Crossed-products by finite index endomorphisms and KMS states

Given a unital -algebra A, an injective endomorphism preserving the unit, and a conditional expectation E from A to the range of α we consider the crossed-product of A by α relative to the transfer operator L=α⁻¹E. When E is of index-finite type we show that there exists a conditional expectation G from the crossed-product to A which is unique under certain hypothesis. We define a “gauge action” on the crossed-product algebra in terms of a central positive element h and study its KMS states. The main result is: if h>1 and E(ab)=E(ba) for all a,b∈A (e.g. when A is commutative) then the KMS_β states are precisely those of the form ψ=φ°G, where φ is a trace on A satisfying the identity

     

KMS states and complex multiplication

We construct a quantum statistical mechanical system which generalizes the Bost–Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explicit class field theory for such fields. This system admits the Dedekind zeta function as partition function and the idèle class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. The geometric notion underlying the construction is that of commensurability of K-lattices.     

On some subalgebras of von Neumann algebras with analyticity

Let (M,α,G) be a covariant system on a locally compact Abelian group G with the totally ordered dual group which admits the positive semigroup . Let H^∞(α) be the associated analytic subalgebra of M; i.e. . Let be the analytic crossed product determined by a covariant system . We give the necessary and sufficient condition that an analytic subalgebra H^∞(α) is isomorphic to an analytic crossed product related to Landstad's theorem. We also investigate the structure of σ-weakly closed subalgebra of a continuous crossed product N?_θR which contains N?_θR₊. We show that there exists a proper σ-weakly closed subalgebra of N?_θR which contains N?_θR₊ and is not an analytic crossed product. Moreover we give an example that an analytic subalgebra is not a continuous analytic crossed product using the continuous decomposition of a factor of type III_λ(0?λ<1).     

Leavitt path algebras with coefficients in a commutative ring

Given a directed graph E we describe a method for constructing a Leavitt path algebra L_R(E) whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and Cuntz-Krieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both generalize and simplify the classical results for Leavitt path algebras over fields. We also analyze the ideal structure of L_R(E), and we prove that if K is a field, then L_K(E)≅K⊗_ZL_Z(E).     

Lyapunov spectra for KMS states on Cuntz-Krieger algebras

We study relations between (H,β)-KMS states on Cuntz-Krieger algebras and the dual of the Perron-Frobenius operator . Generalising the well-studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one-one correspondence between (H,β)-KMS states and eigenmeasures of for the eigenvalue 1. We then apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups G which may have parabolic elements. We show that for the Cuntz-Krieger algebra arising from G there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen-Series map associated with G. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of G. If G has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with G. The second author was supported by the DFG project “Ergodentheoretische Methoden in der hyperbolischen Geometrie”.     

Counterexamples to Kauffman's conjectures on slice knots

In the late 1960s Jerome Levine classified the odd high-dimensional knot concordance groups in terms of a linking matrix associated to an arbitrary bounding manifold for the knot. His proof fails for classical knots in S³$S^3$. Yet this philosophy has remained the only known strategy for understanding the classical knot concordance group. We show that this strategy is fundamentally flawed. Specifically, in 1982, in support of Levine's philosophy, Louis Kauffman conjectured that if a knot in S³$S^3$ is a slice knot then on any Seifert surface for that knot there exists a homologically essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explicitly expressed in terms of invariants of such curves on its Seifert surface. We give counterexamples to Kauffman's conjecture, that is, we exhibit (smoothly) slice knots that admit (unique minimal genus) Seifert surfaces on which every homologically essential simple closed curve of self-linking zero has non-zero Arf invariant and non-zero signatures.     

HNN extensions of von Neumann algebras

Reduced HNN extensions of von Neumann algebras (as well as C^*-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.     

The Ideal Structure of Toeplitz Algebras

We investigate the ideal structure of the Toeplitz algebra of a totally ordered abelian group . We show that the primitive ideals of are parametrised by the disjoint union of the duals of the order ideals of , and identify the hull-kernel topology on when the chain of orderideals in is isomorphic to a subset of      

Semigroups of isometries,Toeplitz algebras and twisted crossed products

We study theC ^*-algebras generated by projective isometric representations of semigroups, using a dilation theorem and the stucture theory of twisted crossed products. These algebras include the Toeplitz algebras of noncommutative tori recently studied by Ji, and similar algebras associated to the twisted group algebras of other groups such as the integer Heisenberg group.     

Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra

Let M be a type I von Neumann algebra with the center Z, and a faithful normal semi-finite trace τ. Consider the algebra L(M, τ) of all τ-measurable operators with respect to M and let S ₀(M, τ) be the subalgebra of τ-compact operators in L(M, τ). We prove that any Z-linear derivation of S ₀(M, τ) is spatial and generated by an element from L(M, τ).      

Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

For discrete Hecke pairs (G,H), we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G/H on c₀(G/H) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of c₀(G/H) which are multiples of the multiplication representation on ?²(G/H), and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from H to G.     

Entropy of automorphisms arising from dynamical systems through discrete groups with amenable actions

Let G⊂Aut(A) be a discrete group which is exact, that is, admits an amenable action on some compact space. Then the entropy of an automorphism of the algebra A does not change by the canonical extension to the crossed product A×G. This is shown for the topological entropy of an exact C∗-algebra A and for the dynamical entropy of an AFD von Neumann algebra A. These have applications to the case of transformations on Lebesgue spaces.     

Bimodules in crossed products of von Neumann algebras

In this paper, we study bimodules over a von Neumann algebra M   in the context of an inclusion M⊆M_?αG$M\subseteq M{?}_{\alpha }G$, where G is a discrete group acting on a factor M by outer ?-automorphisms. We characterize the M  -bimodules X⊆M_?αG$X\subseteq M{?}_{\alpha }G$ that are closed in the Bures topology in terms of the subsets of G  . We show that this characterization also holds for w^?$w^{?}$-closed bimodules when G has the approximation property (AP  ), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain w^?$w^{?}$-continuous surjective isometric maps on X.     

Remarks on ghost projections and ideals in the Roe algebras of expander sequences

We use repeating sequences of expander graphs or small perturbations of expanders to present examples of ideals in the Roe algebras of bounded geometry discrete metric spaces which cannot be expressed as the sum of a ghost ideal and an ideal in which finite propagation operators are dense. This gives a negative answer to a question in [1, 3]. Received: 9 December 2006, Revised: 18 April 2007     

Free probability on algebras with infinitely many states

We study noncommutative probability spaces endowed with infinite sequences of states. Following ideas of Cabanal-Duvillard we extend the notion of conditional freeness. Free product of such spaces is justified by constructing an appropriate ⋆-representation. Finally, we provide limit theorems and describe the sequences of orthogonal polynomials related to the limit measures. Received: 4 November 1998 / Revised version: 22 April 1999     

Approximate AF flows

When is a flow on a unital AF algebra A such that there is an increasing sequence (A_n) of finite-dimensional -invariant C*-subalgebras of A with dense union, we call an AF flow. We show that an approximate AF flow is a cocycle perturbation of an AF flow.     

Mixing and asymptotic properties of Markov semigroups on von Neumann algebras

In the paper, we investigate weak mixing and ‘approach to equilibrium’ of a Markov semigroup, i.e. a semigroup of linear normal positive unital mappings on a von Neumann algebra. In particular, we show that weak mixing is equivalent to ergodictity and triviality of the (point) spectrum of the semigroup, and give conditions assuring that each normal state tends to an equilibrium state under the action of the semigroup. Received May 6, 1999 / in final form October 21, 1999 / Published online July 20, 2000