Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$ -algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$ .

The paper introduces a general method to construct conformal measures for a local homeomorphism on a locally compact non-compact Hausdorff space, subject to mild irreducibility-like conditions. Among other things, the method is used to give necessary and sufficient conditions for the existence of eigenmeasures for the dual Ruelle operator associated to a locally compact non-compact irreducible Markov shift equipped with a uniformly continuous potential function. As an application to operator algebras the results are used to determine for which ${\it\beta}$ there are gauge invariant ${\it\beta}$-KMS weights on a simple graph $C^{\ast }$-algebra when the one-parameter automorphism group is given by a uniformly continuous real-valued function on the path space of the graph.

We consider a construction of ${C}^{\ast } $-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the ${C}^{\ast } $-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the $K$-theory groups and turn them into an algorithmic method for Markov maps.

It is shown that simplicity of the crossed product of a unital $\text{AH}$-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection.

In this paper we investigate the ideal structure of the C*-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, then it is also purely infinite.

D. Ruelle and I. Putnam have constructed three C*-algebras from the homoclinic and heteroclinic structure of a Smale space. This paper gives gives a complete description of these algebras when the Smale space is one of the generalized one-dimensional solenoids studied by R. Williams and I. Yi.

This is a study of the invariant measures of a synchronized subshift (in particular, a sofic shift). Among other results we give necessary and sufficient conditions for the existence of a maximal measure of full support, and a method to determine if the shift is intrinsically ergodic.

We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

We investigate a natural connection between inductive limits of $C^*$-algebras, ergodic theory and the theory of subfactors.

The defect of a factor map out of a totally disconnected dynamical system is a number in $[0,\infty]$ which gives a quantitative measure of how well the dynamical behaviour of the factor agrees with that of the extension. The larger the defect, the more they differ. In this paper we obtain the basic general properties of the defect and calculate it in a series of specific cases.

We consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

We prove that a free discrete transformation group on a compact connected space X is completely determined by the way C(X) lies inside the corresponding crossed product C*-algebra.

For a homogeneous C*-algebra we identify the quotient of the automorphism group by the locally unitary automorphisms as a subgroup of the homeomorphisms of the spectrum. We sharpen a known criterion on the spectrum that ensures that all locally unitary automorphisms of the algebra are inner.

The effect of long-term lithium treatment on the kidneys has generated concern among psychiatrists, and proposals have been made that routine determinations of serum lithium and serum TSH should be supplemented with control of the kidney function through regular determinations of serum creatinine, glomerular filtration rate, and renal concentrating ability, as well as through kidney biopsy in certain circumstances.