Minimal generating reflexive lattices of projections in finite von Neumann algebras

Abstract

We show that the reflexive lattice generated by a double triangle lattice of projections in a finite von Neumann algebra is topologically homeomorphic to the two-dimensional sphere S 2 (plus two distinct points corresponding to zero and I ).

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

“…Our account of the results related to the Kadison-Singer algebras associated with a double triangle lattice in a finite factor is drawn directly from recent literature (see [7,9,17]). We first introduce some standard notation.…”

Derivations of a class of Kadison–Singer algebras

“…By Lemma 2.1 in [8], for each projection P in A with P ∨P 1 = I and P ∧P 1 = 0, there exist a positive contractive operator H in M and a unitary V in M such that I − H is injective and…”

“…Hence the introduction of KS-algebras brings connections between selfadjoint and non-selfadjoint theories, so many techniques and tools in von Neumann algebras can be used to study these non-selfadjoint algebras. In [8], we consider the realizations of the double triangle in abstract lattice theory as a lattice L of orthogonal projections acting on a Hilbert space K, i.e., L = {0, P 1 , P 2 , P 3 , I} with P i ∧ P j = 0 and P i ∨ P j = I for different i and j, where P i 's are orthogonal projections on K and I is the identity operator on K. We proved that, if L generates a finite von Neumann algebra, then it is not reflexive and the reflexive lattice it generates is topologically homeomorphic to the two-dimensional sphere S 2 (plus two distinct points corresponding to zero and I). In general, the reflexive lattice is a KS-lattice, and the corresponding reflexive algebra is a KS-algebra.…”

“…Assume that the von Neumann algebra A generated by L is finite, i.e., A has a faithful normal tracial state τ . Then, by Proposition 2.4 in [8], P 1 is equivalent to I − P 1 in A and A is *-isomorphic to M 2 (P 1 AP 1 ), the matrix algebra of all 2 × 2 matrices with entries in P 1 AP 1 , which acts on P 1 (K) ⊕ P 1 (K). In fact, in this case, the *-isomorphism is spatial.…”

“…As a continuation of our study on the Hochshcild cohomology group and automorphisms for KS-algebras in [7,2], the purpose of this paper is to investigate the automatic continuity and (quasi-)spatiality of derivations of KS-algebras generated by double triangle lattices of projections in finite von Neumann algebras, defined in [8]. We first use the techniques and theory of unbounded operators affiliated with finite von Neumann algebras to characterize the structure of the class of KS-algebras, and relate them with transitive algebras (on different Hilbert spaces).…”