Rohlin flows on the Cuntz algebra $O_\infty$

Semilattices of groups and nonstable K-theory of extended Cuntz limits

We give an elementary characterization of those (abelian) semigroups $M$ that are direct limits of countable sequences of finite direct products of monoids of the form $C\cup\{0\}$ for monogenic groups $C$. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of subgroups of $M$, and it makes it possible to express $M$ as a certain submonoid of a direct product $S\times G$, where $S$ is a distributive semilattice with zero and $G$ is an abelian group. When applied to the monoids $V(A)$ appearing in the nonstable K-theory of…

Cuntz代数中的C*-对角线具有康托尔谱

C*-diagonals with Cantor spectrum in Cuntz algebras

本文证明了Cuntz代数中存在具有康托尔谱的C*-对角线，并构建了相关代数的主以太群状模型。

Hyperrigid subsets of Cuntz-Krieger algebras and the property of rigidity at zero

A subset $\mathcal{G}$ generating a $C^*$-algebra $A$ is said to be hyperrigid if for every faithful nondegenerate $*$-representation $A\subseteq B(H)$ and a sequence $ϕ_n:B(H) \to B(H)$ of unital completely positive maps, we have that \[ \lim_{n\to\infty}ϕ_n(g)= g~~\text{for all } g\in \mathcal{G} ~~ \implies ~~ \lim_{n\to\infty}ϕ_n(a)= a~~\text{for all } a\in A \] where all convergence are in norm. In this paper, we show that for the Cuntz-Krieger algebra $\mathcal{O}(G)$ associated to a row-finite directed graph $G$ with no isolated vertices, the set of partial isometries $\mathcal{E}=\{S_e…

Lifting Commutation Relations in Cuntz Algebras

We examine splitting of the quotient map from the full free product $A*B$, or the unital free product $A*_{\mathbb C}B$, to the (maximal) tensor product $A\otimes B$, for unital C*-algebras $A$ and $B$. Such a splitting is very rare, but we show there is one if $A$ and $B$ are both the Cuntz algebra $O_2$ or $O_\infty$, and in a few other cases. The splitting is not explicit (and in principle probably cannot be). We also describe severe $K$-theoretic obstructions to a splitting.

Cuntz代数主群体模型的动态渐近维数及其动态渐近维数

Principal Groupoid Models for Cuntz Algebras and their Dynamic Asymptotic Dimension

本文研究了Cuntz代数主群体模型的动态渐近维数，证明了O_2存在无限多个非共轭的C*-对角线，并推广了这一结果。

An invariant of states on Cuntz algebras

For an arbitrary state $ω$ on a Cuntz algebra, we define a number $1\leq κ(ω)\leq \infty$ such that if the GNS representations of $ω$ and $ω'$ are unitarily equivalent, then $κ(ω)=κ(ω')$. By using $κ$, we define minimal states and it is shown that the classification problem of states is reduced to that of minimal states. By using results of Dutkay, Haussermann, and Jorgensen, we give a sufficient condition of the minimality of a state. Properties of $κ$ and examples are shown. As an application, a new invariant of a certain class of endomorphisms of ${\cal B}({\cal H})$ is given.

On normalizers of $C^{*}$-subalgebras in the Cuntz algebra $\mathcal{O}_{n}$

In this paper we investigate the normalizer $\mathcal{N}_{\mathcal{O}_{n}}(A)$ of a $C^{*}$-subalgebra $A\subset \mathcal{F}_{n}$ where $\mathcal{F}_{n}$ is the canonical UHF-subalgebra of type $n^{\infty}$ in the Cuntz algebra $\mathcal{O}_{n}$. Under the assumption that the relative commutant $A'\cap \mathcal{F}_{n}$ is finite-dimensional, we show several facts for normalizers of $A$. In particular it is shown that the automorphism group $\{{\rm Ad}u|_{A}\ \ |\ u\in \mathcal{N}_{\mathcal{F}_{n}}(A)\}$ has a finite index in $\{{\rm Ad}U|_{A}\ \ |\ U\in \mathcal{N}_{\mathcal{O}_{n}}(A)\}$.

Loop group actions on Cuntz algebras and geometric higher twists

We consider representations of the Cuntz algebras ${\mathcal O}_N$ as constructed by Bratteli-Jorgensen and use these to define a faithful action of the analytic loop group $L_{\text{an}}U(n)$ on ${\mathcal O}_N$ for $N \geq n$. This extends to a faithful action on the infinite Cuntz algebra ${\mathcal O}_{\infty}$, and we show that this action can be used to explicitly construct ${\mathcal O}_\infty \otimes \mathcal K$-bundles (where $ \mathcal K$ is the algebra of compact operators) over spaces, which are geometric twists in higher twisted $K$-theory.

