Kenneth R. Davidson, Dilian Yang
Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\ca(\Fth)$. The periodic C*-algebras are characterized, and it is shown that $\ca(\Fth) \simeq \rC(\bT) \otimes \fA$ where $\fA$ is a simple C*-algebra.
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We study the KMS states of the C*-algebra of a strongly connected finite k-graph. We find that there is only one 1-parameter subgroup of the gauge action that can admit a KMS state. The extreme KMS states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite-path space of the graph. We deduce that there is a unique KMS state if and only if the k-graph C*-algebra is simple, giving a complete answer to a question of Yang. When the k-graph C*-algebra is not simple, our results reveal a phase change of an unexpected nature…
It is shown that no local periodicity is equivalent to the aperiodicity condition for arbitrary finitely-aligned k-graphs. This allows us to conclude that C*(Λ) is simple if and only if Λis cofinal and has no local periodicity.
We catalogue the primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources. Each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph; the primitive ideals in the Cuntz-Krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group. The Cuntz-Krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic.
We give a proof of Bott periodicity for real graded $C^\ast$-algebras in terms of K- theory and E-theory. Guentner and Higson proved a similar result in the complex graded case but we extend this to cover all graded $C^\ast$-algebras. We obtain the 8-fold periodicity in E-theory by constructing two maps that are inverse to each other.
We give a proof of the Bott periodicity theorem for topological K-theory of C*-algebras based on Loring's treatment of Voiculescu's almost commuting matrices and Atiyah's rotation trick. We also explain how this relates to the Dirac operator on the circle; this uses Yu's localization algebra and an associated explicit formula for the pairing between the first K-homology and first K-theory groups of a (separable) C*-algebra.
For a compact Hausdorff space $X$, the space $SC(X\times X)$ of separately continuous complex valued functions on $X$ can be viewed as a $C^*$-subalgebra of $C(X)^{**}\overline\otimes C(X)^{**}$, namely those elements which slice into $C(X)$. The analogous definition for a non-commutative $C^*$-algebra does not necessarily give an algebra, but we show that there is always a greatest $C^*$-subalgebra. This thus gives a non-commutative notion of separate continuity. The tools involved are multiplier algebras and row/column spaces, familiar from the theory of Operator Spaces. We make some study o…
Let $(M,Γ)$ be a Hopf--von Neumann algebra, so that $M_\ast$ is a completely contractive Banach algebra. We investigate whether the product of two elements of $M$ that are both weakly almost periodic functionals on $M_\ast$ is again weakly almost periodic. For that purpose, we establish the following factorization result: If $M$ and $N$ are injective von Neumann algebras, and if $x, y \in M \bar{\otimes} N$ correspond to weakly compact operators from $M_\ast$ to $N$ factoring through reflexive operator spaces $X$ and $Y$, respectively, then the operator corresponding to $xy$ factors through th…
Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph $Λ$ is a natural generalization of a higher rank graph. A pullback of $Λ$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of $Λ$ is obtained by modding out its periodicity $\PerΛ$, which is deduced from a natural equivalence relation on $Λ$. One of our main results in this paper shows that, for a class of higher rank graphs $Λ$, $Λ$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph C*-algebra can be embedded into the tensor product of the gra…
We describe a class of rank-2 graphs whose C^*-algebras are AT algebras. For a subclass which we call rank-2 Bratteli diagrams, we compute the K-theory of the C*-algebra. We identify rank-2 Bratteli diagrams whose C*-algebras are simple and have real-rank zero, and characterise the K-invariants achieved by such algebras. We give examples of rank-2 Bratteli diagrams whose C*-algebras contain as full corners the irrational rotation algebras and the Bunce-Deddens algebras.
For a row-finite higher-rank graph Λ, we construct a higher-rank graph TΛ such that the Toeplitz algebra of Λ is isomorphic to the Cuntz-Krieger algebra of TΛ. We then prove that the higher-rank graph TΛ is always aperiodic and use this fact to give another proof of a uniqueness theorem for the Toeplitz algebras of higher-rank graphs.
