Portal de Periódicos da CAPES

A II1 Factor Approach to the Kadison–Singer Problem

2014; Springer Science+Business Media; Volume: 332; Issue: 1 Linguagem: Inglês

10.1007/s00220-014-2055-4

1432-0916

Sorin Popa,

Mathematical Analysis and Transform Methods

We show that the Kadison–Singer problem, asking whether the pure states of the diagonal subalgebra $${\ell^\infty\mathbb{N}\subset \mathcal{B}(\ell^2\mathbb{N})}$$ have unique state extensions to $${\mathcal{B}(\ell^2\mathbb{N})}$$ , is equivalent to a similar statement in II1 factor framework, concerning the ultrapower inclusion $${D^\omega \subset R^\omega}$$ , where D is the Cartan subalgebra of the hyperfinite II1 factor R (i.e., a maximal abelian *-subalgebra of R whose normalizer generates R, e.g. $${D=L^\infty([0, 1]^{\mathbb{Z}}) \subset L^\infty([0,1]^{\mathbb{Z}} \rtimes \mathbb{Z} = R)}$$ , and ω is a free ultrafilter. Instead, we prove here that if A is any singular maximal abelian *-subalgebra of R (i.e., whose normalizer consists of the unitary group of A, e.g. $${A=L(\mathbb{Z})\subset L^\infty([0,1]^\mathbb{Z})\rtimes \mathbb{Z}=R}$$ ), then the inclusion $${A^\omega \subset R^\omega}$$ does satisfy the Kadison–Singer property.