Kadison–Singer algebras, II: General case

Abstract

A new class of operator algebras, Kadison-Singer (KS-) algebras, is introduced. These highly noncommutative, non self-adjoint 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. It is shown that these lattices and their reduced forms are often homeomorphic to classical manifolds such as the sphere.Kadison-Singer lattice | reflexive algebra | triangular algebra | von Neumann algebra K S-algebras were previously introduced (1). Examples of KSalgebras with hyperfinite diagonals were given and studied. 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).Suppose H is a Hilbert space and BðHÞ is the algebra of all bounded operators on H. Recall that a KS-algebra is a maximal reflexive subalgebra of BðHÞ with respect to a given von Neumann algebra as its diagonal algebra. Previously (1), we constructed KS-algebras with hyperfinite factors as their diagonal algebras and provided many previously undescribed reflexive lattices. Our main result in this paper is to prove that the reflexive lattice generated by a double triangle (a special lattice with only three nontrivial projections) is, in general, homeomorphic to the two-dimensional sphere S 2 (plus two distinct points corresponding to zero and I), and the corresponding reflexive algebra is a KS-algebra. In particular, we show that the algebra that leaves three free projections invariant is a KS-algebra. This shows that many factors are (minimally) generated by a reflexive lattice of projections that is topologically homeomorphic to S 2 . Noncommutative algebraic structures on S 2 , determined by the projections, give rise to non-isomorphic factors and KS-factors.The paper contains five sections. In section two, maximal triangularity is discussed in different aspects. In section three, we describe the reflexive lattice generated by three free projections and show that it is homeomorphic to S 2 . In section four, we show that this lattice is a KS-lattice and thus the corresponding algebra is a KS-algebra. Certain generalizations of the result is also discussed. In section five, we introduce a notion of connectedness of projections in a lattice of projections in a finite von Neumann algebra and show that all connected components form another lattice, called a "reduced lattice." Reduced lattices of most of our examples were computed. Maximality conditionsIn the definition of KS-algebras, we require that the algebra be maximal in the class of reflexive algebras with the same diagonal. 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...

Minimal generating reflexive lattices of projections in finite von Neumann algebras

“…Let W = A * M 2 (C) be the reduced (von Neumann algebra) free product of A with M 2 (C). Since there exists a trace half projection which is free with Q 1 , it follows that Q 1 is equivalent to I − Q 1 in W [6,18]. Therefore, we may choose a system {E i j } 2 i, j=1 of matrix units in W such that E 11 = Q 1 .…”

“…In [5,6], Ge and Yuan constructed KS-algebras with hyperfinite factors as their diagonals. They also proved that the reflexive lattice determined by three free projections with trace 1 2 is homeomorphic to the two-dimensional sphere S 2 (plus two distinct points corresponding to zero and I ).…”

“…It is easy to see that the lattice generated by three free projections with trace 1 2 is a double triangle lattice, which means that it is a five-element lattice containing three nontrivial elements such that the intersection of any two is zero and the union of any two is I . In [6], the authors claimed that if a double triangle lattice of projections in a type I I 1 factor satisfies some specified conditions, then the reflexive lattice generated by it is also homeomorphic to S 2 (plus zero and I ). Hence, for these cases, Halmos's problem, which asks whether any realization of a double triangle lattice is reflexive, was settled.…”

“…Our methods are different from those in [6] where heavy free probability techniques were involved. In the section following the introduction, by using the theory of unbounded operators affiliated with a von Neumann algebra, we show that the reflexive lattice generated by any double triangle lattice of projections in a finite von Neumann algebra is topologically homeomorphic to S 2 (plus zero and I ).…”

On maximal non-selfadjoint reflexive algebras associated with a double triangle lattice

“…See [1,8,13,15,16]. Recently, a new class of operator algebras, Kadison-Singer algebras, is introduced by Ge and Yuan [6,7]. KadisonSinger (or KS-, for short) algebras combine non-selfadjoint algebras with self-adjoint ones, especially von Neumann algebras, into one consideration.…”