Marie Choda, Ken Dykema
In the reduced free product of C*-algebras (A,phi)=(A_1,phi_1)*(A_2,phi_2), A is shown to be purely infinite and simple under the hypothesis that A_1 is the crossed product of a C*-algebra by a discrete infinite group, phi_1 is well behaved with respect to this crossed product and A_2 is not one dimensional.
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Type III_1 factors arising as (direct summands of) von Neumann algebraic free products are investigated. In particular we compute Connes' Sd- and tau- invariants for those type III_1 factors without any extra assumption.
Certain reduced free products of C*-algebras, (A,phi)=(A_1,phi_1)*(A_2,ϕ_2), taken with respect to faithful states, at least one of which is not a trace, are shown to be purely infinite and simple. It is assumed that one of the A_i contain a partial isometry in the spectral subspace of phi_i corresponding to a positive number not equal to one. For example, if A_1 and A_2 are copies of the two-by-two complex matrices and if phi_1 and phi_2 are not unitarily conjugate, it is shown that A is simple and purely infinite.
Examples of simple, separable, unital, purely infinite $C^*$--algebras are constructed, including: (1) some that are not approximately divisible; (2) those that arise as crossed products of any of a certain class of $C^*$--algebras by any of a certain class of non--unital endomorphisms; (3) those that arise as reduced free products of pairs of $C^*$--algebras with respect to any from a certain class of states.
We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is…
Generalisations of Thompson's group V arising from purely infinite groupoids
研究源于纯无穷极小广群拓扑全群的Thompson群V推广，证明导出子群性质并给出群论刻画，同时完整抽象刻画此类群并描述Brin-Higman-Thompson群的特征。
We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors.
Let A be a simple, sigma-unital, non-unital C*-algebra, with metrizable tracial simplex T(A), which is projection-surjective and injective and has strict comparison of positive elements by traces. Then the following are equivalent: (i) A has quasicontinuous scale; (ii) The multiplier algebra M(A) has strict comparison of positive elements by traces; (iii) The coronal algebra M(A)/A is purely infinite; (iii') The quotient M(A)/Imin is purely infinite; (iv) M(A) has finitely many ideals; (v) Imin=Ifin. If furthermore algebra of n by n matrices of A is projection-surjective and injective for ever…
By using Ocneanu's result on the classification of all irreducible connections on the Dynkin diagrams, we show that the dual principal graphs as well as the fusion rules of bimodules arising from any Goodman-de la Harpe-Jones subfactors are obtained by a purely combinatorial method. In particular we obtain the dual principal graph and the fusion rule of bimodules arising from the Goodman-de la Harpe-Jones subfactor corresponding to the Dynkin diagram $E_8$. As an application, we also show some subequivalence among $A$-$D$-$E$ paragroups.
We prove some unique prime factorization results for tensor products of type $II_1$ factors of the form $Γ_q(\mathbb{C}, S \otimes H)$ arising from symmetric independent copies with sub-exponential dimensions of the spaces $D_k(S)$ and dim$(H)$ finite and greater than a constant depending on $q$.
In this paper, we introduce a $C^{\ast}$-algebra associated with a proper primitive substitution. We show that the $C^{\ast}$-algebra is simple and purely infinite and contains the associated Cuntz-Krieger algebra and the crossed product $C^{\ast}$-algebra of the corresponding Cantor minimal system. We calculate the $K$-groups.
We introduce a new construction of $E_0$-semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of $C_0$-semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III $E_0$-semigroups, which can not be distinguished from $E_0$-semigroups of type I by the invariants introduced by Boris Tsirelson. Finally, by considering the local von Neumann algebras, and by associating a type III facto…
Let ${\cal O}_n$ denote the Cuntz algebra for $n\geq 2$. We introduce an embedding $f$ of ${\cal O}_m$ into ${\cal O}_n$ arising from a geometric progression of Cuntz generaters of ${\cal O}_n$. By identifying ${\cal O}_m$ with $f({\cal O}_m)$, we extend Cuntz states on ${\cal O}_m$ to ${\cal O}_n$. We show (i) a necessary and sufficient condition of the uniqueness of the extension, (ii) the complete classification of all such extensions up to unitary equivalence of their GNS representations, and (iii) the decomposition formula of a mixing state into a convex hull of pure states. The complete …
Let ${\cal O}_{{\cal H}^{A,B}_κ}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^κ$ is transitive, the $C^*$-algebra ${\cal O}_{{\cal H}^{A,B}_κ}$ is simple and purely infinite. In particulr, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra ${\cal O}_{{\cal H}^{[N],[M]}_κ}$ are computed by using the Euclidean algorithm.