The Toeplitz algebra of a Hilbert bimodule

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger algebras O_B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O_\infty.

Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras

Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $λ: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $ϕ_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(ϕ_n)=λ.$ This result leads t…

Morita isomorphism for Cuntz algebras

Let ${\rm Rep}\,{\cal O}_n$ denote the category of all nondegenerate $^*$ representations of the Cuntz algebra ${\cal O}_n$. For any $2\leq n,m<\infty$, we construct an isomorphism functor $F_{n,m}$ from ${\rm Rep}\,{\cal O}_m$ to ${\rm Rep}\,{\cal O}_n$.

Analogs of Cuntz algebras on $L^p$ spaces

For $d = 2, 3, \ldots$ and $p \in [1, \infty),$ we define a class of representations $ρ$ of the Leavitt algebra $L_d$ on spaces of the form $L^p (X, μ),$ which we call the spatial representations. We prove that for fixed $d$ and $p,$ the Banach algebra ${\mathcal{O}_{d}^{p}}$ obtained as the closure of the image of $L_d$ under the representation $ρ$ is the same for all spatial representations $ρ.$ When $p = 2,$ we recover the usual Cuntz algebra ${\mathcal{O}}_{d}.$ We give a number of equivalent conditions for a representation to be spatial. We show that for distinct $p_1$ and $p_2$ in $[1, \…

Equivariant semiprojectivity

We define equivariant semiprojectivity for C*-algebras equipped with actions of compact groups. We prove that the following examples are equivariantly semiprojective: arbitrary finite dimensional C*-algebras with arbitrary actions of compact groups; the Cuntz algebras ${\mathcal{O}}_d$ and extended Cuntz algebras $E_d,$ for finite $d,$ with quasifree actions of compact groups; the Cuntz algebra ${\mathcal{O}}_{\infty}$ with any quasifree action of a finite group. For actions of finite groups, we prove that equivariant semiprojectivity is equivalent to a form of equivariant stability of generat…

无限 Cuntz 代数的表示

Representations of the infinite Cuntz algebra

本文通过定义内蕴 ucp 映射并取归纳极限，构造了无限 Cuntz 代数 $\mathcal O_\infty$ 上的态，进而获得大族两两不等价的不可约表示，其中一族与 Hilbert 空间单位球面双射。

Approximate unitary equivalence of homomorphisms from O_{\infty}

We prove that if two homomorphisms from O_{\infty} to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows that O_{\infty} is classifiable in the sense of Rordam. In particular, Rordam's classification theorem for direct limits of matrix algebras over even Cuntz algebras extends to direct limits involving both matrix algebras over even Cuntz algebras and corners of O_{\infty}, for which the K_0-group can be an arbitrary countable abelian group with no even torsion.

A Poisson Algebra on the Hida Test Functions and a Quantization using the Cuntz Algebra

In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra $\mathcal{O}_{\infty}$ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie brac…

Nonexistence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.

Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers

We show that the group ${\mathbb Q \rtimes \mathbb Q^*_+}$ of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup ${\mathbb N \rtimes \mathbb N^\times}$. The associated Toeplitz $C^*$-algebra ${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$ is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of this Toeplitz algebra in terms of generators and relations, and use this to show that the $C^*$-algebra ${\mathcal Q_\mathbb N}$ recently introduced by Cuntz is the boundary quotien…

Compactly-aligned discrete product systems, and generalizations of O_\infty

The universal C*-algebras of discrete product systems generalize the Toeplitz- Cuntz algebras and the Toeplitz algebras of discrete semigroups. We consider a semigroup P which is quasi-lattice ordered in the sense of Nica, and, for a product system p:E\to P, we study those representations of E, called covariant, which respect the lattice structure of P. We identify a class of product systems, which we call compactly aligned, for which there is a purely C*-algebraic characterization of covariance, and study the algebra C*_{cov}(P,E) which is universal for covariant representations of E. Our mai…

Divisibility properties for C*-algebras

We consider three notions of divisibility in the Cuntz semigroup of a C*-algebra, and show how they reflect properties of the C*-algebra. We develop methods to construct (simple and non-simple) C*-algebras with specific divisibility behaviour. As a byproduct of our investigations, we show that there exists a sequence $(A_n)$ of simple unital infinite dimensional C*-algebras such that the product $\prod_{n=1}^\infty A_n$ has a character.

