Originally Posted by

**Mauritzvdworm** Let $\displaystyle B$ be a C*-algebra and let $\displaystyle p$ be a rank-one projection in $\displaystyle K$, the space of compact operators on some Hilbert space $\displaystyle H$.

I would like to construct a projection $\displaystyle \mathcal{P}$ in the multiplier algebra $\displaystyle M(B\otimes K)$ such that $\displaystyle \mathcal{P}(B\otimes K)\mathcal{P}=B\otimes p$

here is my current idea:

consider the mapping $\displaystyle \eta:B\otimes K\rightarrow B\otimes p$ such that $\displaystyle b\otimes k\mapsto b\otimes p$. Here $\displaystyle b\in B,k\in K$

It can easily be shown that $\displaystyle \eta^2=\eta$, however I'm having some trouble in showing that $\displaystyle \eta^*=\eta$. Is it possible?