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
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.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11801342), Natural Science Foundation of Shaanxi Province (Grant No. 2023-JC-YB-043) and Shaanxi College Students Innovation and Entrepreneurship Training Program (Grant No. S202110708069).
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An, G., Cheng, X. & Sheng, J. Cohomology groups of a new class of Kadison-Singer algebras. Sci. China Math. 67, 593–606 (2024). https://doi.org/10.1007/s11425-022-2107-y
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DOI: https://doi.org/10.1007/s11425-022-2107-y