Bowen–Franks Groups for Subshifts and Ext-Groups for C*-Algebras

We generalize the Bowen–Franks groups for topological Markov shifts to general subshifts as the Ext-groups for the associated C ^*-algebras. The generalized Bowen–Franks groups for subshifts are shown to be invariant under flow equivalence and, hence, invariant under topological conjugacy. They are regarded as the indices of Fredholm operators related to extensions of the associated C ^*-algebras so that they are described in terms of symbolic dynamical systems. In particular, the group for a sofic subshift is determined by the adjacency matrix of its left Krieger cover graph. The Bowen–Franks groups for some non sofic subshifts are calculated, proving that certain subshifts with the same topological entropy are not flow equivalent.     

A Class of <Emphasis Type="Italic">C</Emphasis>*-algebras with non-stable <Emphasis Type="Italic">K</Emphasis><Subscript>1</Subscript>-group property

In this paper, we give a class of C*-algebras with non-stable K ₁-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka.     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.     

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C ^∗-algebra.     

Characterization of Geodesic Flows on T 2 with and without Positive Topological Entropy

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T ² for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \mathbbR²{\mathbb{R}^{2}} . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T ².     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

The K-Theory of C *-Algebras with Finite Dimensional Irreducible Representations

We study the K-theory of unital C^*-algebras A satisfying the condition that all irreducible representations are finite and of some bounded dimension. We construct computational tools, but show that K-theory is far from being able to distinguish between various interesting examples. For example, when the algebra A is n-homogeneous, i.e., all irreducible representations are exactly of dimension n, then K_*(A) is the topological K-theory of a related compact Hausdorff space, this generalises the classical Gelfand-Naimark theorem, but there are many inequivalent homogeneous algebras with the same related topological space. For general A we give a spectral sequence computing K_*(A) from a sequence of topological K-theories of related spaces. For A generated by two idempotents, this becomes a 6-term long exact sequence.     

The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C^*-algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero. Dedicated to the memory of Peter Slodowy     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

Sur la classification des automorphismes périodiques des C-algèbres UHF

We obtain a complete classification up to conjugacy and up to outer conjugacy of finite tensor product type automorphisms of UHF C^*-algebras which are periodic of period N (N prime).     

A complex semigroup approach to group algebras of infinite dimensional Lie groups

A host algebra of a topological group G is a C ^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C ^*-algebra C ^*(S,α), and we describe a setting which permits us to conclude that this C ^*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future. To K.H. Hofmann on the occasion of his 75th birthday     

Complete positivity,tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C^*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology v on A⊗B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C^*-algebraA is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C^*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.     

K-theory for rickart C*-algebras

We give an explicit index map for any properly infinite closed ideal of a Rickart C ^*-algebra. This generalizes Olsen's work on von Neumann algebras. We use our results to compute the topological and the algebraic K ₁-groups of any quotient algebra of a Rickart C ^*-algebra.     

Bounded and Unitary Elements in Pro-C*-algebras

A pro-C*-algebra is a (projective) limit of C*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C*-algebras can be seen as non-commutative k-spaces. An element of a pro-C*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C*-algebra. In this paper, we investigate pro-C*-algebras from a categorical point of view. We study the functor (−) _b that assigns to a pro-C*-algebra the C*-algebra of its bounded elements, which is the dual of the Stone-Čech-compactification. We show that (−) _b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand duality for commutative unital pro-C*-algebras is also presented.     

Full groups of Cantor minimal systems

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K ⁰-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC ^*-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics. Research supported in part by operating grants from NSERC (Canada). Research supported in part by the Norwegian Research Council for Science and Humanities.     

The Adjoint Action of an Expansive Algebraic -Action

 Let , and let α be an expansive -action by continuous automorphisms of a compact abelian group X with completely positive entropy. Then the group of homoclinic points of α is countable and dense in X, and the restriction of α to the α-invariant subgroup is a -action by automorphisms of . By duality, there exists a -action by automorphisms of the compact abelian group : this action is called the adjoint action of α. We prove that is again expansive and has completely positive entropy, and that α and are weakly algebraically equivalent, i.e. algebraic factors of each other. A -action α by automorphisms of a compact abelian group X is reflexive if the -action on the compact abelian group adjoint to is algebraically conjugate to α. We give an example of a non-reflexive expansive -action α with completely positive entropy, but prove that the third adjoint is always algebraically conjugate to . Furthermore, every expansive and ergodic -action α is reflexive. The last section contains a brief discussion of adjoints of certain expansive algebraic -actions with zero entropy. Received 11 June 2001; in revised form 29 November 2001     

Entropy structure

Investigating the emergence of entropy on different scales, we propose an “entropy structure” as a kind of master invariant for the entropy theory of topological dynamical systems. An entropy structure is a sequence of functionsh _k on the simplex of invariant measures which converges to the entropy functionh and which falls into a distinguished equivalence class defined by a natural equivalence relation capturing the “type of nonuniformity in convergence”. An entropy structure recovers several existing invariants, including the symbolic extension entropy h_sex and the Misiurewicz parameter h^*. Entropy theories of Misiurewicz, Katok, Brin—Katok, Newhouse, Romagnoli, Ornstein—Weiss and others all yield candidate sequences (h _k); we determine which of these exhibit the correct type of convergence and hence become entropy structures. One of the satisfactory sequences arises from a new treatment of entropy theory strictly in terms of continuous functions (in place of partitions or covers). The results allow the computation of symbolic extension entropy without reference to zero dimensional extensions. New light is shed on the property of asymptotich-expansiveness. Supported by the KBN grant 2 P03 A 04622.     

关于连续映射半群拓扑熵的一点注记

本文研究了度量空间中连续映射构成半群的拓扑熵.利用Patr′ao~([8])的方法,给出了度量空间中两种有限个连续映射构成的半群的拓扑d-熵的定义,比较了两种拓扑d-熵的大小.证明了局部紧致可分度量空间上有限个真映射构成的半群的拓扑d-熵和它的一点紧化空间上对应的拓扑熵相等.上面结果推广了Patr′ao的相应结论.     

Linear functionals on nonlinear spaces and applications to problems from viscoplasticity theory

A classical result in the theory of monotone operators states that if C is a reflexive Banach space, and an operator A: C→C^* is monotone, semicontinuous and coercive, then A is surjective. In this paper, we define the ‘dual space’ C^* of a convex, usually not linear, subset C of some Banach space X (in general, we will have C^*⊃X^*) and prove an analogous result. Then, we give an application to problems from viscoplasticity theory, where the natural space to look for solutions is not linear. Copyright © 2007 John Wiley & Sons, Ltd.