Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.

Let $p,q \in M_{\infty}(A)$ be (Murray–von Neumann) equivalent projections.

Let $U_{p,q} = \{u \in M \otimes B(H) \text{ partial isometry } \vert uu^*=p$, $u^*u=q \}$

*Question*: Is is true that $U_{p,q} \cap M_{\infty}(A) \neq \emptyset$ ?

*Application*: let $\Gamma$ a countable torsion-free ICC group, $H=l^2(H)$, $A=C^*_r(\Gamma)$, $M=L\Gamma = A''$.

$M \otimes B(H)$ is a ${\rm II}_\infty$-factor with a trace $Tr$. Two projections $p,q$ are equivalent iff $Tr(p)=Tr(q)$.

If the question above admits a positive answer for $A$: could we deduce that $K_0(A)$ is a subgroup of $\mathbb{R}$?

The problem is that $K_0$ is defined (here p8) for idempotents (more general that projections).