Unbounded derivations and invariant states

Let be a von Neumann algebra with a cyclic and separating vector . Let =i[H, ·] be the spatial derivation implemented by a selfadjoint operatorH, such thatH=0. Let be the modular operator associated with the pair (, ). We prove the equivalence of the following three conditions:1)H is essential selfadjoint onD(), andH commutes strongly with .2) The restriction ofH toD() is essential selfadjoint onD(^1/2) equipped with the inner product(|)_#=(|)+(^1/2|^1/2), , D(^1/2).3) exp (itH) exp (–itH)= for anyt.We show by an example, that the first part of 1),H is essential selfadjoint onD(), does not imply 3). This disproves a conjecture due to Bratteli and Robinson [3].Part of this work was done while O.B. was a member of Zentrum für interdisziplinäre Forschung der Universität Bielefeld     

Unbounded derivations of C*-algebras

We study unbounded derivations ofC*-algebras and characterize those which generate one-parameter groups of automorphisms. We also develop a functional calculus for the domains of closed derivations and develop criteria for closeability. Some specialC*-algebras are considered \(\mathfrak{B}\mathbb{C}(\mathfrak{H}),\mathfrak{B}(\mathfrak{H})\) , UHF algebras, and in this last context we prove the existence of non-closeable derivations.     

Unbounded derivations and invariant trace states

Let \(\mathfrak{M}\) be a von Neumann algebra with cyclic trace vector ?. Let δ(A)=i[H, A] be a spatial derivation of \(\mathfrak{M}\) implemented by an operatorH such thatH?=0 andH is essentially self-adjoint onD(δ)?. It follows that: $$e^{itH} \mathfrak{M}e^{ - itH} = \mathfrak{M},t \in \mathbb{R}.$$      

Faithful compact quantum group actions on connected compact metrizable spaces

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.     

Unbounded derivations of commutativeC*-algebras

It is shown that an unbounded *-derivation of a unital commutativeC*-algebraA is quasi well-behaved if and only if there is a dense open subsetU of the spectrum ofA such that, for anyf in the domain of , (f) vanishes at any point ofU wheref attains its norm. An example is given to show that even if is closed it need not be quasi well-behaved. This answers negatively a question posed by Sakai for arbitraryC*-algebras.It is also shown that there are no-zero closed derivations onA if the spectrum ofA contains a dense open totally disconnected subset.     

Unbounded derivations ofC*-algebras II

It is demonstrated that a closed symmetric derivation δ of aC?-algebra \(\mathfrak{A}\) generates a strongly continuous one-parameter group of automorphisms of aC?-algebra \(\mathfrak{A}\) if and only if, it satisfies one of the following three conditions

1. (αδ+1)(D(δ))= \(\mathfrak{A}\) , α∈?\{0}.
2. δ possesses a dense set of analytic elements.
3. δ possesses a dense set of geometric elements.

Together with one of the following two conditions

1. ∥(αδ+1)(A)∥≧∥A∥, α∈IR,A∈D(δ).
2. If α∈IR andA∈D(δ) then (αδ+1)(A)≧0 impliesA≧0.

Other characterizations are given in terms of invariant states and the invariance ofD(δ) under the square root operation of positive elements.     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

Spectral subspaces for compact actions

We study spectral subspaces under the action of compact groups on Banach spaces, with particular reference to operator algebras.     

The inhomogeneous quantum invariance group of commuting fermions

We consider a model of d fermions where creation and annihilation operators of different fermions commute. We show that this particle algebra is invariant under an inhomogeneous quantum group.      

自均衡相位特性的超导滤波器小型化设计

提出了一种具有多个传输零点的高温超导滤波器小型化设计方法。该滤波器由十个慢波交叉指结构谐振器和三个自均衡CQ跨线结构组成,其中一个CQ结构通过产生一对实轴上的传输零点实现相位均衡,另外两个CQ结构通过产生两对虚轴上的传输零点实现滤波器的高频率选择性。滤波器中心频率为1,720MHz,带宽为10MHz,通带外陡峭度为40d B/MHz,通带内回波损耗为23d B,且在60%的通带内具有平坦的群时延响应。在Mg O衬底上采用DBCO高温超导薄膜制作十阶超导滤波器,并在77K温度下进行测试,最终得到了良好的带外抑制性能和平坦的群时延特性。测试结果与仿真结果基本一致,说明所提出的设计方法是有效的。     

Hamiltonian group actions and integrable systems

The purpose of this paper is to show how some ideas from the theory of Hamiltonian systems with symmetry can be applied to dynamical systems like the nonperiodic Toda Lattice of Moser to yield striking information about their trajectories.Our presentation of these ideas is as self-contained as possible. We try to keep abstract machinery in its proper place, and emphasize the concrete computations which are implicit in some parts of modern symplectic geometry. Most of the results we shall derive are known in one form or another. The only novelties, perhaps, are some proofs and examples.     

Unbounded functionals on a group algebra and applications to quantum theory

A certain class of positive functionals on a group algebra is examined that is pertinent to the induced representations of Frobenius and Mackey. Though these functionals are not bounded in theL ¹ norm, continuity still persists to an extent that secures the existence of a continuous group representation obtained from Gelfand's construction. The theory thus developed provides a new aspect of both the improper states in quantum theory and the induced representations of groups. The method is applied to the Poincaré group and it is shown that the representations, in which particles can be accommodated, are determined up to unitary equivalence by unbounded functionals of a simple structure. It is stressed that representations describing an infinitely degenerate vacuum emerge from mass nonzero representations as the mass tends to zero.     

Faithful actions of locally compact quantum groups on classical spaces

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.     

Mappin class group actions on quantum doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS ^±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofU _q(sl₂) forq anl=2m+1^st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofU _q(sl₂).     

Superconformal algebras and transitive group actions on quadrics

A classification of “physical” superconformal algebras is given. The list consists of seven algebras: the Virasoro algebra, the Neveu-Schwarz algebra, theN = 2,3 and 4 algebras, the superalgebra of all vector fields on theN = 2 supercircle, and a new algebraCK ₆ constructed in [3]. The proof relies heavily on the classification of all connected subgroups ofSO _n(C) which act transitively on the quadric (v, v) = 1. To Ernest Borisovich Vinberg on his 60_th birthday Partially supported by NSF grant DMS-9622870     

Landau Hamiltonians with Unbounded Random Potentials

We prove the almost sure existence of a pure point spectrum for the two-dimensional Landau Hamiltonian with an unbounded Anderson-like random potential, provided that the magnetic field is sufficiently large. For these models, the probability distribution of the coupling constant is assumed to be absolutely continuous. The corresponding densityg has support equal to , and satisfies , for some > 0. This includes the case of Gaussian distributions. We show that the almost sure spectrum is , provided the magnetic field B0. We prove that for each positive integer n, there exists a field strength B _n , such that for all B>B _n , the almost sure spectrum is pure point at all energies except in intervals of width about each lower Landau level , for m < n. We also prove that for any B0, the integrated density of states is Lipschitz continuous away from the Landau energiesE _n (B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.