Guyan Robertson, Tim Steger
This is an appendix to the paper {\bf Asymptotic K-theory for groups acting on $\tA_2$ buildings}, and contains the results of the computations performed by the authors.
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Let $X$ be a finite connected graph, each of whose vertices has degree at least three. The fundamental group $Γ$ of $X$ is a free group and acts on the universal covering tree $Δ$ and on its boundary $\partial Δ$, endowed with a natural topology and Borel measure. The crossed product $C^*$-algebra $C(\partial Δ) \rtimes Γ$ depends only on the rank of $Γ$ and is a Cuntz-Krieger algebra whose structure is explicitly determined. The crossed product von Neumann algebra does not possess this rigidity. If $X$ is homogeneous of degree $q+1$ then the von Neumann algebra $L^\infty(\partial Δ)\rtimes Γ$…
Both boundary maps in K-theory are expressed in terms of surjections from projective C*-algebras to semiprojective C*-algebras.
For any reduced free product $\mathrm{C}^*$-algebra $(A, φ) =(A_1, φ_1) \star (A_2, φ_2)$, we prove a boundary rigidity result for the embedding of $A$ into its associated $\mathrm{C}^*$-algebra $Δ\mathbf{T}(A, φ)$. This provides new examples of rigid embeddings of exact $\mathrm{C}^*$-algebras into purely infinite simple nuclear $\mathrm{C}^*$-algebras.
The K-theory of boundary C*-algebras of symmetric spaces
研究对称空间最大Furstenberg边界上的晶格作用交叉乘积C*-代数的K理论，提供新的非空间同构的边界C*-代数例子
In this paper, we study the boundary quotient C*-algebras associated to products of odometers. One of our main results shows that the boundary quotient C*-algebra of the standard product of $k$ odometers over $n_i$-letter alphabets ($1\le i\le k$) is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph C*-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz-Pimsner C*-algebra is isomorphic to the…
We develop a theory of boundary functions for ideals in trivially analytic subalgebras of simple AF C*-algebras with an injective 0-cocycle, a class which includes all full nest algebras. Boundary functions are maps from the spectrum of the diagonal of the analytic subalgebra to itself. The relation between boundary functions and ideal sets is explored and a description is given of meet and join irreducible boundary functions.
Let $Δ$ be a building of type $\widetilde A_2$ and order $q$, with maximal boundary $Ω$. Let $Γ$ be a group of type preserving automorphisms of $Δ$ which acts regularly on the chambers of $Δ$. Then the crossed product $C^*$-algebra $C(Ω) \rtimes Γ$ is isomorphic to $M_{3(q+1)} \otimes\cl O_{q^2}\otimes\cl O_{q^2}$, where $\cl O_n$ denotes the Cuntz algebra generated by $n$ isometries whose range projections sum to the identity operator.
Let A be a unital separable simple infinite-dimensional nuclear C*-algebra with at least one tracial state. We prove that if the trace space of A has compact finite-dimensional extreme boundary then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A which is determined by the uniform topology on the trace space. As an application, it is shown that if furthermore A has strict comparison then A absorbs the Jiang-Su algebra tensorially.
All operator algebras have (not necessarily irreducible) boundary representations. A unital operator algebra has enough such boundary representations to generate its C*-envelope.
We consider the semigroup crossed product of the additive natural numbers by the multiplicative natural numbers. We study its Toeplitz C*-algebra generated by the right-regular representation, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. We identify the Crisp-Laca boundary quotient as the C*-algebra of the corresponding g…
This note presents an analysis of a class of operator algebras constructed as cross-sectional algebras of flat holomorphic matrix bundles over a finitely bordered Riemann surface. These algebras are partly inspired by the bundle shifts of Abrahamse and Douglas. The first objective is to understand the boundary representations of the containing $C^*$-algebra, i.e. Arveson's noncommutative Choquet boundary for each of our operator algebras. The boundary representations of our operator algebras for their containing $C^*$-algebras are calculated, and it is shown that they correspond to evaluations…
We initiate a study of non-commutative Choquet boundary for spaces of unbounded operators. We define the notion of local boundary representations for local operator systems in locally C$^*$-algebras and prove that local boundary representations provide an intrinsic invariant for a particular class of local operator systems. An appropriate analog of purity of local completely positive maps on local operator systems is used to characterize local boundary representations for local operator systems in Frechet locally C$^*$-algebras.
For a smooth manifold with boundary we construct a semigroupoid and a continuous field of C*-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of \hbar-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.
Let $Δ$ be an infinite, locally finite tree with more than two ends. Let $Γ<\aut(Δ)$ be an acylindrical uniform lattice. Then the boundary algebra $\cl A_Γ= C(\partialΔ)\rtimes Γ$ is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.
