Local automorphisms of the unitary group and the general linear group on a Hilbert space

2-Local derivations and automorphisms on $B(H)$

The paper is devoted to 2-local derivations and 2-local automorphisms on the algebra $B(H)$ of all bounded linear operators on a Hilbert space $H.$ We prove that every 2-local derivation on $B(H)$ is a derivation. A similar result is obtained for automorphisms.

Structure of the group of automorphisms of C$^{*}$-algebras

We obtain a kind of structure theorem for the automorphism group ${\rm Aut}{\cal A}$ of a unital C$^{*}$-algebra ${\cal A}$. According to it, ${\rm Aut}{\cal A}$ can be regarded as a subgroup of the semi-direct product of direct product group consisting of some family of projective unitary groups and some permutation group on the spectrum of ${\cal A}$.

Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

An \textit{ideal} of $N$-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for $N$-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $CDD_N$ of all unitary equivalence classes of such $N$-tuples are established and certain ideals in $CDD_N$ are distinguished. It is proved that infinite…

Automorphisms of Cuntz-Krieger algebras

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_Λ (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results o…

Local automorphisms of operator algebras on Banach spaces

In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras of which statement we present a short proof. The need to get such a proof was formulated in Semrl's paper.

几乎自同构群的酉 $L^{p+}$-表示

Unitary $L^{p+}$-representations of almost automorphism groups

本文研究具有 Kunze-Stein 开子群的局部紧群，证明诱导表示保持 $L^{p+}$ 性质，并据此构造树的几乎自同构群及 Neretin 群中不同 $p$ 阈值的酉表示。

Asymptotically unitary equivalence and asymptotically inner automorphisms

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple \CA with tracial rank zero. Suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(a)u_t=ϕ_2(a)\tforal a\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $τ\circ ϕ_1=τ\circ ϕ_2$ for all $τ\in T(A)$ and the rotation map ${\tildeη}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. In particular, an automorphism $\af$ on a unital separable simple \CA $A$ in ${\cal N}$ with tracial rank zero…

算子代数与波兰群的酉表示

Operator algebras and unitary representations of Polish groups

本文研究希尔伯特空间上算子代数与拓扑群酉表示之间的对应关系，证明范数可分 $C^*$ 代数的阿贝尔子代数研究可归结为波兰群酉表示研究，并应用于阿贝尔子代数的构造与分类。

Some Calkin algebras have outer automorphisms

We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, consistently, there are many outer automorphisms.

显式幺正的高阶希尔伯特空间

Manifestly unitary higher Hilbert spaces

该研究提出低维高阶希尔伯特空间的显式幺正构造，从Baez的2-希尔伯特空间构建出3-希尔伯特空间的C*-3-范畴，并证明其到3-向量空间的遗忘函子是满忠实的。

Hilbert模的Co-actions、等距与同构类

Co-actions, Isometries and isomorphism classes of Hilbert modules

本文证明了Hilbert A-模的A-线性映射由酉算子诱导的条件，并基于乘法酉算子理论计算了C*-代数量子群上Hilbert模的等价类，发展了可用于证明co-actions不存在性的理论。

A generalization of Wigner's unitary-antiunitary theorem to Hilbert modules

Let H be a Hilbert $C^*$-module over a matrix algebra A. It is proved that any function $T:H\to H$ which preserves the absolute value of the (generalized) inner product is of the form $Tf=ϕ(f)Uf$ $(f\in H)$, where $ϕ$ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner's classical unitary-antiunitary theorem for Hilbert modules.

What type of dynamics arise in E_0-dilations of commuting quantum Markov process?

Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $ϕ$ and $θ$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $α$ and $β$ such that $ϕ_s \circ θ_t (P_H A P_H) = P_H α_s \circ β_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $ϕ$ is not an automorphism semigroup then $α$ is cocycle conjugate to the minimal *-endomorphic dilation of $ϕ$, and that if $ϕ$ is an automorphism semigroup then $α$ is also an automorphism semigroup. In particular, we conclude that if $ϕ$ is not an automorphism semigroup a…

Multiplicity freeness of unitary representations in sections of holomorphic Hilbert bundles

We prove several results asserting that the action of a Banach-Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity free. These results require the existence of compatible antiholomorphic bundle maps and certain multiplicity freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,σ)$-weakly visible action (generalizing T.Koboyashi's visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for gro…

Structured Parseval Frames in Hilbert $C^*$-modules

We investigate the structured frames for Hilbert $C^{*}$-modules. In the case that the underlying $C^{*}$-algebra is a commutative $W^*$-algebra, we prove that the set of the Parseval frame generators for a unitary operator group can be parameterized by the set of all the unitary operators in the double commutant of the group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also obtain the existence and uniquenes…

Kaplansky-Hilbert 模上酉群表示的分解定理与 Furstenberg-Zimmer 结构定理

A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem

本文建立了 Stone 代数上 Kaplansky-Hilbert 模的酉群表示分解定理，并基于算子理论将其应用于推广著名的 Furstenberg-Zimmer 结构定理至任意群在任意概率空间上的保测作用情形。

All automorphisms of all Calkin algebras

The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce the notion of Polish $ω_1$-trees and cohrerent families of Polish spaces and prove some uniformization results.

Unitary Continuous Representations of Compact Quantum Groups

Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous correpresentations of $C^*$-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is proved that the unitary continuous correpresentations decompose in finite dimensional irreducible correpresentations.

On 2-local *-automorphisms and 2-local isometries of B(H)

It is an important result of \v Semrl which states that every 2-local automorphism of the full operator algebra over a separable Hilbert space is necessarily an automorphism. In this paper we strengthen that result quite substantially for *-automorphisms. Indeed, we show that one can compress the defining two equations of 2-local *-automorphisms into one single equation, hence weakening the requirement significantly, but still keeping essentially the conclusion that such maps are necessarily *-automorphisms.

Hilbert von Neumann modules

We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike [Skei], but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of [Skei], by verifying the `Riesz lemma' or what is called `self-duality' in [Skei]. An advantage with our approach is that we can totally side-step the need to go through $C^*$-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach. We establish the analogue of the …

Some automorphism invariance properties for multicontractions

In the theory of row contractions on a Hilbert space, as initiated by Popescu, two important objects are the Poisson kernel and the characteristic function. We determine their behaviour with respect to the action of the group of unitarily implemented automorphisms of the algebra generated by creation operators on the Fock space. The case of noncommutative varieties, introduced recently by Popescu, is also discussed.

Multiplicative maps on ideals of operators which are local automorphisms

We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra B(H), H being a separable Hilbert space. Let $ϕ:B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then $ϕ$ is an automorphism of the algebra B(H).

Unitary orbits in a full matrix algebra

The Hilbert manifold $Σ$ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits $Ω\subset Σ$ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set $Ω$, and we study the convexity of this set for the geodesic structures that arise when we give $Σ$ two Riemannian metrics.

Admissible vectors and Hilbert algebras

Admissible vectors for unitary representations of locally compact groups are the basis for group-frame and covariant coherent state expansions. Main tools in the study of admissible vectors have been Plancherel and central integral decomposition, of applicability only under certain separability and semifiniteness restrictions. In this work we present a study of admissible vectors in terms of convolution Hilbert algebras valid for arbitrary unitary representations of general locally compact groups.

酉群与矩阵代数约化积的迹范数刚性

Trace-norm rigidity for reduced products of unitary groups and matrix algebras

研究有限维酉群和矩阵代数的迹度量约化积间同态，证明酉群相关稳定性变体等，表明乘积形式同构由特定方式诱导，还得到坐标识别及自同构群的刚性与分类结果，矩阵代数在更一般背景下有类似刚性结果。

Characteristic functions for multicontractions and automorphisms of the unit ball

A \emph{multicontraction} on a Hilbert space $\HH$ is an $n$-tuple of operators $T=(T_1,...,T_n)$ acting on $\HH$, such that $\sum_{i=1}^n T_i T_i^*\le \1_\HH$. We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball.

