Some new classes of Kadison-Singer lattices in Hilbert spaces. (English) Zbl 1331.47089
Let \(\mathcal {B}(\mathcal {H})\) be the algebra of all bounded linear operators on Hilbert space \(\mathcal H\). Recall that a lattice \(\mathcal {L}\) of projections in \(\mathcal {B}(\mathcal {H})\) is called a Kadison-Singer lattice if \(\mathcal {L}\) is a minimal reflexive lattice that generates the von Neumann algebra \(\mathcal {L}^{\prime \prime }\) or, equivalently, \(\mathcal {L}\) is reflexive and Alg(\(\mathcal {L}\)) is a Kadison-Singer algebra. In the paper under review, the authors construct some Kadison-Singer lattices in separable Hilbert spaces as well as complex matrix algebras \(M_{n}(\mathbb {C})\).
Reviewer: Wu Jing (Fayetteville)
MSC:
| 47L75 | Other nonselfadjoint operator algebras |
| 51D25 | Lattices of subspaces and geometric closure systems |
| 15A30 | Algebraic systems of matrices |
| 47L30 | Abstract operator algebras on Hilbert spaces |
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