Minimal sequences and the Kadison-Singer problem

Abstract

The Kadison-Singer problem asks: does every pure state on the C*-algebra ℓ∞ (Z) admit a unique extension to the C*-algebra β (ℓ2 (Z))? A yes answer is equivalent to several open conjectures including Feichtinger's: every bounded frame is a finite union of Riesz sequences. We prove that for measurable S ⊂ T, {χS e2πikt}k∈Z is a finite union of Riesz sequences in L2 (T) if and only if there exists a nonempty A ⊂ Z such that χA is a minimal sequence and {χS e2πikt} k∈A is a Riesz sequence. We also suggest some directions for future research.