  We investigate angles between Haagerup--Schultz projections of operators
belonging to finite von Neumann algebras, in connection with a property
analogous to Dunford's notion of spectrality of operators. In particular, we
show that an operator can be written as the sum of a normal and an
s.o.t.-quasinilpotent operator that commute if and only if the angles between
its Haagerup--Schultz projections are uniformly bounded away from zero (and we
call this the uniformly nonzero anlges property). Moreover, we show that
spectrality is equivalent to this uniformly nonzero angles property plus
decomposability. Finally, using this characterization, we construct an easy
example of an operator which is decomposable but not spectral, and we show that
Voiculescu's circular operator is not spectral (nor are any of the circular
free Poisson operators).
