Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one

The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then reverse-engineering a proof from the hints given.

For a preprint I'm working on, it would be preferable to give a precise citation from a "standard text", rather than spend time giving the proof "for the reader's convenience". Any suggestions?

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If anyone's interested, an outline of a proof is as follows: consider an idempotent P in B(H), with H a Hilbert space, and note that we can always decompose H as an orthogonal sum with respect to which P has the block matrix form

$$ P= left(begin{matrix} I & R \\ 0 & 0 end{matrix}right) $$

Then it's not hard to see that conjugating $P$ by the invertible operator

$$ S= left( begin{matrix} I & R \\ 0 & I end{matrix} right) $$

will give

$$ E = left(begin{matrix} I & 0 \\ 0 & 0 end{matrix} right) $$

Since $S= I+P-E$, it suffices to show that $E$ is in the C*-algebra generated by I and P (for then S will also lie in that algebra, and then we're done). This follows by messing around with various combinations of P, its adjoint, and their products.