We present a generalization of bilateral weighted shift operators for the noncommutative multivariable setting. We discover a notion of periodicity for these shifts, which has an appealing diagramatic interpretation in terms of an infinite tree structure associated with the underlying Hilbert space. These shifts arise naturally through weighted versions of certain representations of the Cuntz C$^*$-algebras $O_n$. It is convenient, and equivalent, to consider the weak operator topology closed algebras generated by these operators when investigating their joint reducing subspace structure. We p…
We study the general and connected stable ranks for C*-algebras. We estimate these ranks for pullbacks of C*-algebras, and for tensor products by commutative C*-algebras. Finally, we apply these results to determine these ranks for certain commutative C*-algebras, and non-commutative CW-complexes.
Complexity Rank of AT Algebras of Real Rank Zero
本文证明实秩零的可分单AT代数复杂度秩至多为一，并解答了Jaime和Willett关于无理旋转代数复杂度秩的问题。
We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
We study the general and connected stable ranks for $C^{\ast}$-algebras. We estimate these ranks for certain $C(X)$-algebras, and use that to do the same for certain group $C^{\ast}$-algebras. Furthermore, we also give estimates for the ranks of crossed product $C^{\ast}$-algebras by finite group actions with the Rokhlin property.
It is well-known that every commutative separable unital C*-algebra of real rank zero is a quotient of the C*-algebra of all compex continous functions defined on the Cantor cube. We prove a non-commutative version of this result by showing that the class of all separable unital C*-algebras of real rank zero concides with the class of quotients of a certain separable unital C*-algebra of real-rank zero.
We introduce the notion of a topological higher-rank graph, a unified generalization of the higher-rank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs, and show that the $C^*$-algebras defined are coherent with the existing theory.
Suppose that $A,B$ are nuclear, separable ${\rm C}^*$-algebras of stable rank one and real rank zero, $A$ is unital simple, $B$ is stable and $({\rm K}_0(B),{\rm K}_0^+(B))$ is weakly unperforated in the sense of Elliott \cite{Ell}. We show that any unital extension with trivial index maps of $A$ by $B$ is absorbing.
We define the relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs. We prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for relative Cuntz-Krieger algebras.
Let $\Fth$ be a $\Bk$-graph on a single vertex. We show that every irreducible atomic $*$-representation is the minimal $*$-dilation of a group construction representation. It follows that every atomic representation decomposes as a direct sum or integral of such representations. We characterize periodicity of $\Fth$ and identify a symmetry subgroup $H_θ$ of $\bZ^\Bk$. If this has rank $s$, then $\ca(\Fth) \cong \rC(\bT^s) \otimes \fA$ for some simple C*-algebra $\fA$.
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 $\mathcal{C}$.
Real rank of some multiplier algebras
本研究构造出可分核实秩零平凡K-理论C*-代数，推翻Brown和Pedersen两个猜想，还计算了两类可数分解因子的稳定乘子与冠代数实秩，结合Zhang结果完成相关实秩计算。
We introduce a class of recursive subhomogeneous algebras that we call diagonal subhomogeneous and we give a notion of diagonal maps between these algebras. We show that any simple limit of diagonal subhomogeneous algebras with diagonal maps has stable rank one. As an application we show that for any minimal homeomorphism of a compact Hausdorff space the associated crossed product has stable rank one.
We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group determines a continuous bundle of twisted higher-rank graph algebras over the dual group. We use this to show that for a circle-valued 2-cocycle on a higher-rank graph obtained by exponentiating a real-valued cocycle, the K-theory of the twisted higher-rank graph algebra coincides with that of the untwisted one.
We introduce a concept of the bounded rank (with respect to a positive constant) for unital C*-algebras as a modification of the usual real rank and present a series of conditions insuring that bounded and real ranks coincide. These observations are then used to prove that for a given $n$ and $K > 0$ there exists a separable unital C*-algebra $Z_{n}^{K}$ such that every other separable unital C*-algebra of bounded rank with respect to $K$ at most $n$ is a quotient of $Z_{n}^{K}$.
Let $A$ be a separable, unital, simple C*-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of $A$ is realized as the rank of an operator in the stabilization of $A$. Assuming moreover that $A$ has locally finite nuclear dimension, we deduce that $A$ is $\mathcal{Z}$-stable if and only if it has strict comparison of positive elements. In particular, the Toms-Winter conjecture holds for separable, unital, simple, approximately subhomogeneous C*-algebras with stable rank one.
We show that every separable C*-algebra of real rank zero that tensorially absorbs the Jiang-Su algebra contains a dense set of generators. It follows that in every classifiable, simple, nuclear C*-algebra, a generic element is a generator.