The inner structure of boundary quotients of right LCM semigroups

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,θ)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture al…

Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O_2

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}_2 \otimes A \cong {\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\cal O}_{\infty} \otimes A \cong A.$ The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \in X,$ such that the algebra of continuous sections of $A$ is separable…

$L^p$-Cuntz代数与加权复合算子的谱

$L^p$-Cuntz algebras and spectrum of weighted composition operators

本文定义了由转移算子构成的 $L^p$-算子代数交叉积，证明其与 $L^p$-Cuntz代数等距同构，并据此给出了抽象加权移位算子谱半径的计算公式，推广了 $p=2$ 的经典结果。

$L^p$-operator algebras associated with oriented graphs

For each $1\le p<\infty$ and each countable oriented graph $Q$ we introduce an $L^p$-operator algebra $\mathcal{O}^p(Q)$ which contains the Leavitt path $\mathbb{C}$-algebra $L_Q$ as a dense subalgebra and is universal for those $L^p$-representations of $L_Q$ which are spatial in the sense of N.C. Phillips. For $\mathcal{R}_n$ the graph with one vertex and $n$ loops ($2\le n\le \infty$), $\mathcal{O}^p(\mathcal{R}_n)=\mathcal{O}^p_n$, the $L^p$-Cuntz algebra introduced by Phillips. If $p\notin\{1,2\}$ and $\mathcal{S}(Q)$ is the inverse semigroup generated by $Q$, $\mathcal{O}^p(Q)=F_{\operato…

Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces

For $p \in [1, \infty),$ we define and study full and reduced crossed products of algebras of operators on $σ$-finite $L^p$ spaces by isometric actions of second countable locally compact groups. We give universal properties for both crossed products. When the group is abelian, we prove the existence of a dual action on the full and reduced $L^p$ operator crossed products. When the group is discrete, we construct a conditional expectation to the original algebra which is faithful in a suitable sense. For a free action of a discrete group on a compact metric space $X,$ we identify all traces on…

Jump transformations and an embedding of ${\cal O}_{\infty}$ into ${\cal O}_{2}$

A measurable map $T$ on a measure space induces a representation $Π_{T}$ of a Cuntz algebra ${\cal O}_{N}$ when $T$ satisfies a certain condition. For such two maps $τ$ and $σ$ and representations $Π_τ$ and $Π_σ$ associated with them, we show that $Π_τ$ is the restriction of $Π_σ$ when $τ$ is a jump transformation of $σ$. Especially, the Gauss map $τ_1$ and the Farey map $σ_1$ induce representations $Π_{τ_1}$ of ${\cal O}_{\infty}$ and that $Π_{σ_1}$ of ${\cal O}_{2}$, respectively, and $Π_{τ_1}=Π_{σ_1}|_{{\cal O}_{\infty}}$ with respect to a certain embedding of ${\cal O}_{\infty}$ into ${\ca…

A purely infinite AH-algebra and an application to AF-embeddability

We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit of C*-algebras of the form C_0([0,1),M_k) and it absorbs the Cuntz algebra O_\infty tensorially. Thus one can reach an O_\infty-absorbing C*-algebra as an inductive limit of the finite and elementary C*-algebras C_0([0,1),M_k). As an application we give a new proof of a recent theorem of Ozawa that the cone over any separable exact C*-algebra is AF-embeddable, and we exhibit a concrete AF-algebra into which this class of C*-algebras can be embedded.

Families of Type {\rm III KMS} States on a Class of $C^*$-Algebras containing $O_n$ and $\mathcal{Q}_\N$

We construct a family of purely infinite $C^*$-algebras, $\mathcal{Q}^λ$ for $λ\in (0,1)$ that are classified by their $K$-groups. There is an action of the circle $\T$ with a unique ${\rm KMS}$ state $ψ$ on each $\mathcal{Q}^λ.$ For $λ=1/n,$ $\mathcal{Q}^{1/n}\cong O_n$, with its usual $\T$ action and ${\rm KMS}$ state. For $λ=p/q,$ rational in lowest terms, $\mathcal{Q}^λ\cong O_n$ ($n=q-p+1$) with UHF fixed point algebra of type $(pq)^\infty.$ For any $n>0,$ $\mathcal{Q}^λ\cong O_n$ for infinitely many $λ$ with distinct KMS states and UHF fixed-point algebras. For any $λ\in (0,1),$ $\mathca…

无限图代数的保对角同构

Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs

本文建立了Cuntz代数、图C*-代数、Leavitt路径代数、拓扑全群、群胚及K-理论中若干陈述之间的逻辑等价性，为图代数保对角同构的几何分类研究提供了关键判定依据。

