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Is there any $*$ homomorphism $T$ from $A$ to $B$, wherein $B$ is a $*$ closed subalgebra of $C^*$ algebra $A$, containing the unit of $A$, such that $T(b)=b$ for all $b\in B$ and $\|T(a)\|=\|a\|$ for all $a\geq 0$ ?

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    I'm not quite sure what you're asking: Are you asking whether this situation can occur at all or are you asking whether such a $T$ exists for every $\ast$-closed subalgebra $B$ of $A$ containing the identity of $A$? If the latter is intended: why would you think this is the case? (it's not true: there need not even be an idempotent continuous linear map with range $B$)2012-06-25

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