Minimal Sequences and the Kadison-Singer Problem

The Kadison-Singer problem asks: does every pure state on the C*-algebra ℓ^∞(Z) admit a unique extension to the C*-algebra B(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 subset T, {Xs e^2πikt} k element of Z is a finite union of Riesz sequences in L²(T) if and only if there exists a nonempty subset Z such that X Lambda is a minimal sequence and {Xs e^2πikt} k element of Lambda is a Riesz sequence. We also suggest some directions for future research.