{\mathbb{Z}}-Actions on AH Algebras and {\mathbb{Z}^2}-Actions on AF Algebras

We consider \mathbbZ{\mathbb{Z}}-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two \mathbbZ{\mathbb{Z}}-actions with the Rohlin property on such a C*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer \mathbbZ²{\mathbb{Z}^2}-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants.     

Outer automorphisms and reduced crossed products of simpleC*-algebras

Every outer automorphism of a separable simpleC*-Algebra is shown to have a pure state which is mapped into an inequivalent state under this automorphism. The reduced crossed product of a simpleC*-algebra by a discrete group of outer automorphisms is shown to be simple.     

Cyclic cocycles from graded KMS functionals

Each graded KMS functional of aZ/2-gradedC*-algebra with respect to a supersymmetric one-parameter automorphism group gives rise to a cyclic cocycle.     

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C ^*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor ^*-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C ^*-algebra. As an application, we associate an ergodic G-action on a unital C ^*-algebra to an inclusion of Rep(G) into an abstract tensor C ^*-category . If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G. If a tensor C ^*-category has a Hecke symmetry making an object ρ of dimension d and μ-determinant 1, then there is an ergodic action of S _μ U(d) on a unital C ^*-algebra having the as its spectral subspaces. The special case of is discussed.     

On the possible temperatures of a dynamical system

A simpleC*-algebra and a continuous one-parameter automorphism group are constructed such that the set of inverse temperatures at which there exist equilibrium states (i.e., KMS states, or, for =±, ground or ceiling states) is an arbitrary closed subset of IR{±}.With partial support of the National Science Foundation     

The automorphism group of the irrational rotation C*-algebra

The structure of the automorphism group of a simple C^*-algebra of real rank zero which is an inductive limit of circle algebras is described. In particular, it is proved that the automorphism group of the irrational rotation C^*-algebra,A , for any irrational number , is an extension of a topologically simple group by GL₂().Dedicated to Huzihiro Araki     

Effect Algebras with the Riesz Decomposition Property and AF C*-Algebras

Relations between effect algebras with Riesz decomposition properties and AF C^*-algebras are studied. The well-known one-one correspondence between countable MV-algebras and unital AF C^*-algebras whose Murray-von Neumann order is a lattice is extended to any unital AF C^* algebras and some more general effect algebras having the Riesz decomposition property. One-one correspondence between tracial states on AF C^*-algebras and states on the corresponding effect algebras is proved. In particular, pure (faithful) tracial states correspond to extremal (faithful) states on corresponding effect algebras.     

Dimension and Dynamical Entropy for Metrized <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-Algebras

 We introduce notions of dimension and dynamical entropy for unital C ^* -algebras ``metrized' by means of , which are complex-scalar versions of the Lip-norms constitutive of Rieffel's compact quantum metric spaces. Our examples involve the UHF algebras and noncommutative tori. In particular we show that the entropy of a noncommutative toral automorphism with respect to the canonical coincides with the topological entropy of its commutative analogue. Received: 13 February 2002 / Accepted: 20 August 2002 Published online: 22 November 2002     

Interactions in Noncommutative Dynamics

A mathematical notion of interaction is introduced for noncommutative dynamical systems, i.e., for one parameter groups of *-automorphisms of endowed with a certain causal structure. With any interaction there is a well-defined “state of the past” and a well-defined “state of the future”. We describe the construction of many interactions involving cocycle perturbations of the CAR/CCR flows and show that they are nontrivial. The proof of nontriviality is based on a new inequality, relating the eigenvalue lists of the “past” and “future” states to the norm of a linear functional on a certain C ^*-algebra. To the memory of Irving Segal Received: 12 October 1999 / Accepted: 21 October 1999     

A note on regular states and supplementary conditions

We show that linear Hermitian supplementary conditions can never be imposed in a representation associated with a regular state on the C ^*-algebra of the CCRs. Nevertheless, there is a well-defined method for imposing the constraints in an abstract C ^*-framework, which yields as its final physical algebra a CCR C ^*-algebra, on which one can again require its physical states to be regular. These states derive from states on the original C ^*-algebra which are regular up to nonphysical quantities.     

