Kadison–Singer algebras: Hyperfinite case

Abstract

A new class of operator algebras, Kadison-Singer algebras (KSalgebras), is introduced. These highly noncommutative, non-selfadjoint algebras generalize triangular matrix algebras. They are determined by certain minimally generating lattices of projections in the von Neumann algebras corresponding to the commutant of the diagonals of the KS-algebras. A new invariant for the lattices is introduced to classify these algebras.Kadison-Singer lattice | reflexive algebra | triangular algebra | von Neumann algebra I n ref. 1, Kadison and Singer initiate the study of non-selfadjoint algebras of bounded operators on Hilbert spaces. They introduce a class of algebras they call "triangular operator algebras." An algebra T is triangular (relative to a factor M) when T ∩ T Ã is a maximal abelian (self-adjoint) algebra in the factor M (see Definitions). When the factor is the algebra of all n × n complex matrices, this condition guarantees that there is a unitary matrix U such that the mapping A → UAU Ã transforms T onto a subalgebra of the upper triangular matrices.Beginning with ref. 1, the theory of non-self-adjoint operator algebras has undergone a vigorous development parallel to, but not nearly as explosive as, that of the self-adjoint theory, the C*-algebra and von Neumann algebra theories. Of course, the selfadjoint theory began with the 1929-1930 von Neumann article (2), well before the 1960 (1) article appeared. Surprisingly, to the present authors, and apparently to Kadison and Singer as well (from private conversations), this parallel development has not produced the synergistic interactions we would have expected from subjects that are so closely and naturally related, and thus likely to benefit from cross-connections with one another.Considerable effort has gone into the study of triangular-operator algebras (see refs. 3 and 4), and another class of non-self-adjoint operator algebras, the "reflexive algebras" (see refs. 5-8). Many definitive and interesting results are obtained during the course of these investigations. For the most part, these more detailed results rely on relations to compact, or even finite-rank, operators. This direction is taken in the seminal article (1), as well. In Section 3.2 of ref 1, a detailed and complete classification is given for an important class of (maximal) triangular algebras; but much depends on the analysis of those T for which (the "diagonal") T ∩ T Ã is generated by one-dimensional projections. On the other hand, the emphasis of C*-algebra and von Neumann algebra theory is on those algebras where compact operators are (almost) absent.One of our main goals in this article is to recapture the synergy that should exist between the powerful techniques that have developed in self-adjoint operator-algebra theory and those of the non-self-adjoint theory by conjoining the two theories. We do this by embodying those theories in a single class of algebras. For this, we mimic the defining relation for the triangular algebra, removing the commutativity assumption on the diagonal suba...

Kadison–Singer algebras, II: General case

“…The following result follows easily from the above lemma and shows that all nontrivial projections in LatðAlgðF 3 ÞÞ have trace 1 2 .…”

“…Our examples of KS-algebras given previously (1) are "maximal triangular" in the class of all algebras with the same diagonal, that is, an algebraic maximality without reflexiveness or closedness assumptions. In general, the algebraic maximality assumption is a much stronger requirement.…”

“…Proof: We may assume that Q i ∈ LatðAlgðF 3 ÞÞ \ f0; I; P 1 g and Q i is given [1] in the proof of lemma four, for i ¼ 1; 2; 3 and a 1 , a 2 , and a 3 are distinct scalars.…”

“…In this article, we shall continue our study of KS-algebras and mostly deal with the case when the diagonal is a finite von Neumann algebra. We shall use the notation and definitions previously introduced (1).…”

Minimal generating reflexive lattices of projections in finite von Neumann algebras

“…Recently, motivated by the work of Kadison and Singer on maximal triangular algebras, Ge and Yuan [5] introduced a new class of non selfadjoint algebras which they called Kadison-Singer algebras, or KS-algebras for simplicity. KS-algebras combine triangularity [11], reflexivity [3,7] and von Neumann algebra properties into one package.…”