Alin Ciuperca, George A. Elliott, Luis Santiago
The class of separable C*-algebras which can be written as inductive limits of continuous-trace C*-algebras with spectrum homeomorphic to a disjoint union of trees and trees with a point removed is classified by the Cuntz semigroup.
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It is shown that the Cuntz semigroup is a complete invariant for the C*-algebras that can be realized as an inductive limit of a sequence of finite direct sums of splitting interval algebras.
A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely, the class of separable simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have spectrum homeomorphic to the closed interval [0,1], or to a disjoint union of copies of this space. Also, the range of the invariant is calculated.
We prove closure properties for the class of C*-algebras that are inductive limits of semiprojective C*-algebras. Most importantly, we show that this class is closed under shape domination, and so in particular under shape and homotopy equivalence. It follows that the considered class is quite large. It contains for instance the stable suspension of any nuclear C*-algebra satisfying the UCT and with torsion-free $K_0$-group. In particular, the stabilized C*-algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives.
We show that a separable C*-algebra is an inductive limits of projective C*-algebras if and only if it has trivial shape, that is, if it is shape equivalent to the zero C*-algebra. In particular, every contractible C*-algebra is an inductive limit of projectives, and one may assume that the connecting morphisms are surjective. Interestingly, an example of Dadarlat shows that trivial shape does not pass to full hereditary sub-C*-algebra. It then follows that the same fails for projectivity. To obtain these results, we develop criteria for inductive limit decompositions, and we discuss the relat…
Let $A=\underrightarrow{\lim}{A_n}$ be an AF algebra, $G$ be a compact group. We consider inductive limit actions of the form $α=\underrightarrow{\lim}{α_n}$, where $α_n\colon G\curvearrowright A_n$ is an action on the finite dimensional C*-algebra $A_n$ which fixes each matrix summand. If each $α_n$ is inner, such actions are classified by equivariant K-theory by Handelman and Rossmann. However, if the actions $α_n$ are not inner, we show that such actions are not classifiable by equivariant K-theory. We give a complete classification of such actions using twisted equivariant K-theory.
An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},ϕ_{nm})$, where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all A…
We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form $(Z/nZ)\sqcup\{0\}$ (where 0 is a new zero element), for positive integers $n$. The key properties are the Riesz refinement property and the requirement that each element $x$ has finite order, that is, $(n+1)x=x$ for some positive integer $n$. Such monoids are necessarily semilattices of abelian groups, and part of our approach yields a characterization of the Riesz refinement property among semilattices of abelian groups. Further, we describe the monoids…
Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a compact quantum metric of Rieffel, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov-Hausdorff propinquity of Latremoliere on compact quantum metric spaces. This allows us to place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state. As a consequen…
We consider unital simple inductive limits of generalized dimension drop C*-algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop $C^*$-algebras. Suppose that $A$ is one of these C*-algebras. We show that $A\otimes Q$ has tracial rank no more than one, where $Q$ is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let $A$ and $B$ be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then $A\cong B$ if and only if they have the same Elliott invariant.
In the program to classify C$^*$-algebras, it is very important to find abstract conditions which are sufficient to imply that a given algebra has tracial rank zero, in the sense of Huaxin Lin. Even in the presence of a unique trace, we show that the union of the known necessary conditions is not enough.
