The basic theory of projections was worked out by . Two subspaces belonging to "M" are called (Murray–von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if "M" "knows" that the subspaces are isomorphic). This induces a natural equivalence relation on projections by defining "E" to be equivalent to "F" if the corresponding subspaces are equivalent, or in other words if there is a partial isometry of "H" that maps the image of "E" isometrically to the image of "F" and is an element of the von Neumann algebra. Another way of stating this is that "E" is equivalent to "F" if "E=uu*" and "F=u*u" for some partial isometry "u" in "M".