeq \rho_z$. – ABB Dec 31 '15 at 16:31
  • Your notation doesn't completely make sense to me. The conclusion is false, however; let $A = K(H)$ and $A^{**} = B(H)$. The only central projections in $B(H)$ are $0$ and $I$, but $K(H)$ has lots of inequivalent representations --- one of each multiplicity. – Nik Weaver Dec 31 '15 at 16:47

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