The Ordered K-Theory of C/*-Algebras¶Associated with Substitution Tilings

We consider the C ^*-algebra, A _T , constructed from a substitution tiling system which is primitive, aperiodic and satisfies the finite pattern condition. Such a C ^*-algebra has a unique trace. We show that this trace completely determines the order structure on the group K ₀(A _T ); a non-zero element in K ₀(A _T ) is positive if and only if its image under the map induced from the trace is positive. Received: 12 August 1999 / Accepted: 2 May 2000     

CQG algebras: A direct algebraic approach to compact quantum groups

The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to aC ^*-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.     

Poisson Lie groups and pentagonal transformations

Symplectic pentagonal transformations are intimately related to global versions of Poisson Lie groups (Manin groups, S ^*-groups, or symplectic pseudogroups). Symplectic pentagonal transformations of cotangent bundles, preserving the natural polarization, are shown to be in one to one correspondence with pentagonal transformations of the base manifold with a cocycle (if the base is connected and simply connected). By the results of Baaj and Skandalis, this allows to quantize (at the C ^*-algebra level!) those Poisson Lie groups, whose associated symplectic pentagonal transformation admits an invariant polarization. The (2n)²-parameter family of Poisson deformations of the (2n+1)-dimensional Heisenberg group described by Szymczak and Zakrzewski is shown to fall into this case.Supported by Alexander von Humboldt Foundation. On leave from Department of Mathematical methods in Physics, Warsaw University, Poland.     

Deformation Quantization for Hilbert Space Actions

Rieffel's theory of deformations of C^*-algebras for -actions can be extended to actions of infinite-dimensional Hilbert spaces. The CCR algebra over a Hilbert space H can be exhibited in this manner as a deformation of a commutative C^*-algebra of almost periodic functions on H. Received: 26 August 1996 / Accepted: 28 January 1997     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

Dynamics and potentials

A dynamics (i.e. a one-parameter group of automorphisms) of a system described by a C*-algebra with a local structure in terms of C*-subalgebras A(I) for local domains I of the physical space (a discrete lattice) is normally constructed out of potentials P(I), each of which is a self-adjoint element of the subalgebra A(I), such that the the first time derivative of the dynamical change of any local observable A is i times the convergent sum of the commutator [P(I), A] over all finite regions I. We will invert this relation under the assumption (obviously assumed in the usual approach) that local observables all have the first time derivative, i.e. we prove the existence of potentials for any given dynamics satisfying the above-stated condition. Furthermore, by imposing a further condition for the potential P(I) to be chosen for each I that it does not have a portion which can be shifted to potentials for any proper subset of I, we also show (1) the existence, (2) uniqueness, (3) an automatic convergence property for the sum over I, and (4) a quite convenient property for the chosen potential. The so-obtained properties (3) and (4) are not assumed and are very useful, though they were never noticed nor used before.     

C*-Algebras associated with one dimensional almost periodic tilings

For each irrational number, 0<α<1, we consider the space of one dimensional almost periodic tilings obtained by the projection method using a line of slope α. On this space we put the relation generated by translation and the identification of the “singular pairs”. We represent this as a topological spaceX _α with an equivalence relationR _α. OnR _α there is a natural locally Hausdorff topology from which we obtain a topological groupoid with a Haar system. We then construct the C^*-algebra of this groupoid and show that it is the irrational rotation C^*-algebra,A _α. Research supported by the Natural Sciences and Engineering Research Council of Canada and the Fields Institute for Research in Mathematical Sciences.     

Clustering for algebraicK-systems

We prove that for a von Neumann algebra that is an algebraicK system with respect to some automorphism, the invariant state isK-clustering andr-clustering. Further, we study by using examples how far the von Neumann algebra inherits theK property from the underlyingC ^* algebra.     

A remark onC*-algebras

It is shown that any complex Banach algebra with hermitean involution and the weakC*-property |x|²=|x ²| for allx=x* is aC*-algebra.The research in this paper was partially supported by the U. S. Army Research Office, Durham.     

Some Remarks on Group Bundles and <Emphasis Type="Italic">C</Emphasis>* Dynamical Systems

We introduce the notion of fibred action of a group bundle on a C(X)-algebra. By using such a notion, a characterization in terms of induced C*-bundles is given for C*-dynamical systems such that the relative commutant of the fixed-point C*-algebra is minimal (i.e., it is generated by the centre of the given C*-algebra and the centre of the fixed-point C*-algebra). A class of examples in the setting of the Cuntz algebra is given, and connections with superselection structures with nontrivial centre are discussed. The author was partially supported by the European Network “Quantum Spaces - Noncommutative Geometry” HPRN-CT-2002-00280.