Profiling the jazz singer

The Kadison-Singer Conjecture

We are given a large integer N , and a fixed basis {ei : 1 ≤ i ≤ N} for the space H = C . We are also given an N -by-N matrix H that has all its diagonal entries zero, and has norm one. If σ ⊆ {1, 2, . . . , N}, let Pσ be the projection onto the space Hσ := ∨{ei : i ∈ σ}. Call such a projection basic, and the range of a basic projection a basic subspace. For each constant 0 < γ < 1, we want to ...

Wolf Singer

On Jazz

The question of what is meant by "jazz" seems to mock the clear-cut definitive answer. Just as the historical origins of the form arc disappearing into the fog of the recent past, so its range is disappearing within its ambivalent use at the present moment. For the purpose of providing a crude orientation, one could concede that it is that type of dance music--whether it be used in an unmediate...

The Bebop Revolution in Jazz

The bebop style of jazz is a pivotal invention in twentieth-century American popular music an outgrowth of the rhythmic and harmonic experiments of young African-American jazz musicians. At first a source of controversy because of its unorthodox approach, bebop eventually gained widespread acceptance as the foundation of "modern" jazz, and it continues to influence jazz musicians. The process o...

The Other Kadison–singer Problem

Let H be `2 (over C) and let B(H) denote the C*-algebra of all bounded operators on H. Fix an orthonormal basis (en) for H. The atomic masa corresponding to this basis is `∞; equivalently, the algebra of all operators that are diagonalized by the basis (en). The projections in `∞ are exactly the projections onto subspaces spanned by a subset of {en}. That is, P(`∞) ∼= P(N) (here P(A) denotes th...