The Terms I grew up with:
A bounded linear operator $U$ on a Hilbert space $H$ is a partial isometry if there exists a subspace $M$ of $H$ such that $\|Ux\| = \|x\|$ for all $x\in M$, and $Ux = 0$ for all $x\in M^{\perp}$.
$M$ is called the initial space of $U$ and $V(M)$ is called the final space.
Am I correct to say that the definition (particularly line 3 and 4) just require $U$ to have the same initial and final space as $P\otimes 1$ for some projection $P$ on $H_{\mu}$?
I didn't give much background but I can supply more if it would help, just let me know.
The source is Enock & Schwartz - Kac Algebras and Duality of Locally Compact Groups.