C^*-algebras associated with complex dynamical systems

Iteration of a rational function $R$ gives a complex dynamical system on the Riemann sphere. We introduce a $C^*$-algebra ${\mathcal O}_R$ associated with $R$ as a Cuntz-Pimsner algebra of a Hilbert bimodule over the algebra $A = C(J_R)$ of continuous functions on the Julia set $J_R$ of $R$. The algebra ${\mathcal O}_R$ is a certain analog of the crossed product by a boundary action. We show that if the degree of $R$ is at least two, then $C^*$-algebra ${\mathcal O}_R$ is simple and purely infinite. For example if $R(z) = z^2 - 2$, then the Julia set $J_R = [-2,2]$ and the restriction $R : J_R…

紧量子群在图C*-代数上的等距作用

Isometric actions of compact quantum groups on graph C*-algebras

研究了紧量子群在图C*-代数上的等距作用，证明了量子自同构群的等距性质及U_n^+在Cuntz代数上的非等距性。

可计算强自吸收C*-代数

Computably strongly self-absorbing C*-algebras

本文引入可计算强自吸收C*-代数的概念，证明了Cuntz代数、特定UHF代数及Jiang-Su代数等满足该性质，并作为Elliott近似交织论证的可计算版本特例。

$L^p$-算子交叉积的$p$-核性

$p$-nuclearity of $L^p$-operator crossed products

本文研究$L^p$-算子交叉积的$p$-核性问题，证明在$p$-完全等距作用下全$L^p$-算子交叉积的$p$-核性等价于原代数的$p$-核性，并应用该结论证明$L^p$-Cuntz代数等的$p$-核性，解决了Phillips提出的相关问题。

On the decomposition into Discrete, type II and type III $C^*$-algebras

We obtained a "decomposition scheme" of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsido), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to "classify" C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking "essential extension" and "normal quotient". Furthermore, there exist the largest discrete finite ideal $A_{d,1}$, the largest discrete essentially infinite ideal $A_{d,\infty}$, the largest type II f…

Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers

We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $β\in[1,2]$, there is a unique KMS$_β$ state, and we prove that it is a factor state of type III$_1$. There is a phase transition at $β=2:$ For each $β\in (2,\infty]$, the set of extremal KMS$_β$ states decomposes as a disjoint union over a quotient of a ray class group in which the fibers are extremal traces on certain group C*-algebras associated with the ideal classes. Moreover, in most cases, there is a further phas…

Colocalizations of noncommutative spectra and bootstrap categories

We construct a compactly generated and closed symmetric monoidal stable $\infty$-category $\mathtt{NSp'}$ and show that $\mathtt{hNSp'}^{op}$ contains the suspension stable homotopy category of separable $C^*$-algebras $\mathtt{ΣHo^{C^*}}$ constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of $\mathtt{NSp'}$, namely, $\mathtt{NSp'}[\mathbb{K}^{-1}]$ and $\mathtt{NSp'}[\mathcal{Z}^{-1}]$, both of which are shown to be compactly generated and closed symmetric monoidal. We prove that Kasparov $KK$-category of separable $C^*$-al…

Boundary operator algebras for free uniform tree lattices

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 $L^\infty(\partial Δ)\rtimes Γ$…

动力 Kirchberg–Phillips 定理

The dynamical Kirchberg-Phillips theorem

本文建立任意局部紧群在 Kirchberg 代数上一类可和作用的余圈共轭分类，证明其由等变 Kasparov 理论决定，并解决及恢复相关猜想与结果。

Holomorphic automorphisms of noncommutative polyballs

In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of aut…

离散可逆群的一致Roe代数中的严格比较性质

Strict comparison holds in the uniform Roe algebra of a discrete amenable group

本文证明离散可逆群的一致Roe代数中，正元素迹态下的严格不等式蕴含Cuntz次等价，并证明其通用极小集交叉积C*-代数是强AH代数。

Representation Theory and Numerical AF-invariants: The representations and centralizers of certain states on O_d

Let O_d be the Cuntz algebra on generators S_1,...,S_d, 2 \leq d < \infty, and let D_d \subset O_d be the abelian subalgebra generated by monomials S_αS_α^* =S_{α_{1}}...S_{α_{k}}S_{α_{k}}^*...S_{α_{1}}^* where α=(α_1...α_k) ranges over all multi-indices formed from {1,...,d}. In any representation of O_d, D_d may be simultaneously diagonalized. Using S_i(S_αS_α^*) =(S_{iα}S_{iα}^*)S_i, we show that the operators S_i from a general representation of O_d may be expressed directly in terms of the spectral representation of D_d. We use this in describing a class of type III representations of O_d…