Minimal sequences and the Kadison-Singer problem

TL;DR

This paper explores the Kadison-Singer problem and its connection to Feichtinger's conjecture for exponentials, establishing conditions under which certain sets of exponential functions can be decomposed into Riesz sequences.

Contribution

It proves a new equivalence condition involving minimal sequences and Riesz subsequences for the Feichtinger conjecture in the context of exponential functions.

Findings

Set of projections onto measurable subsets of the circle group can be expressed as finite unions of Riesz sequences.

Existence of a Riesz subsequence with a nonzero minimal characteristic function is equivalent to the union property.

Provides a new criterion linking minimal sequences to the decomposition of exponential sets into Riesz sequences.

Abstract

The Kadison-Singer problem asks: does every pure state on the diagonal sublgebra of the C*-algebra of bounded operators on a separable infinite dimensional Hilbert space admit a unique extension? A yes answer is equivalent to several open conjectures including Feichtinger's: every bounded frame is a finite union of Riesz sequences. We consider the special case: Feichtinger's conjecture for exponentials and prove that the set of projections onto a measurable subset of the circle group of the set of exponential functions equals a union of a finite number of Reisz sequences if and only if there exists a Reisz subsequence corresponding to integers whose characteristic function is a nonzero minimal sequence.