Boris Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gausian spaces, measure type spaces and `slightly coloured noises', using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the `Shale map' respects compositions (This settles an old conjecture of K. R. Parthasarathy). Using this we associate a product…
The Selfless Dichotomy
本文填补了无私C*-概率空间中稳定有限/纯无限二分法的空白，证明非忠实无私C*-概率空间是纯无限且单的，从而完成二分法，并改进永久性质，推进Choda-Dykema猜想。
In this paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff étale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff étale minimal topological principal groupoids. This generalizes the dichotomy obtained by Bönicke-Li and Rainone-Sims. We a…
General solidity phenomena and anticoarse spaces for type $\mathrm{III}_1$ factors
本研究构建纯无限情形下的反粗空间理论，结合1-有界熵与随机矩阵强收敛结果，证明自由Araki–Woods因子是首类满足广泛不可分解性的$\rm III$型因子，涵盖强坚实性与Peterson–Thom性质。
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…
Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger $C^*$-algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the $C^*$-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their $K$-theory. We find examples whose $C^*$-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic …
We present a class of subshifts $Z_N, N = 1,2,...$ whose associated $C^*$-algebras ${\cal O}_{Z_N}$ are simple, purely infinite and not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The class of the subshifts is the first examples whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The subshifts $Z_N$ are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that KMS-state for gauge action on the associated $C^*$-algebra ${\cal O}_{Z_N}$ exists if and only if …
We investigate the ideal structures of the C^*-algebras arising from topological graphs. We give the complete description of ideals of such C^*-algebras which are invariant under the so-called gauge action, and give the condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our C^*-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph such that the associated C^*-algebra is prime but not primitive.
Let $A$ be an $N \times N $ irreducible matrix with entries in $\{0,1\}$. We define the topological Markov Dyck shift $D_A$ to be a nonsofic subshift consisting of the $2N$ brackets $(_1,...,(_N,)_1,...,)_N$ with both standard bracket rule and Markov chain rule coming from $A$. The subshift is regarded as a subshift defined by the canonical generators $S_1^*,..., S_N^*, S_1,..., S_N $ of the Cuntz-Krieger algebra ${\Cal O}_A$. We construct an irreducible $λ$-graph system ${{\frak L}^{Ch(D_A)}}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^*$-algebra …
Von Neumann algebras arising from Bost-Connes type systems
该研究证明数域Bost-Connes型系统与Connes-Marcolli GL₂系统特定β区间的KMS_β态为III₁型，等价于阿代尔空间上相关作用的遍历性，并给出GL₂系统β=2时的对应实例。
Let $α: G\curvearrowright X$ be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action $α$ is topologically free and has no $G$-invariant regular Borel probability measure on $X$, dynamical comparison implies that the reduced crossed product of $α$ is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and t…
Factorizing random sets and type III Arveson systems
本文提出通过无限乘积构造类型III阿瓦索系统的方法，并应用布朗运动零集分析验证了该方法的有效性。
In the given article it is introduced new notions of a C$^*$-algebra of von Neumann type I and C$^*$-algebras of types I$_n$, II, II$_1$, II$_\infty$ and III. It is proved that any GCR-algebra is a C$^*$-algebra of von Neumann type I, and a C$^*$-algebra is an NGCR-algebra if and only if this C$^*$-algebra does not have a nonzero Abelian annihilator. Also an analog of the theorem on decomposition of a von Neumann algebra to subalgebras of types I, II and III is proved. In the final part it is proved that every C$^*$-factor of von Neumann type I is a C$^*$-algebra of type I$_n$ for some cardina…
We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III_1) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel-Lepowsky-Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is…
We compute the index of the inclusions of type $III_{1/2}$ factors arising from endomorphisms of the Cuntz algebra ${\mathcal O}_2$ associated to the rank-two permutation matrices.
We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras $M = B \rtimes Γ$ arising from arbitrary actions $Γ\curvearrowright B$ of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra $N' \cap M^ω$ of any nonamenable von Neumann subalgebra with normal expectation $N \subset M$. We use this result to show that for any strongly ergodic essentially free nonsingular action $Γ\curvearrowright (X, μ)$ of any bi-exact countable discrete group on a standard proba…
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…
Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
本文通过引入空间方向分解，将Smale空间étale群胚的同调不变量研究推广到超越稳定不连通情形，证明Putnam同调与Crainic-Moerdijk群胚同调同构，并给出不稳定C*-代数K群的谱序列计算方法。
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined…
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