Astrid an Huef, Iain Raeburn
Suppose that $G$ is a locally compact group and $π$ is a (not necessarily irreducible) unitary representation of a closed normal subgroup $N$ of $G$ on a Hilbert space $H$. We extend results of Clifford and Mackey to determine when $π$ extends to a unitary representation of $G$ on the same space $H$ in terms of a cohomological obstruction.
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Structural Chirality from Inverse Semigroups to Twisted Groupoid $C^*$-Algebras
本文提出通过逆半群的结构手性理论，构建扭曲群oid $C^*$-代数的镜像对应关系，并揭示其在非交换算子代数中的统一桥梁作用。
Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions
本文研究了 $C^*$-对应物乘积系统的扭 representations 的 Wold 分解和幺正扩展，提出了双扭共变表示，并通过直接极限构造了幺正扩展。
The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either the matrices over A for A the real numbers, A the complex numbers or A the algebra of quaternions. There are potential obstructions …
On almost commuting unitary matrices
研究当某类近似可交换矩阵接近精确可交换矩阵的问题，在缠绕数障碍消失时，依据原始矩阵换位子得出到可交换酉矩阵距离的有效界。
On the diversity of twisted commuting operators
本文研究扭曲交换算子的多样性约束，发现投影等算子无法成为扭曲交换算子，多数扭曲要求算子对含非平凡核，下方有界算子无法形成此类对，扭曲交换酉算子由特定酉对建模。
A duality is discussed for Lie group bundles vs. certain tensor categories with non-simple identity, in the setting of Nistor-Troitsky gauge-equivariant K-theory. As an application, we study C*-algebra bundles with fibre a fixed-point algebra of the Cuntz algebra: a classification is given, and a cohomological invariant is assigned, representing the obstruction to perform an embedding into a continuous bundle of Cuntz algebras. Finally, we introduce the notion of twisted equivariant K-theory.
We describe the tube algebra and its representations in the cases of diagonal and Bisch-Haagerup subfactors possibly with a scalar 3-cocycle obstruction. We show that these categories are additively equivalent to the direct product over conjugacy classes of representation category of a centralizer subgroup (corresponding to the conjugacy class) twisted by a scalar 2-cocycle obtained from the 3-cocycle obstruction.
We show that the Baum-Connes morphism twisted by a non-unitary representation, defined in [GA08], is an isomorphism for a large class of groups satisfying the Baum-Connes conjecture. Such class contains all the real semi-simple Lie groups, all hyperbolic groups and many infinite discret groups having Kazhdan's property (T). We define a tensorisation by a non-unitary finite dimensional representation on the left handside of the Baum-Connes morphism and we show that its analogue in K-theory must be defined on the K-theory of the twisted group algebras introduced in [GA07b].
We consider converses to the density theorem for irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that under the right assumptions, the restriction of a $σ$-projective unitary representation $π$ of a group $G$ to a lattice $Γ$ extends to a Hilbert module over the twisted group von Neumann algebra $\text{L}(Γ,σ)$. We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that t…
A discrete countable group G is matricially stable if the finite dimensional approximate unitary representations of G are perturbable to genuine representations in the point-norm topology. For large classes of groups G, we show that matricial stability implies the vanishing of the rational cohomology of G in all nonzero even dimensions. We revisit a method of constructing almost flat K-theory classes of BG which involves the dual assembly map and quasidiagonality properties of G. The existence of almost flat K-theory classes of BG which are not flat represents an obstruction to matricial stabi…
Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point of X. We use obstruction theory to establish a necessary and sufficient condition for A and B to be unitarily equivalent. We also determine bounds on the number of possible unitary equivalence classes in terms of cohomological invariants of X.
$K$-Theoretic Obstructions to Linearizing QCA Representations
本文针对量子元胞自动机表示，利用代数$K$-理论谱发展障碍理论，研究其线性化问题，并计算了点、线和平面上QCA空间的同伦类型。
We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps. Detailed computations are presented in the case where G is a finitely generated free abelian group. We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role.
A unitary invariant for $C^*$-algebras generated by isometries with a twisted commutation relation
本文研究满足$V_1V_2=qV_2V_1$（$q$为无理旋转）的等距算子对生成的$C^*$-代数的酉分类，通过函数模型和半Fredholm指标构造酉不变量，扩展了Berger-Coburn-Lebow定理至非交换情形。
Cartan-preserving *-automorphism groups: realization and obstructions for compact abelian groups
研究 C*-代数中保卡坦子代数的自同构群，通过扭曲 étale 群胚描述，得到相关定理类似物。研究卡坦固定自同构群，证明 UCT 基尔希贝格代数的相关性质，及扩张有效群胚产生的 C*-代数的性质，还给出不等价卡坦子代数存在性。
C*-algebras form a 2-category with \Star{}homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions, and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby-Smith twisted actions and equivalence of such actions, covariant representations, and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green and Busby-Smith. The Packer-Raeburn Stabilisation Tr…
We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional irreducible unitary representations being invariant for the dual of this automorphism. Also, we prove that any finitely generated residually finite non-amenable group has the R-infinity property (any automorphism has infinitely many twisted conjugacy classes). This gives a lot of new examples and covers many known classes of such groups.
Let $R(ϕ)$ be the number of $ϕ$-conjugacy (or Reidemeister) classes of an endomorphism $ϕ$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(ϕ)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(ϕ)<\infty$. Applying the result to iterations of $ϕ$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ pr…
We study the $H^3$ invariant of a group homomorphism $ϕ:G \rightarrow \mathrm{Out}(A)$, where $A$ is a classifiable C$^*$-algebra. We show the existence of an obstruction to possible $H^3$ invariants arising from considering the unitary algebraic $K_1$ group. In particular, we prove that when $A$ is the Jiang--Su algebra $\mathcal{Z}$ this invariant must vanish. We deduce that the unitary fusion categories $\mathrm{Hilb}(G, ω)$ for non-trivial $ω\in H^3(G, \mathbb{T})$ cannot act on $\mathcal{Z}$.
QWEP stability under twisted crossed products
本文研究扭曲交叉积的QWEP性质，证明其不被超线性群交叉积保持，给出非QWEP例子，并建立标量扭曲下QWEP的等价条件及正则圈积的限制定理，为Fourier乘子转移提供输入并识别范数障碍。
We use a noncommutative generalization of Fourier analysis to define a broad class of pseudo-probability representations, which includes the known bosonic and discrete Wigner functions. We characterize the groups of quantum unitary operations which correspond to phase-space transformations, generalizing Gaussian and Clifford operations. As examples, we find Wigner representations for fermions, hard-core bosons, and angle-number systems.
Fusion Bialgebras and Fourier Analysis
本文引入融合双代数及其对偶并建立傅里叶分析框架，利用 Schur 乘积等解析不等式识别融合环酉范畴化与子因子化障碍，并分类排除了大批 Frobenius 型整融合环。
A Cup Product Obstruction to Frobenius Stability
本文证明：若有限生成离散群的二阶上同调中某非挠元可表为两个一阶上同调类的杯积，则该群非Frobenius稳定，并推广至Schatten p-范数情形。
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