Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑^(∞)_(n=1)en/2n be a separating vector for the commutant N',E_(ξ)be the projection from H onto the subspace[Cξ]spanned ...Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑^(∞)_(n=1)en/2n be a separating vector for the commutant N',E_(ξ)be the projection from H onto the subspace[Cξ]spanned by the vectorξ,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) and 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(AlgL,B(K))with coefficients in B(K)is trivial for n≥1.展开更多
Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspac...Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).展开更多
Let L be the complete lattice generated by a nest N on an infinite-dimensional separable Hilbert space H and a rank one projection P ξ given by a vector ξ in H. Assume that ξ is a separating vector for N , the core...Let L be the complete lattice generated by a nest N on an infinite-dimensional separable Hilbert space H and a rank one projection P ξ given by a vector ξ in H. Assume that ξ is a separating vector for N , the core of the nest algebra Alg(N ). We show that L is a Kadison-Singer lattice, and hence the corresponding algebra Alg(L) is a Kadison-Singer algebra. We also describe the center of Alg(L) and its commutator modulo itself, and show that every bounded derivation from Alg(L) into itself is inner, and all n-th bounded cohomology groups H n (Alg(L), B(H)) of Alg(L) with coefficients in B(H) are trivial for all n≥1.展开更多
This article is part exposition of a recent rather technical paper of the last two authors on matrix pavings related to the 1959 Kadison-Singer Extension Problem and part a report on further computational results prov...This article is part exposition of a recent rather technical paper of the last two authors on matrix pavings related to the 1959 Kadison-Singer Extension Problem and part a report on further computational results providing new bounds on the paving parameters for classes of small matrices investigated there and subsequently. A website maintained by the authors provides to all interested the matrices experimentally discovered that yield these bounds along with the proprietary MATLAB software with simple operational directions to load them, pave them, and perform paving searches. The convergence to 1 or not of the infinite sequences of these paving parameters in most cases is equivalent to the Kadison-Singer Extension Problem, and in all cases convergence to 1 negates the problem. The last two sections describe the search process and an interpretation of the data integrated with the results of the precursor to this paper.展开更多
The Kadison-Singer problem has variants in different branches of the sciences and one of these variants was proved in 2013. Based on the idea of “sparsification” and with its origins in quantum physics, at the sixti...The Kadison-Singer problem has variants in different branches of the sciences and one of these variants was proved in 2013. Based on the idea of “sparsification” and with its origins in quantum physics, at the sixtieth anniversary of the problem, we revisit the problem in its original formulation and also explore its transition to a result with wide ranging applications. We also describe how the notion of “sparsification” transcended various fields and how this notion led to resolution of the problem.展开更多
We show that many Kadison-Singer algebras are maximal triangular in all algebras containing them although their definition requires the maximality taken in the class of reflexive algebras. Diagonal-trivial maximal non...We show that many Kadison-Singer algebras are maximal triangular in all algebras containing them although their definition requires the maximality taken in the class of reflexive algebras. Diagonal-trivial maximal non self-adjoint subalgebras of matrix algebras with lower dimensions are classified.展开更多
We show that the reflexive algebra Alg(L) given by a double triangle lattice L in a finite factor M(with L" = M) is maximal non-selfadjoint in the class of all weak operator closed subalgebras with the same diago...We show that the reflexive algebra Alg(L) given by a double triangle lattice L in a finite factor M(with L" = M) is maximal non-selfadjoint in the class of all weak operator closed subalgebras with the same diagonal subalgebra Alg(L) ∩ Alg(L)^+.Our method can be used to prove similar results in finite-dimensional matrix algebras.As a consequence,we give a new proof to the main theorem by Hou and Zhang(2012).展开更多
基金supported by National Natural Science Foundation of China(Grant No.11801342)Natural Science Foundation of Shaanxi Province(Grant No.2023-JC-YB-043)Shaanxi College Students Innovation and Entrepreneurship Training Program(Grant No.S202110708069)。
文摘Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑^(∞)_(n=1)en/2n be a separating vector for the commutant N',E_(ξ)be the projection from H onto the subspace[Cξ]spanned by the vectorξ,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) and 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(AlgL,B(K))with coefficients in B(K)is trivial for n≥1.
基金supported by National Natural Science Foundation of China(Grant No.11271390)Natural Science Foundation Project of ChongQing,Chongqing Science Technology Commission(Grant No.2010BB9318)
文摘Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).
基金supported by National Natural Science Foundation of China(Grant No. A0324614, 10971117)the Natural Science Foundation of Shandong Province (Grant No. Y2006A03,ZR2009AQ005)
文摘Let L be the complete lattice generated by a nest N on an infinite-dimensional separable Hilbert space H and a rank one projection P ξ given by a vector ξ in H. Assume that ξ is a separating vector for N , the core of the nest algebra Alg(N ). We show that L is a Kadison-Singer lattice, and hence the corresponding algebra Alg(L) is a Kadison-Singer algebra. We also describe the center of Alg(L) and its commutator modulo itself, and show that every bounded derivation from Alg(L) into itself is inner, and all n-th bounded cohomology groups H n (Alg(L), B(H)) of Alg(L) with coefficients in B(H) are trivial for all n≥1.
基金supported by Naval Academy Research Council seed grants
文摘This article is part exposition of a recent rather technical paper of the last two authors on matrix pavings related to the 1959 Kadison-Singer Extension Problem and part a report on further computational results providing new bounds on the paving parameters for classes of small matrices investigated there and subsequently. A website maintained by the authors provides to all interested the matrices experimentally discovered that yield these bounds along with the proprietary MATLAB software with simple operational directions to load them, pave them, and perform paving searches. The convergence to 1 or not of the infinite sequences of these paving parameters in most cases is equivalent to the Kadison-Singer Extension Problem, and in all cases convergence to 1 negates the problem. The last two sections describe the search process and an interpretation of the data integrated with the results of the precursor to this paper.
文摘The Kadison-Singer problem has variants in different branches of the sciences and one of these variants was proved in 2013. Based on the idea of “sparsification” and with its origins in quantum physics, at the sixtieth anniversary of the problem, we revisit the problem in its original formulation and also explore its transition to a result with wide ranging applications. We also describe how the notion of “sparsification” transcended various fields and how this notion led to resolution of the problem.
基金Supported by National Natural Science Foundation of China(Grant No.11371290)
文摘We show that many Kadison-Singer algebras are maximal triangular in all algebras containing them although their definition requires the maximality taken in the class of reflexive algebras. Diagonal-trivial maximal non self-adjoint subalgebras of matrix algebras with lower dimensions are classified.
基金supported by National Natural Science Foundation of China(Grant No.11371290)
文摘We show that the reflexive algebra Alg(L) given by a double triangle lattice L in a finite factor M(with L" = M) is maximal non-selfadjoint in the class of all weak operator closed subalgebras with the same diagonal subalgebra Alg(L) ∩ Alg(L)^+.Our method can be used to prove similar results in finite-dimensional matrix algebras.As a consequence,we give a new proof to the main theorem by Hou and Zhang(2012).