Periodicity, type $II_1$ factors and free Poisson laws in interacting Fock spaces
本文证明2-周期交互Fock空间中位置算子生成的冯·诺伊曼代数是$II_1$型因子，并证明平方位置算子在真空态下服从Marchenko-Pastur分布，从而在该框架下自然实现自由泊松律。
We consider a family of operator-algebraic dynamical systems involving the Toeplitz algebras of higher-rank graphs. We explicitly compute the KMS states (equilibrium states) of these systems built from small graphs with up to four connected components.
Let $A$ be a unital simple AH-algebra with stable rank one and real rank zero such that $kx=0$ for all $x\in {\rm ker}ρ_A,$ the subgroup of infinitesmal elements in $K_0(A),$ and for the same integer $k\ge 1.$ We show that $A$ has tracial rank zero and is isomorphic to a unital simple AH-algebra with no dimension growth.
We explore various limit constructions for C*-algebras, such as composition series and inverse limits, in relation to the notions of real rank, stable rank, and extremal richness. We also consider extensions and pullbacks. We identify some conditions under which the constructions preserve low rank for the C*-algebras or their multiplier algebras. We also discuss the version of topological dimension theory appropriate for primitive ideal spaces of C*-algebras and provide an analogue for rank of the countable sum theorem of dimension theory. As an illustration of how the main results can be appl…
We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well.
The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.
We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar algebra motivated by Kuperberg's A_2-spider. This A_2-planar algebra contains a subfamily of vector spaces which will capture the double complex structure pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system, including both the periodicity three coming from the A_2-Temperley-Lieb algebra as well as the periodicity two coming from the subfactor basic construction. We use an A_2-planar algebra to obtain a description of the …
Orbit equivalence of higher-rank graphs
本文研究了高秩图的连续轨道等价、最终单侧共轭和双侧共轭，证明了保持周期性的连续轨道等价等价于边界路径群胚同构，并用$C^*$-代数和Kumjian-Pask代数刻画了这些等价关系。
We introduce higher-rank analogues of the Leavitt path algebras, which we call the Kumjian-Pask algebras. We prove graded and Cuntz-Krieger uniqueness theorems for these algebras, and analyze their ideal structure.
For a finite, strongly connected $k$-graph $Λ$, an Huef, Laca, Raeburn and Sims studied the KMS states associated to the preferred dynamics of the $k$-graph $C^*$-algebra $C^*(Λ)$. They found that these KMS states are determined by the periodicity of $Λ$ and a certain Borel probability measure $M$ on the infinite path space $Λ^\infty$ of $Λ$. Here we consider different dynamics on $C^*(Λ)$, which arise from a functor $y: Λ\to \mathbb{R}_+$ and were first proposed by McNamara in his thesis. We show that the KMS states associated to McNamara's dynamics are again parametrized by the periodicity g…
We show that the class of C*-algebras with stable rank greater than a given positive integer is axiomatizable in logic of metric structures. As a consequence we show that the stable rank is continuous with respect to forming ultrapowers of C*-algebras, and that stable rank is Kadison--Kastler stable.
We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.
A necessary and sufficient condition for the simplicity of the C*-algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C*-algebras are proved.
Various notions of stable ranks are studied for topological algebras. Some partial answers to R.G. Swan's problem (Have two Banach or good Fréchet algebras as in the density theorem in K-theory the same stable rank ?) are obtained. For example, a Fréchet dense $\ast$-subalgebra $A$ of a $C^\ast$-algebra $B$, closed under $C^\infty$-functional calculus of self-adjoint elements, has the same Bass stable rank as $B$.
We show that, if A is a separable simple unital C*-algebra which absorbs the Jiang-Su algebra Z tensorially and which has real rank zero and finite decomposition rank, then A is tracially AF in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of C*-algebras as above which, additionally, satisfy the Universal Coefficients Theorem. In particular, such algebras are completely determined by their ordered K-theory. They are approximately homogeneous of topological dimension less than or equal to 3, approximately sub…
We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C*-algebras. The concept is particularly relevant for Elliott's program to classify nuclear C*-algebras by K-theory data. We study some of its properties and show that a simple unital C*-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the Jiang-Su algebra Z tensorially, has tracial rank zero in the sense of Lin. As a consequence, any such C*-algebra, if it additionally satisfies the Universal Coefficients Theorem, is approximately homog…
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