酉TQFT、酉圆盘状$n$-范畴与更高希尔伯特空间

Unitary TQFTs, unitary disk-like $n$-categories, and higher Hilbert spaces

本文提出用完备有限酉圆盘状$n$-范畴定义有限$(n+1)$-希尔伯特空间，验证了$n=1,2$情形，并由此分类了低维酉TQFT。

Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms

In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.

A note on equicontinuity of families of operators and automorphisms

This note concerns uniform equicontinuity of families of operators on a separable Hilbert space H, and of families of maps on B(H). It is shown that a one parameter group of automorphisms is uniformly equicontinuous if and only if the group of unitaries which implements it is so. A "geometrical" necessary and sufficient condition is given for a family of operators to be uniformly equicontinuous.

Quantum automorphism groups of finite quantum groups are classical

In a recent paper of Bhowmick, Skalski and Sołtan the notion of a quantum group of automorphisms of a finite quantum group was introduced and, for a given finite quantum group G, existence of the universal quantum group acting on G by automorphisms was proved. We show that this universal quantum group is in fact a classical group. The key ingredient of the proof is the use of multiplicative unitary operators, and we include a thorough discussion of this notion in the context of finite quantum groups.

边界作用于索恩-汤普森群的外自同构群

Boundary actions of outer automorphism groups of Thompson-like groups

本文研究了索恩-汤普森群的外自同构群的边界作用，并证明这些群是C*-简单。

Local automorphisms of some quantum mechanical structures

Let H be a separable infinite dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented.

Order automorphisms of the set of bounded observables

Let H be a complex Hilbert space and denote by Bs(H) the set of all self-adjoint bounded linear operators on H. In this paper we describe the form of all bijective maps (no linearity or continuity is assumed) on Bs(H) which preserve the usual order in both directions.

Positive Voiculescu-Brown entropy in noncommutative toral automorphisms

We show that the Voiculescu-Brown entropy of a noncommutative toral automorphism arising from a matrix S in GL(d,Z) is at least half the value of the topological entropy of the corresponding classical toral automorphism. We also obtain some information concerning the positivity of local Voiculescu-Brown entropy with respect to single unitaries. In particular we show that if S has no roots of unity as eigenvalues then the local Voiculescu-Brown entropy with respect to every product of canonical unitaries is positive, and also that in the presence of completely positive CNT entropy the unital ve…

关于局部紧压缩群酉表示理论的研究

On the unitary representation theory of locally compact contraction groups

本文研究局部紧压缩群及其半直积的酉表示，提出刻画type I或CCR群的问题，分类无挠群的不可约表示，并给出正则树自同构群的新例子。

等变 Tannaka-Krein 重构与离散结构的量子自同构群

Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

本文通过推广 Tannaka-Krein 重构定理，为一大类离散结构构造量子自同构群，并给出从连通局部有限离散结构得到相应范畴的方法，以量子 Cayley 图为例加以说明。

Unitary equivalence of representations of graph algebras and branching systems

In this paper we show that, for a class of countable graphs, every representation of the associated graph algebra in a separable Hilbert space is unitarily equivalent to a representation obtained via branching systems.

连通局部有限图的量子自同构群与离散群的量子化

Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups

本文为连通局部有限图构造量子自同构群，引入酉张量范畴以定义Haar泛函，并证明图的量子同构等价于其量子自同构群的幺半范畴等价。

Coexistency on Hilbert space effect algebras and a characterisation of its symmetry transformations

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig's formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper's first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the…

关于建筑的自同构群的Howe-Moore性质

On the Howe--Moore property for automorphism groups of buildings

研究局部有限非球型建筑上的强传递自同构群，证明在足够厚度下，弱混合酉表示是C0的，从而得到Howe-Moore性质，并应用于Kac-Moody格点得到特征刚性。