Motivated by algebraic quantum field theory and our previous work we study properties of inductive systems of \ $C^*$-algebras over arbitrary partially ordered sets. A partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of a certain set. We consider a topology on the set of indices generated by a base of neighbourhoods. Examples of those topologies with different properties are given. An inductive system of $C^*$-algebras and its inductive limit arise naturally over each maximal upward directed subset. Using those induc…
A classification result is obtained for the C*-algebras that are (stably isomorphic to) inductive limits of 1-dimensional noncommutative CW complexes with trivial $K_1$-group. The classifying functor Cu is defined in terms of the Cuntz semigroup of the unitization of the algebra. For the simple C*-algebras covered by the classification, Cu reduces to the ordered $K_0$-group, the cone of traces, and the pairing between them. As an application of the classification, it is shown that the crossed products by a quasi-free action $O_2\rtimes_λ\mathbb{R}$ are all isomorphic for a dense set of positiv…
Completely positive approximations and inductive systems
本文提出C*-编码条件刻画归纳系统极限完全序同构于C*-代数，并证明可分核C*-代数恰为C*-编码系统极限，推广了NF与CPC*-系统，为所有可分核C*-代数提供无零阶映射的归纳极限描述。
K-theoretic classification of inductive limit actions of fusion categories on AF-algebras
本文通过K理论不变量，对融合范畴在AF代数上的诱导极限作用进行分类，并应用于有限深度强AF包含的分类。
This article deals with inductive systems of Toeplitz algebras over arbitrary directed sets. For such a system the family of its connecting injective $*$-homomorphisms is defined by a set of natural numbers satisfying a factorization property. The motivation for the study of those inductive systems comes from our previous work on the inductive sequences of Toeplitz algebras defined by sequences of numbers and the limit automorphisms for the inductive limits of such sequences. We show that there exists an isomorphism in the category of unital $C^*$-algebras and unital $*$-homomorphisms between …
In this paper, we consider the real rank zero $\mathrm{C}^*$-algebras which can be written as an inductive limit of the Elliott-Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.
Non-commutative multivariable versions of weighted shift operators arise naturally as `weighted' left creation operators acting on the Fock space Hilbert space. We identify a natural notion of periodicity for these $N$-tuples, and then find a family of inductive limit algebras determined by the periodic weighted shifts which can be regarded as non-commutative multivariable generalizations of the Bunce-Deddens C*-algebras. We establish this by proving that the C*-algebras generated by shifts of a given period are isomorphic to full matrix algebras over Cuntz-Toeplitz algebras. This leads to an …
The note is concerned with inductive systems of Toeplitz algebras and their $*$-homomorphisms over arbitrary partially ordered sets. The Toeplitz algebra is the reduced semigroup $C^*$-algebra for the additive semigroup of non-negative integers. It is known that every partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of some set. In our previous work we have studied a topology on this set of indexes. For every maximal upward directed subset we consider the inductive system of Toeplitz algebras that is defined by a giv…
We present the first range result for the total K-theory of C*-algebras. This invariant has been used successfully to classify certain separable, nuclear C*-algebras of real rank zero. Our results complete the classification of the so-called AD algebras of real rank zero.
A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely the class of simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have their spectrum homeomorphic to the interval [0,1] or to a finite disjoint union of closed intervals. In particular, a classification of those stably AI algebras which are inductive limits of hereditary sub-C*-algebras of interval algebras is obtained. Also, the range of the invariant is calculated.
The strongly Leibniz property and the Gromov--Hausdorff propinquity
本研究构造了对强莱布尼茨性质敏感的对偶格罗莫夫-豪斯多夫邻近度新版本，证明其在强莱布尼茨量子紧度量空间类上完备，给出归纳极限上强莱布尼茨L-半范数的存在充分条件，并应用于AF代数得到埃弗罗斯-沈代数的收敛性结果。
Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C*-algebra is the stationary inductive limit of a C*-stable Fell algebra that has compact spectrum and trivial Dixmier-Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to in principle compute the K-theory of the stable C*-algebra. A specif…
It is shown that a C*-algebra of the form C(X,U), where U is a UHF algebra, is not an inductive limit of subhomogeneous C*-algebras of topological dimension less than that of X. This is in sharp contrast to dimension-reduction phenomenon in (i) simple inductive limits of such algebras, where classification implies low-dimensional approximations, and (ii) when dimension is measured using decomposition rank, as the author and Winter proved that dr(C(X,U)) $\leq$ 2.
