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    于嘉琪, 李建奎. 一些Kadison-Singer代数的例子[J]. 华东理工大学学报(自然科学版), 2018, 44(6): 945-949. DOI: 10.14135/j.cnki.1006-3080.20171227003
    引用本文: 于嘉琪, 李建奎. 一些Kadison-Singer代数的例子[J]. 华东理工大学学报(自然科学版), 2018, 44(6): 945-949. DOI: 10.14135/j.cnki.1006-3080.20171227003
    YU Jia-qi, LI Jian-kui. Some Examples of Kadison-Singer Algebras[J]. Journal of East China University of Science and Technology, 2018, 44(6): 945-949. DOI: 10.14135/j.cnki.1006-3080.20171227003
    Citation: YU Jia-qi, LI Jian-kui. Some Examples of Kadison-Singer Algebras[J]. Journal of East China University of Science and Technology, 2018, 44(6): 945-949. DOI: 10.14135/j.cnki.1006-3080.20171227003

    一些Kadison-Singer代数的例子

    Some Examples of Kadison-Singer Algebras

    • 摘要:H是一个复的Hilbert空间。L=(0),L,M,K,H是一个五角子空间格,满足MLx0的闭线性扩张,其中非零元x0属于K但不属于K+L。本文证明了Alg L是一个半单的Kadison-Singer代数。同时还给出两个不是套代数的CSL代数的例子,一个相似于某个Kadison-Singer代数,另一个却与任何一个Kadison-Singer代数都不相似。

       

      Abstract: Kadison-Singer algebras are a new class of non-self-adjoint reflexive operator algebras on a Hilbert space, which make connections between theories of self-adjoint algebras and non-self-adjoint algebras. Kadison-Singer algebras are closely associated with von Neumann algebras, and they are full of topological structures. Such class of operator algebras provide a new way to develop the theories of self-adjoint algebras. Pentagon subspace lattices and double triangle lattices are playing important roles in theories of lattices, and most of the focus was only put on double triangle lattices before. So we begin the study in pentagon subspace lattices to show more examples of Kadison-Singer algebras. Since any pentagon subspace lattice with gap-dimension 1 is reflexive, we construct a pentagon subspace lattice with gap-dimension 1 to ensure the reflexivity of the algebra. Let H be a complex Hilbert space. Suppose L=(0), L, M, K, H is a pentagon subspace lattice satisfying that M is the closed linear span of L and x0, where x0 is a nonzero vector in K but not in K+L. We prove AlgL is a semi-simple Kadison-Singer algebra. Besides, we know that nest algebras are Kadison-Singer algebras and nest algebras are invariant under similar transformation. However, any CSL algebra, not nest algebra, is not a Kadison-Singer algebra, and an algebra, which is similar to a Kadison-Singer, is not necessarily a Kadison-Singer algebra. So there are two natural questions:whether there exist a CSL algebra, not nest algebra, which can be similar to a Kadison-Singer algebra, and whether a CSL algebra exist but is not similar to any Kadison-Singer algebra.

       

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