Cohomology groups of a new class of Kadison-Singer algebras

Abstract

Let ${\cal N}$ be a maximal discrete nest on an infinite-dimensional separable Hilbert space ${\cal H},\xi = \sum\nolimits_{n = 1}^\infty {{{{e_n}} \over {{2^n}}}}$ be a separating vector for the commutant ${{\cal N}^\prime},\,\,{E_\xi}$, be the projection from ${\cal H}$ onto the subspace $[\mathbb{C}\xi]$ spanned by the vector ξ, and Q be the projection from ${\cal K} = {\cal H} \oplus {\cal H} \oplus {\cal H}$ onto the closed subspace $\{{(\eta,\eta,\eta)^{\rm{T}}}:\eta \in {\cal H}\}$. Suppose that ${\cal L}$ is the projection lattice generated by the projections

$$\left({\matrix{{{E_\xi}} \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill \cr}} \right),\,\,\,\,\,\left\{{\left({\matrix{E \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill \cr}} \right):E \in {\cal N}} \right\},\,\,\,\,\,\left({\matrix{I \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & I \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill \cr}} \right)\,\,\,\,\,{\rm{and}}\,\,\,\,\,Q.$$

We show that ${\cal L}$ is a Kadison-Singer lattice with the trivial commutant. Moreover, we prove that every n-th bounded cohomology group ${H^n}({\rm{Alg}}{\cal L},\,B({\cal K}))$ with coefficients in $B({\cal K})$ is trivial for n ⩾ 1.