# How rich the group of unitary elements in a von Neumann algebra to get “Murray-von Neumann” equivalence?

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Neumann algebra. It is well known that in a finite von Neumann algebra the equivalence of projections can be obtained by unitary elements, i.e. if $e\sim f$ in a finite von Neumann algebra then there exists an unitary element $u$ such that $u^*eu=f$. What can we say in general cases; let $e\sim f$, if there exists projections $e_i,f_i$ with $e_1e_2=0=f_1f_2$, $e=e_1+e_2$, $f=f_1+f_2$ along with unitaries $u_i$ such that $u_i^*e_iu_i=f_i$?

• In an infinite von Neumann algebra the projection 1=e is MvN-equivalent to many other projections f≠1. But u*eu=u*1u=u*u=1≠f, so there is no such u. – Dmitri Pavlov Jul 5 '18 at 18:08
• @DmitriPavlov Does this immediately show that you cannot write $e$ as the sum of two orthogonal projects, and the same for $1$, with those subprojections unitarily equivalent? – Matthew Daws Jul 7 '18 at 6:55
• @MatthewDaws: You can always do it if you are willing to subdivide. For two projections e,f to be unitarily equivalent it is necessary and sufficient that e~f and 1−e~1−f. Furthermore, in the σ-finite case e~f if and only if tr(e)=tr(f), where tr denotes the canonical center-valued trace. Thus it suffices to show that if tr(e)=tr(f), we can find e_i,f_i such that tr(e_i)=tr(f_i), tr(1−e_i)=tr(1−f_i), e=e_1+e_2, f=f_1+f_2. It suffices to deal with the case of an infinite factor and tr(e)=tr(f)=∞. In this case we take e_i and f_i such that tr(e_i)=tr(f_i)=∞. – Dmitri Pavlov Jul 7 '18 at 17:12
• what about $e_1=f_1=0$? Moreover, if $e$ is a rank-one projection in $B(H)$, one of $e_i$'s must be 0. – C.Ding Jul 12 '18 at 1:08
• @C.Ding: What about them? I require tr(e_i)=tr(f_i)=∞, but tr(0)=0. – Dmitri Pavlov Jul 19 '18 at 14:23

If two projections in a von Neumann algebra are related in the sense of the question then they are Murray-von Neumann equivalent. In Theorem 4.1 of "Equivalence in operator algebras", by Kadison and Pedersen, it is shown more generally that if $e,f$ are projections in a von Neumann algebra such that $e=\sum_i x_i^*x_i$ and $f=\sum_i x_ix_i^*$ for some $x_i$ (the sums are allowed to be infinite) then $e$ and $f$ are Murray-von Neumann equivalent.