The notion of bounded element of C*-inductive locally convex spaces (or C*-inductive partial *-algebras) is introduced and discussed in two ways: the first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space.
Rokhlin Dimension and Inductive Limit Actions on AF-algebras
本文研究了具有有限 Rokhlin 维度的有限生成交换群在 AF-代数上的归纳极限作用，并证明了在特定条件下交叉乘积 C*-代数为 A𝕋-代数。
In this article we study property (T) groups arising from Rips construction in geometric group theory in the spirit of \cite{CDK19} and certain inductive limit groups from this class. Using interplay between Popa's deformation/rigidity and methods in geometric group theory we are able to extend the class of groups considered in \cite{CDK19} that remembers semidirect product features while passing to the group von Neumann algebras. Combining these results with the method developed in \cite{CDHK20} we are able to produce more examples of property (T) group factors with trivial fundamental group.…
We consider II$_1$ factors $M$ which can be realized as inductive limits of subfactors, $N_n \nearrow M$, having spectral gap in $M$ and satisfying the bi-commutant condition $(N_n'\cap M)'\cap M=N_n$. Examples are the enveloping algebras associated to non-Gamma subfactors of finite depth, as well as certain crossed products of McDuff factors by amenable groups. We use deformation/rigidity techniques to obtain classification results for such factors.
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.
We obtain a complete classification of the locally finite algebras and the operator algebras, given as algebraic inductive limits and Banach algebraic inductive limits respectively, of direct systems: A_1 contained in A_2 contained in A_3 and so on. Here the A_k are 2n-cycle algebras, where n is at least 3 and the inclusions are of rigid type. The complete isomorphism invariant is essentially the triple (K_0(A), H_1(A), Sigma(A)) where K_0(A) is viewed as a scaled ordered group, H_1(A) is a partial isometry homology group and Sigma(A), contained in the direct sum of K_0(A) and H_1(A), is the 2…
Convergence of inductive sequences of spectral triples for the spectral propinquity
本文为量子紧度量空间归纳序列在Gromov-Hausdorff近似下的收敛引入了新的充分必要条件，并推广至谱三元组在谱近似下的收敛，应用于非交换螺线管和Bunce-Deddens C*-代数。
A translation invariant state $ω$ on $C^*$-algebra $\clb=\otimes_{k \in \IZ}M^{(k)}$, where $M^{(k)}=M_d(\IC)$ is the $d-$dimensional matrices over field of complex numbers, give rises a stationary quantum Markov chain and associates canonically a unital completely positive normal map $τ$ on a von-Neumann algebra $\clm$ with a faithful normal invariant state $ϕ$. We give an asymptotic criteria on the Markov map $(\clm,τ,ϕ)$ for purity of $ω$. Such a pure $ω$ gives only type-I or type-III factor $ω_R$ once restricted to one side of the chain $\clb_R=\otimes_{\IZ_+}M^{(k)}$. In case $ω_R$ is typ…
The noncommutative Fourier transform of the irrational rotation C*-algebra is shown to have a K-inductive structure (at least for a large concrete class of irrational parameters, containing dense $G_δ$'s). This is a structure for automorphisms that is analogous to Huaxin Lin's notion of tracially AF for C*-algebras, except that it requires more structure from the complementary projection.
In this paper, a classification is given of real rank zero $C^*$-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras $\mathcal{C}$.
We show that if A is a separable, nuclear, O_infty-absorbing (or strongly purely infinite) C*-algebra, which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form M_k(C_0(G,v)), where G is a finite graph (and C_0(G,v) is the algebra of continuous functions on G that vanish at a distinguished point v in G). We show further that any separable, nuclear, stable, O_2-absorbing C*-algebra is isomorphic to a crossed product of a C*-algebra D with the integers by an action alpha, where D is an inductive limit of C*-algebras of the form M_k(C_0…
A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits.