Using ultrapowers of $C^{\ast}$-algebras we provide a new construction of the multiplier algebra of a $C^{\ast}$-algebra. This extends the work of Avsec and Goldbring [Houston J. Math., to appear, arXiv:1610.09276.] to the setting of noncommutative and nonseparable $C^{\ast}$-algebras. We also extend their work to give a new proof of the fact that groups that act transitively on locally finite trees with boundary amenable stabilizers are boundary amenable.
We study the Toeplitz algebra $\TT(\N\rtimes\N^\times)$ and three quotients of this algebra: the $C^*$-algebra $\qn$ recntly introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of $\N\rtimes\N^\times$ satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on $\TT(\nxnx)$ to describe the KMS sta…
Given a manifold with boundary endowed with an action of a discrete group on it, we consider the algebra of operators generated by elements in the Boutet de Monvel algebra of pseudodifferential boundary value problems and shift operators acting on functions on the manifold and its boundary. Under certain conditions on the group action, we construct a Chern character for symbols of elliptic elements in this algebra with values in a de Rham type cohomology of the fixed point manifolds and obtain an index formula in terms of this Chern character.
Let $\G$ be a group of type rotating automorphisms of an affine building $\cB$ of type $\wt A_2$. If $\G$ acts freely on the vertices of $\cB$ with finitely many orbits, and if $Ω$ is the (maximal) boundary of $\cB$, then $C(\Om)\rtimes \G$ is a p.i.s.u.n. $C^*$-algebra. This algebra has a structure theory analogous to that of a simple Cuntz-Krieger algebra and is the motivation for a theory of higher rank Cuntz-Krieger algebras, which has been developed by T. Steger and G. Robertson. The K-theory of these algebras can be computed explicitly in the rank two case. For the rank two examples of t…
Local boundary representations for local operator systems
本文证明Fréchet局部C*-代数中局部算子系统的局部边界表示等价于希尔伯特空间上的局部边界表示，从而将研究简化为算子系统的边界表示。
$\mathcal{Z}$-stability for $\mathrm C^*$-algebras of minimal line-bundle-twisted homeomorphisms with the small boundary property
本文研究了具有小边界性质的最小线束扭曲同胚的C*-代数的Z稳定性，并展示了其分类及张量积的稳定性。
A subgroup of an amenable group is amenable. The $C^*$-algebra version of this fact is false. This was first proved by M.-D. Choi who proved that the non-nuclear $C^*$-algebra $C^*_r(\ZZ_2*\ZZ_3)$ is a subalgebra of the nuclear Cuntz algebra ${\cal O}_2$. A. Connes provided another example, based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same. In fact he proved that certain of the $C^*$-algebras studied by J. Cuntz and W. Krieger [10] can be constructed naturally as crossed product algebras. For example if the group $Γ$ a…
We study C*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. If $P$ is such a semigroup, its C*-algebra admits a natural boundary quotient $\mathcal{Q}(P)$. We show that $\mathcal{Q}(P)$ is isomorphic to the tight C*-algebra of a certain inverse semigroup associated to $P$, and thus is isomorphic to the C*-algebra of an étale groupoid. We use this to give conditions on $P$ which guarantee that $\mathcal{Q}(P)$ is simple and purely infinite,…
A Fejér-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick.
Boundary quotients of C$^*$-algebras of left cancellative monoids and their groupoid models
本文研究了左可消幺半群约化半群C$^*$-代数的边界商，提出了两种由Paterson和Spielberg广群约化得到的潜在广群模型，给出了保证其成为广群模型的条件，并构造了满足新条件但非C$^*$-正则的例子。
We extend applications of Furstenberg boundary theory to the study of $C^*$-algebras associated to minimal actions $Γ\!\curvearrowright\! X$ of discrete groups $Γ$ on locally compact spaces $X$. We introduce boundary maps on $(Γ,X)$-$C^*$-algebras and investigate their applications in this context. Among other results, we completely determine when $C^*$-algebras generated by covariant representations arising from stabilizer subgroups are simple. We also characterize the intersection property of locally compact $Γ$-spaces and simplicity of their associated crossed products.
Note on C*-algebras associated to boundary actions of hyperbolic 3-manifold groups
本文利用Kirchberg-Phillips分类证明双曲3-流形群边界交叉积C*-代数的同构类型仅由流形同调决定，并构造出无穷多个不同构但交叉积同构的双曲群，且这些同构不具有动力学来源。
We investigate the tracial and ideal structures of $C^*$-algebras of quasi-regular representations of stabilizers of boundary actions. Our main tool is the notion of boundary maps, namely $Γ$-equivariant unital completely positive maps from $Γ$-$C^*$-algebras to $C(\partial_FΓ)$, where $\partial_FΓ$ denotes the Furstenberg boundary of a group $Γ$. For a unitary representation $π$ coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on $C^*_π(Γ)$. Consequently, we completely describe the tracial structure of the $C^*$-algebras $C^*_π(Γ)$, and for a…
We study the C*-algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C*-algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.
