Cohomology groups of a new class of Kadison-Singer algebras

Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space ℋ , ξ = ∑ n = 1 ∞ e n 2 n be a separating vector for the commutant N ′ , E ξ , be the projection from ℋ onto the subspace [ ℂ ξ ] spanned by the vector ξ, and Q be the projection from K = ℋ ⊕ ℋ ⊕ ℋ onto the closed subspace { ( η , η , η ) T : η ∈ H } . Suppose that ℒ is the projection lattice generated by the projections ( E ξ 0 0 0 0 0 0 0 0 ) , { ( E 0 0 0 0 0 0 0 0 ) : E ∈ N } , ( I 0 0 0 I 0 0 0 0 ) a n d Q . We show that ℒ is a Kadison-Singer lattice with the trivial commutant. Moreover, we prove that every n -th bounded cohomology group H n ( Alg ℒ , B ( K ) ) with coefficients in B ( K ) is trivial for n ⩾ 1.