# On existence of Singular value decomposition in von Neumann algebras

For which class of von Neumann algebras we will have singular value decomposition of each element of the algebra?

• You would have to define what you mean by "singular value decomposition". – Martin Argerami Oct 2 '18 at 0:54
• The question of Martin is pertinent. It is not clear (to me) what do you want your diagonal algebra to be in a general von Neumann algebra. If you pick a generic Abelian subalgebra $D$ you would need that any selfadjoint element $x \in M$ lies in $U^\ast D U$ or some unitary. Therefore, a natural condition would be that all unital Abelian subalgebras are conjugate. – Adrián González-Pérez Oct 2 '18 at 10:01
• I forgot the word "maximal". – Adrián González-Pérez Oct 2 '18 at 11:01
• Beside finite dimensional algebras I cannot give an example from the top of my head. I think that there are nonequivalent maximal Abelian subalgebras (MASA's) in the hyperfinite II$_1$ factor: arxiv.org/pdf/math/0602155.pdf – Adrián González-Pérez Oct 2 '18 at 11:05
• There are non-equivalent masas in $B(H)$, namely the discrete and continuous masas. – Martin Argerami Oct 3 '18 at 3:10