The Kadison-Singer problem in mathematics and engineering: a detailed account

Abstract

We will show that the famous, intractible 1959 Kadison-Singer problem in C * -algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer.

The Kadison–Singer Problem in mathematics and engineering

“…This theorem gave rise to a problem in the area that has received a great deal of attention (3,14). It had been ''folklore'' for years that KS and BT must be equivalent.…”

“…Sometimes BT is called ''strong BT,'' since there is a weakening of it called ''weak BT.'' In weak BT we allow A to depend on the norm of the operator T. A significant amount of effort has been invested in trying to show that strong and weak BT are equivalent (3,6,11 …”

“…. , n} so that for all j ϭ 1, 2 In his work on time-frequency analysis, Feichtinger (3,11) noted that all of the Gabor frames he was using (see Section 7) had the property that they could be divided into a finite number of subsets which were Riesz basic sequences. This observation led to the conjecture:…”

“…For many more equivalences of KS and a much more detailed discussion, see ref. 3. To reduce the redundancy of statements of theorems, we adopt the notation: Problem A (or Conjecture A) implies Problem B (or Conjecture B) means that a positive solution to the former implies a positive solution to the latter.…”

“…We will say that an arbitrary operator T satisfies PC if T Ϫ D(T) satisfies PC where D(T) is the diagonal of T. It is known that the class of operators satisfying PC (the ''pavable operators'') is a closed subspace of B(ᐉ 2 ). Also, to verify PC we only need to verify it for any one the following classes of operators (3,5). (i) unitary operators, (ii) positive operators, (iii) orthogonal projections, (iv) Gram operators of the form T*T ϭ (͗f i , f j ͘) i,jʦI where ʈf i ʈ ϭ 1 and Te i ϭ f i is a bounded operator.…”