Murray–von Neumann equivalence

Remarks

Such a $v$ is a partial isometry with initial projection $p$ and final projection $q$. The relation is reflexive ($v = p$), symmetric ($v^*$ in place of $v$), and transitive (if $v^*v = p$, $vv^* = q$, $w^*w = q$, $ww^* = r$, then $wv$ works for $p$ and $r$). Example: in $B(\ell^2(\N))$ let $s e_k = e_{2k}$; then $s^* s = 1$ and $s s^*$ is the projection onto the span of the $e_{2k}$, so this projection is equivalent to $1$. Non-example: in $M_2(\C)$, the projections $\operatorname{diag}(1, 0)$ and $1$ are not equivalent, since $v^*v$ and $vv^*$ always have the same rank. For K-theory one uses the same relation for projections in matrix algebras of different sizes, with $v$ a rectangular matrix over $A$. Unitarily equivalent projections are Murray–von Neumann equivalent, but the converse fails in general, as the example above shows.