Reference for “the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner”

I read a paper of Alain Connes on "Duality between shapes and spectra" and in page 4, he says

Due to a theorem of von Neumann the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner, independent of the geometry one starts with.

Question.

• What is a precise statement and reference for this "mysterious" theorem of von Neumann ?
• What are nontechnical explanations and justifications of this phenomenon ?

One possible interpretation of Connes's statement is that up to an isomorphism, there is a unique faithful indecomposable representation of any commutative von Neumann algebra on a Hilbert space.

Indeed, the category of von Neumann algebras is contravariantly equivalent to the category of compact strictly localizable enhanced measurable spaces.

After extracting an enhanced measurable space $$(X,M,N)$$ from a commutative von Neumann algebra $$A$$ in this manner, elements of $$A$$ can be identified with equivalence classes of bounded measurable functions on $$X$$ modulo equality almost everywhere.

We can now easily describe isomorphism classes of representations of $$A$$ on a Hilbert space. Such an isomorphism class is specified by partitioning $$X$$ into almost disjoint (up to a negligible set) nonnegligible measurable subsets $$\{X_i\}_{i∈I}$$, and assigning a distinct cardinal number $$a_i$$ to each element of the partition. The corresponding Hilbert space is $$\bigoplus_{i∈I} {\rm L}^2(X_i,M_{X_i},N_{X_i})⊗{\bf C}^{a_i}$$ and $$A$$ acts on each summand by restricting the corresponding bounded measurable function on $$X$$ to $$X_i$$ and then acting via multiplication on the corresponding $${\rm L}^2$$-space. Here $${\bf C}^{a_i}$$ denotes any complex Hilbert space of dimension $$a_i$$.

Such a representation is faithful if $$a_i≥1$$ for all $$i$$. It is indecomposable if $$a_i≤1$$ for all $$i$$. Thus, a faithful indecomposable representation must have $$a_i=1$$ for all $$i$$, and there is a unique such a representation, namely $${\rm L}^2(X,M,N)$$, also known as the Haagerup standard from of $$A$$.

The point of all this is that although the (abstract) commutative von Neumann algebra appears to know nothing about an enhanced measurable space or its $${\rm L}^2$$-space, all this data can be reconstructed in a unique manner, i.e., it is unique up to a unique isomorphism.

If, furthermore, we know the Dirac operator, we can proceed to refine $$X$$ to a smooth manifold, as described by Connes.

• Thanks for the detailed answer. – dohmatob Sep 10 at 2:49