The Kadison-Singer Property

Abstract

What soon became the Kadison-Singer conjecture was formulated by Kadison and Singer in 1959 and was proved (against the negative advice on its validity by the originators!) by Marcus, Spielman, & Srivastava in 2014, after important earlier contributions by Anderson (1979, [1]), Weaver (2004, [26]), and others. Despite its seemingly technical setting within operator (algebra) theory, the conjecture and its resolution have generated considerable interest from the mathematical community, as exemplified by e.g. specialized conferences, a Seminar Bourbaki by Valette, a widely read blog by Tao, coverage by the Quanta magainze, and even by the press. This interest may be explained by the unexpectedly large scope of the conjecture (see Casazza et al, 2005, [4]) as well as by the closely related depth of its proof, which used techniques from diverse fields of mathematics (it may also have helped that Singer shared the 2004 Abel Prize with Atiyah, though for unrelated work).Despite this interest, a relatively elementary account of the conjecture and its proof was lacking so far. This monograph, which is a revised version of the author's M.Sc Thesis at Radboud University Nijmegen, fills this gap. In fact, it does far more than that; for example, it includes a clean proof that in the so-called 'continuous' case the conjecture (which indeed was never posited for that case) would be false, which is perhaps as surprising as its truth in the 'discrete case' (see below for this terminology). This was already established by Kadison and Singer themselves, but in a very contrived way. Furthermore, this book contains a detailed proof of the classification of maximal abelian subalgebras of the algebra B(H) of all bounded operators on a separable Hilbert space H that are closed under hermitian conjugation (i.e., MASA's), which lies at the basis of the Kadison-Singer conjecture. There are many other results like those, which make this treatise as complete and self-contained as can be expected given its modest length.All that remains to be added is a brief account of the historical context of the Kadison-Singer conjecture, which, as the originators acknowledge, was at least in part inspired by quantum mechanics. At the time, the Hilbert space approach to quantum mechanics proposed by von Neumann in 1932 was about 25 years old. In the meantime, von Neumann, Gelfand, and Naimark had created the new mathematical discipline of operator algebras, to which Kadison (who had been a student v vi Foreword This leaves the discrete case, about which the Kadison-Singer conjecture claims that every pure state on ∞ (N) has a unique extension to a pure state on 2 (N). So this conjecture is now a theorem and the best way to find out about it is to continue reading.

Randomness? What Randomness?

“…This discussion is closely related to the so-called Kadison-Singer conjecture in operator algebras, which however is nontrivial only for non-normal states. SeeStevens (2016) and Landsman (2017), §2.6 and §4.3…”

Bohrification: From classical concepts to commutative algebras

“…The second part of the theorem is the path-breaking solution of the Kadison-Singer conjecture due to Marcus, Spielman, & Srivastava (2014a,b), with important earlier contributions by Weaver (2004); see also Tao (2013) and Stevens (2015) for lucid expositions of the proof. Though less well known, the third part, due to Kadison & Singer themselves, whose arguments were later simplified by Anderson (1979) and Stevens (2015), is as remarkable than the second; it shows that Dirac's notation |λ may be ambiguous, or, equivalently, that maximal commutative C*-subalgebras of B(H) that are unitarily equivalent to L ∞ (0, 1) (like the one generated by the position operator or the momentum operator) do not suffice to characterize pure states. What to make of this is unclear.…”

“…This classification was stated without proof in Kadison & Singer (1959); the details appeared later in Kadison & Ringrose (1986, §9.4), based on von Neumann (1931), who initiated the study of commutative von Neumann algebras. See also Stevens (2015).…”

Finite Parseval frames in Walsh analysis

“…В 1959 г. эта проблема была сформулирована следующим образом (см. [27]): «Does every pure state on the (abelian) von Neumann algebra D of bounded diagonal operators on the Hilbert space 2 have a unique extension to a state on B( 2 ), the von Neumann algebra of all bounded linear operators on 2 ?» Проблема Кадисона-Зингера решена в 2013 г. и эквивалентна ряду других известных задач: о базисах гильбертова пространства, о замощении бесконечных матриц, о разбиениях фреймов, об обратимости конечных матриц с доминирующей диагональю, о тригонометрических суммах на канторовых множествах, о комбинаторных свойствах систем векторов в R d (см., например, [19,33]). Основные результаты о конечномерных интерпретациях проблемы Кадисона-Зингера анализируются в [18] (см.…”