Path-connected Closures of Unitary Orbits

Unitary group orbits versus groupoid orbits of normal operators

We study the unitary orbit of a normal operator $a\in \mathcal B(\mathcal H)$, regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the differing ambient spaces if and only if the spectrum of $a$ is finite, and in that case it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of $a$…

Approximate Unitary Equivalence in Simple C^*-algebras of Tracial Rank One

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $ϕ, ψ: C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $ϕ$ and $ψ$ are approximately unitarily equivalent if and only if [ϕ]=[ψ] in KL(C,A) τ\circ ϕ=τ\circ ψfor all tracial states of A and ϕ^‡=ψ^‡, here ϕ^‡ and ψ^‡ are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A) induced by ϕand ψ, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.

Schur-Horn theorems in II$_\infty$-factors

We describe majorization between selfadjoint operators in a $σ$-finite II$_\infty$ factor $(\mathcal{M},τ)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$ with trace-preserving conditional expectation $E_{\mathcal{A}}$, we characterize the closure in the measure topology of the image through $E_{\mathcal{A}}$ of the unitary orbit of a selfadjoint operator in $\mathcal{M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in $\mathcal{M}$ and …

Unitary spherical representations of Drinfeld doubles

We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of $SU_q(3)$, we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group $SU_q(2n+1)$ has property (T), which also implies central property (T) of the dual of $SU_q(2n+1)$.

Closed Unitary and Similarity Orbits of Normal Operators in Purely Infinite C*-Algebras

We will investigate the norm closure of the unitary and similarity orbits of normal operators in unital, simple, purely infinite C*-algebras. An operator theoretic proof will be given to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial K_1-group. Some upper and lower bounds for the distance between unitary orbits will be obtained based on these methods. In addition, a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital, simple, purely infin…

The Higson-Mackey analogy for finite extensions of complex semisimple groups

In the 1970's, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey's analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum-Connes conjecture with trivial coefficients for such groups. Here we extend Higson's results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our m…

与酉算子本质交换

Essentially Commuting with a Unitary

针对谱为圆周的酉算子 $R$，本文证明与其本质交换的酉算子集合道路连通，并确定满足非平凡条件的正交投影的道路连通分支与整数集 $\mathbb{Z}$ 双射。

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH--Lie group. On the complex enveloping algebra $U_\C(\g)$ of its Lie algebra $\g$ we define the concept of an analytic functional and show that every positive analytic functional $λ$ is integrable in the sense that it is of the form $λ(D) = \la \ddπ(D)v, v\ra$ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of *-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations. For the matrix coefficient …

Bounded normal generation for projective unitary groups of certain infinite operator algebras

We study the question how quickly products of a fixed conjugacy class cover the entire group in the projective unitary group of the connected component of the identity of the Calkin algebra, as well as the projective unitary group of a factor von Neumann algebra of type III. Our result is that the number of factors that are needed is as small as permitted by the (essential) operator norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups and analogous results for unitary groups of II_1-factors.

双交换范畴中有限深度对象的分类：基于锚定平面代数

Classification of finite depth objects in bicommutant categories via anchored planar algebras

本文通过锚定平面代数将双交换范畴与平面代数分类联系起来，证明有限深度对象可由中心范畴中的连通有限深度酉锚定平面代数分类，推广了已有结果。