$C^*$-algebras of transmission problems and elliptic boundary value problems with shift operators
研究群作用在光滑流形上的一类新型非局部边值问题的Fredholm可解性，通过边界轨道分解流形并利用$C^*$-代数技术进行分析。
Boundary algebras of the Kitaev Quantum Double model
本文证明了Kitaev量子双重模型满足局部拓扑序公理，识别其边界代数网为两类融合范畴网，并建立了与锥von Neumann代数及平凡$G$-SPT的联系，还计算了阿贝尔群(3+1)D情形的边界代数。
Simplicity and pure infiniteness for $\text{C}^*$-algebras associated with stabilizers of boundary actions
该研究针对离散群作用于紧空间的稳定子相关C*-代数，利用Rieffel诱导方法，证明当空间为极端边界时，对应群C*-代数纯无限，回答了汤普森群相关C*-代数的部分问题
In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system $(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$. For the first groupoid, we associate an inverse semigroup to a generalized Boolean dynamical system and use the tight spectrum $\mathsf{T}$ as the unit space of a groupoid $Γ(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$ that is isomorphic to the tight groupoid $\mathcal{G}_{tight}$. The other one is defined as the Renault-Deaconu groupoid $Γ(\partial E, σ_E)$ arising from a topological correspondence $E$ associated with a generalized Boolean d…
We classify operator systems $S\subseteq \mathcal B(H)$ that act on finite dimensional Hilbert spaces by making use of the noncommutative Choquet boundary. S is said to be {\em reduced} when its boundary ideal is 0. In the category of operator systems, that property functions as semisimplicity does in the category of complex Banach algebras. We construct explicit examples of reduced operator systems using sequences of "parameterizing maps" $Γ_k: \mathbb C^r\to \mathcal B(H_k)$, $k=1,..., N$. We show that every reduced operator system is isomorphic to one of these, and that two sequences give r…
We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C$^*-$modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C$^*-$algebras of an operator space, which generalize the algebras of adjointable operators on a C$^*-$module, and the `imprimitivity C$^*-$algebra'. It also generalizes a classical Banach space notio…
We study reproducing kernel Hilbert spaces on the unit ball with the complete Nevanlinna-Pick property through the representation theory of their algebras of multipliers. We give a complete description of the representations in terms of the reproducing kernels. While representations always dilate to $*$-representations of the ambient $C^*$-algebra, we show that in our setting we automatically obtain coextensions. In fact, we show that in many cases, this phenomenon characterizes the complete Nevanlinna-Pick property. We also deduce operator theoretic dilation results which are in the spirit of…
Frustration-Free Models for Topological Holography: Fusion Spin Chains as Boundary Algebras
本文提出一类基于幺正球面融合范畴的无阻挫二维量子自旋模型，推广Levin-Wen弦网模型，并利用arXiv:2510.20662的理论将Temperley-Lieb-Jones代数及融合自旋链实现为边界代数。
We study boundaries for unital operator algebras. These are sets of irreducible $*$-representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our an…
Local topological order and boundary algebras
本文提出了一套关于局部拓扑有序量子自旋系统的公理，基于局部基态投影的网络，并展示了这些公理在凯塔夫的扭码和李文-温类型模型中得到满足。对于在$\mathbb{Z}^{k}$上局部拓扑有序的自旋系统，定义了一个在$\mathbb{Z}^{k-1}$上的局部边界代数网络，为bulk拓扑序的双曲对偶提供了数学精确的代数描述。构造了一个规范量子通道，使得边界上的准局部代数状态可以参数化bulk-边界状态，而无需参考边界哈密顿量。作为推论，我们获得了奥加塔[Ann. H. Poincaré 25, 2024]最近结果的新证明，即扭码中的bulk锥von Neumann代数为类型II，且显示李文-温模型可以有锥代数为类型III。最后，我们论证了边界代数网络的DHR双模的辫积张量范畴可以表征(2+1)D中的bulk拓扑序，并可用于表征边界态的拓扑序。
The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, corresponding to the two filtrations by interior order and vanishing degree at the boundary, together with the exterior derivations of the algebra. This leads to an index formula which is a pseudodifferential extension of that of Atiyah, Patodi and Singer for Dirac operators; togeth…
Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras
本文通过验证 Myrberg 点的自由性，证明 CAT(0) 空间的多类边界作用（视觉、horofunction、Roller）是拓扑自由的强边界作用，推广了已知结果，并应用于构造新的 $C^*$-无自群及精确纯无限单的约化交叉积 $C^*$-代数。
Holography for bulk-boundary local topological order
本文扩展局域拓扑序公理至带拓扑边界的量子自旋系统，通过边界DHR双模范畴恢复拓扑边界序，并详细分析Levin-Wen和Walker-Wang模型。
In this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]) and the theory of boundary quotients of $C^\ast$-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar M-fractals for a given monoid M, gives rise to examples of $C^\ast$- algebras generalizing the boundary quotients discussed by X. Li in [4, §7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries natur…
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