In von Neumann algebras, the Connes embedding problem or conjecture, due to Alain Connes, asks whether every type II1 factor on a separable Hilbert space can be embedded into the ultrapower of the hyperfinite type II1 factor by a free ultrafilter. The problem admits a number of equivalent formulations.
Let ω be a free ultrafilter on the natural numbers and let R be the hyperfinite type II1 factor with trace τ . One can construct the ultrapower R ω as follows: let l ∞ ( R ) = { ( x n ) n ⊆ R : s u p n | | x n | | < ∞ } be the von Neumann algebra of norm-bounded sequences and let I ω = { ( x n ) ∈ l ∞ ( R ) : l i m n → ω τ ( x n ∗ x n ) 1 2 = 0 } . The quotient l ∞ ( R ) / I ω turns out to be a II1 factor with trace τ R ω ( x ) = l i m n → ω τ ( x n + I ω ) , where ( x n ) n is any representative sequence of x .
Connes' Embedding Conjecture asks whether every type II1 factor on a separable Hilbert space can be embedded into some R ω .
The isomorphism class of R ω is independent of the ultrafilter if and only if the continuum hypothesis is true (Ge-Hadwin and Farah-Hart-Sherman), but such an embedding property does not depend on the ultrafilter because von Neumann algebras acting on separable Hilbert spaces are, roughly speaking, very small.
