Let τ be a faithful normal semifinite trace on a von Neumann algebraM, and M^u be a unitary part of M. We prove a new property of rearrangements of some tripotents in M. If V ∈M is an isometry (or a coisometry) and U - V is τ-compact for some U ∈M^u then V ∈M^u. Let M
be a factor with a faithful normal trace τ on it. If
V ∈M^{is} an isometry (or a coisometry) and U - V
is compact relative toMfor some U ∈M^u then V ∈M^u. We also obtain some corollaries.