Soft inductive limits of operator systems and a noncommutative Lazar-Lindenstrauss theorem
本研究提出算子系统软归纳极限的推广概念，放宽传递性条件并支持任意有向集指标，证明可分算子系统为核算子系统当且仅当其是矩阵代数的归纳极限，将经典Lazar-Lindenstrauss定理推广至非交换Choquet理论领域。
We present a systematic development of inductive limits in the categories of ordered *-vector spaces, Archimedean order unit spaces, matrix ordered spaces, operator systems and operator C*-systems. We show that the inductive limit intertwines the operation of passing to the maximal operator system structure of an Archimedean order unit space, and that the same holds true for the minimal operator system structure if the connecting maps are complete order embeddings. We prove that the inductive limit commutes with the operation of taking the maximal tensor product with another operator system, a…
Inductive limits of compact quantum metric spaces
本文研究紧量子度量空间的归纳极限问题，通过态空间的逆极限构造了归纳极限上的紧量子度量空间结构，并给出了两个归纳极限Lipschitz同构的充分条件。
We introduce a topology on the ideal space of inductive limits of C*-algebras built by a topological inverse limit of the Fell topologies on the C*-algebras of the given inductive sequence and we produce conditions for when this topology agrees with the Fell topology of the inductive limit. With this topology, we impart criteria for when convergence of ideals of an AF algebra can provide convergence of quotients in the quantum Gromov-Hausdorff propinquity building off previous joint work with Latremoliere. These findings bestow a continuous map from a class of ideals of the Boca-Mundici AF alg…
Let $G$ be an inductive limit of finite cyclic groups and let $A$ be a unital simple projectionless C*-algebra with $K_1(A) \cong G$ and with a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in $\Aut(A)/\WInn(A)$ are conjugate, where $\WInn(A)$ means the subgroup of $\Aut(A)$ consisting of automorphisms which are inner in the tracial representation. In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple AT-algeb…
In this paper, we give a K-theoretic classification of real ranks zero inductive limits of generalized dimension drop interval algebras.
A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper, we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.
We show that finitely generated subhomogeneous C*-algebras have finite decomposition rank. As a consequence, any separable ASH C*-algebra can be written as an inductive limit of subhomogeneous C*-algebras each of which has finite decomposition rank. It then follows from work of H. Lin and of the second named author that the class of simple unital ASH algebras which have real rank zero and absorb the Jiang-Su algebra tensorially satisfies the Elliott conjecture.
The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $\mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.
For an arbitrary countable directed graph E we show that the only possible values of the stable rank of the associated Cuntz-Krieger algebra C*(E) are 1, 2 or \infty. Explicit criteria for each of these three cases are given. We characterize graph algebras of type I, and graph algebras which are inductive limits of C*-algebras of type I. We also show that a gauge-invariant ideal of a graph algebra is itself isomorphic to a graph algebra.
For unital AF algebras with faithful tracial states, we provide criteria for convergence in the quantum Gromov-Hausdorff propinquity. For any unital AF algebra, we introduce new Leibniz Lip-norms using quotient norms. Next, we show that any unital AF algebra is the limit of its defining inductive sequence of finite dimensional C*-algebras in quantum propinquity given any Lip-norm defined on the inductive sequence. We also establish sufficient conditions for when a *-isomorphism between two AF algebras produces a quantum isometry in the context of various Lip-norms.
We construct spectral triples on a class of particular inductive limits of matrix-valued function algebras. In the special case of the Jiang-Su algebra we employ a particular $AF$-embedding.
We study the limits of inductive sequences (A_i,ϕ_i) where each A_i is a direct sum of full matrix algebras over compact metric spaces and each partial map of ϕ_i is diagonal. We give a new characterisation of simplicity for such algebras, and apply it to prove that the said algebras have stable rank one whenever they are simple and unital. Significantly, our results do not require any dimension growth assumption.
The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $\mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.
Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence
本文在Latremoliere的propinquity框架下，从度量角度研究了谱三元组的等距群与双Lipschitz自同构集，证明了其紧致性，并给出了归纳极限下等距群作用在协变Gromov-Hausdorff propinquity意义下收敛的充要条件。
Let $A$ be an $AH$ algebra $A=\lim\limits_{n\to \infty}(A_{n}=\bigoplus\limits_{i=1}\limits^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m})$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. In this article, we will classify all $AH$ algebras with ideal property of no dimension growth---that is, $sup_{n,i}dim(X_{n,i})<+\infty$. This result gener…
We show that Bunce-Deddens algebras, which are AT-algebras, are also limits of circle algebras for Rieffel's quantum Gromov-Hausdorff distance, and moreover, form a continuous family indexed by the Baire space. To this end, we endow Bunce-Deddens algebras with a quantum metric structure, a step which requires that we reconcile the constructions of the Latremoliere's Gromov-Hausdorff propinquity and Rieffel's quantum Gromov-Hausdorff distance when working on order-unit quantum metric spaces. This work thus continues the study of the connection between inductive limits and metric limits.
A C*-diagonal in the Jiang-Su algebra via entangled matrix cones
通过将蒋-苏代数构造为维数下降代数的归纳极限，把维数下降代数描述为纠缠矩阵锥来定义连接*同态，得到有一维非局部连通谱的C* - 对角线，还给出卡当对中正则化子的新刻画。
We show that it is consistent with ZFC that there is a simple nuclear non-separable C*-algebra which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O_2, or of the CAR algebra.
C$^*$-diagonals in AH-algebras arising from generalized diagonal connecting maps: spectrum and uniqueness
本文将 Bratteli 型图关联到源于广义对角连接映射的 AH-代数，通过引入谱不完备概念刻画谱连通分量，证明了部分 AI-代数不具备唯一的归纳极限 Cartan 子代数，并探讨了 AF-代数的唯一性。
Let $(\clb,λ_t,ψ)$ be a $C^*$-dynamical system where $(λ_t: t \in \IT_+)$ be a semigroup of injective endomorphism and $ψ$ be an $(λ_t)$ invariant state on the $C^*$ subalgebra $\clb$ and $\IT_+$ is either non-negative integers or real numbers. The central aim of this exposition is to find a useful criteria for the inductive limit state $\clb \raro^{λ_t} \clb$ canonically associated with $ψ$ to be pure. We achieve this by exploring the minimal weak forward and backward Markov processes associated with the Markov semigroup on the corner von-Neumann algebra of the support projection of the state…
K-Theory for hermitian symmetric spaces of non-compact type, as developed recently by the authors, allows to put Cartan's classification into a homological perspective. We apply this method to the case of inductive limits of finite dimensional hermitian symmetric spaces. This might be seen as an indication of how much more powerful the homological theory is in comparison to the more classical approach. When seen from high above, we follow the path laid out by a similar result in the theory of C*-algebras. Important is a clear picture of the behavior of morphisms between bounded symmetric domai…
Spectral Triples on a non-standard presentation of Effros-Shen AF algebras
本文针对 Mitscher-Spielberg 以路径范畴给出的 Effros-Shen AF 代数非标准归纳极限表示，借鉴 Christensen-Ivan 构造定义并研究相应的谱三元组。
A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C*-algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra $A$ of $B(\ell_2)$, which is the inductive limit of length $ω_1$ of its separable stable ideals $I_α$ ($α<ω_1$) satisfying $I_{α+1}/I_α\cong K(\ell_2)$ for each $α<ω_…
We show that unital simple C*-algebras with tracial topological rank zero which are locally approximated by subhomogeneous C^-algebras can be classified by their ordered $K$-theory. We apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered $K$-theory. In particular, irrational higher dimensional non-commutative tori of the form $C({\mathbb T}^k)\times_θ{\mathbb Z}$ are in fact inductive limits of circle algebras.
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