Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices

In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras, J. Math. Phys. 46 (2005)], for Cuntz-Krieger algebras OA for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of OA. We use these representations to describe the Perron-Frobenius operator, associated to a nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.     

Interval maps from Cuntz-Krieger algebras

Markov interval maps f naturally produce transition aperiodic 0-1 matrices with extra features. We characterize the 0-1 matrices that can be realized as Markov transition matrices of interval maps, and parametrize the orbit representations (yielded in Correia Ramos et al. (2008) [2]) of the Cuntz-Krieger algebra OA obtained from interval maps with the same matrix A.     

Classification of Cuntz-Krieger algebras

A classification is given of the simple Cuntz-Krieger algebras . It is proved that these algebras are classified up to stable isomorphism by their K₀-group. Thus the sign of the determinant of 1 —A is not an isomorphism invariant. The (non-stabilized) isomorphism type of is determined by K₀( ) together with the position of the class of the unit of .     

KMS States for generalized Gauge actions on Cuntz-Krieger algebras

Given a zero-one matrix A we consider certain one-parameter groups of automorphisms of the Cuntz-Krieger algebra , generalizing the usual gauge group, and depending on a positive continuous function H defined on the Markov space _A. The main result consists of an application of Ruelles Perron-Frobenius Theorem to show that these automorphism groups admit a single KMS state.*Partially supported by CNPq.     

Cuntz-Krieger algebras representations from orbits of interval maps

Let f be an expansive Markov interval map with finite transition matrix Af. Then for every point, we yield an irreducible representation of the Cuntz-Krieger algebra OAf and show that two such representations are unitarily equivalent if and only if the points belong to the same generalized orbit. The restriction of each representation to the gauge part of OAf is decomposed into irreducible representations, according to the decomposition of the orbit.     

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”.     

Generalized Cuntz-Krieger algebras

To a special embedding of circle algebras having the same spectrum, we associate an r-discrete, locally compact groupoid, similar to the Cuntz-Krieger groupoid. Its -algebra, denoted , is a continuous version of the Cuntz-Krieger algebras . The algebra is generated by an AT-algebra and a nonunitary isometry. We compute its K-theory under the assumption that the AT-algebra is simple.

     

Type III factors arising from Cuntz-Krieger algebras

We determine the types of factors arising from GNS-representations of quasi-free KMS states on Cuntz-Krieger algebras. Applying our result to the Cuntz-Krieger algebras arising from the boundary actions of some amalgamated free product groups, we also determine the types of harmonic measures on the boundaries.

     

Graph C *-Algebras and Their Ideals Defined by Cuntz-Krieger Family of Possibly Row-Infinite Directed Graphs

Let E be a possibly row-infinite directed graph. In this paper, first we prove the existence of the universal C^*-algebra C^*(E) of E which is generated by a Cuntz-Krieger E-family {s_e, p_v}, and the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the ideal of C^*(E). Then we get our main results about the ideal structure of Finally the simplicity and the pure infiniteness of is discussed.     

Ternary Derivations of Generalized Cayley–Dickson Algebras

Abstract

Ternary derivations, ternary Cayley derivations and ternary automorphisms are computed over fields of characteristic ≠ 2, 3 for the algebras A _t obtained by the Cayley–Dickson duplication process. While the derivation algebra of A _t stops growing after t = 3, the ternary derivation algebra significantly decreases in the step from the octonions A ₃ to the sedenions A ₄, revealing the symmetry lost on that stage.     

The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A _k k[x]_(x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]_(x).     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$algebras

Let G be a group and assume that (A _p ) _p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δ _p,q : A _pq → A _p ⊗ A _q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ _p,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras. Presented by Ken Goodearl.     

Formal Expansion of Harish-Chandra Homomorphism

Abstract

Let 𝔤 be a complex semisimple Lie algebra with adjoint group G and let 𝔥 be a Cartan subalgebra of 𝔤. Let Â(𝔤) and Â(𝔥) denote the algebra of differential operators with formal power series coefficients on 𝔤 and 𝔥 respectively. We construct a subalgebra A _𝔤 of Â(𝔤) containing all the pull-backs of the differential operators in G attached to any element x in 𝔤. We also consider the projection P: A _𝔤 → Â _𝔥. Then, we calculate explicity the pull-back of the differential operator in G attached to an element h in 𝔥 modulo Ker P.     

The classification of two-component Cuntz-Krieger algebras

Cuntz-Krieger algebras with exactly one nontrivial closed ideal are classified up to stable isomorphism by the Cuntz invariant. The proof relies on Rørdam's classification of simple Cuntz-Krieger algebras up to stable isomorphism and the author's classification of two-component reducible topological Markov chains up to flow equivalence.

     

Quotients of Incidence Algebras and the Euler Characteristic

In this note, we show that, if A ? kQ _A /I _A is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with Q _A , then A ? kΣ if and only if I _A has a generating set consisting of exactly χ(Q _A ) elements, where χ(Q _A ) is the Euler characteristic of Q _A . We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected.     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

Weakly strongly singular integral operators on anisotropic Hardy spaces and their dual operators

Let A be an expansive dilation. We define weakly strongly singular integral kernels and study the action of the operators induced by these kernels on anisotropic Hardy spaces associated with A.     

A Decomposition Theorem for CK1 of Central Simple Algebras

Let A be a central simple algebra over its center F. Define CK₁ A = Coker(K₁ F → K₁ A). We prove that if A and B are F-central simple algebras of coprime degrees, then CK₁(A? _F B) = CK₁ A × CK₁ B.     

Cuntz-Krieger algebras of infinite graphs and matrices

We show that the Cuntz-Krieger algebras of infinite graphs and infinite -matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their -theory. Since the finite approximating graphs have sinks, we have to calculate the -theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.