Worked solution: Marcus–Spielman–Srivastava proof via interlacing families (2013)

Step 8 of 8: Conclusion: tracing

KS_2

back to Kadison and Singer's original question

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In plain words

With $KS_2$ now a proven theorem, all that is left is to walk back down the chain of equivalences that Nik Weaver and earlier researchers (Anderson, Akemann, Casazza and collaborators) had already assembled over the preceding decades: $KS_2$ implies a form of the paving conjecture, which implies the Kadison–Singer problem has an affirmative answer.

The net result, resolved in 2013 and published in the Annals of Mathematics in 2015: every pure state on the diagonal operators of $B(\ell^2)$ does extend uniquely to the whole algebra. A single question from 1959, once reduced to elementary linear algebra, was answered using ideas from a seemingly unrelated corner of mathematics: the theory of stable polynomials from statistical physics.

KS_2 \implies \text{Paving Conjecture (Akemann--Anderson)} \implies \text{Kadison--Singer: unique extension holds}

Detailed analysis

Weaver (2004) proved that $KS_2$ implies a version of the projection paving conjecture of Akemann and Anderson (1991), which in turn was already known (through work summarized in Casazza, Fickus, Tremain and Weber's 2006 survey "The Kadison–Singer problem in mathematics and engineering") to imply an affirmative solution of the Kadison–Singer problem itself: every pure state on the diagonal masa of $B(\ell^2)$ extends to a unique pure state on $B(\ell^2)$ . With $KS_2$ now proved unconditionally by Marcus, Spielman and Srivastava, this entire fifty-four-year-old chain closes.

As a byproduct, the same interlacing-families machinery gives Marcus, Spielman and Srivastava's Theorem 1.5, an explicit quantitative paving bound for arbitrary Hermitian matrices with zero diagonal, which independently resolves the finite-dimensional version of the Bourgain–Tzafriri conjecture that had been studied since the 1980s and is closely related to problems in frame theory and signal processing (relevant to the Kadison–Singer problem's applications noted by Casazza and Tremain).

As the earlier steps of this proof already cautioned, the 2013 paper is the final link of a long chain of reductions assembled by many mathematicians over decades — Marcus, Spielman and Srivastava's real contribution is the interlacing-families technology that finally cracked the hardest remaining link, $KS_2$ itself.

Knowledge used in this step

Operator theory