arXiv subjects
Publications and source records attributed to Klaus Thomsen.
The paper describes a cover of the future cover of a sofic shift which is canonical in the same way as the future cover itself. In some cases the cover is isomorphic to the future cover and in other it is a genuine extension. The extended future cover is then used to introduce and describe the desynchronization of sofic shifts.
The paper contains a new proof of the theorem by Krieger which establishes the canonicity of the future cover of a sofic shift. In addition the paper describes a method to produce new canonical covers from a given one, resulting in canonical covers related to, but generally different from the future cover.
Let $A$ and $B$ be $C^*$-algebras with $A$ separable, let $I$ be an ideal in $B$, and let $ψ\colon A\to B/I$ be a completely positive contractive linear map. We show that there is a continuous family $Θ_t\colon A\to B$, for $t\in [1,\infty)$, of lifts of $ψ$ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If $ψ$ is of order zero, then $Θ_t$ can be chosen to have this property asymptotically. If $A$ and $B$ carry continuous actions of a second countable locally compact group $G$ such that $I$ is $G$-invariant and $ψ$ is equivariant, we show that the family $Θ_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for $ψ$ exists, we can arrange that $Θ_t$ is linear and completely positive for all $t\in [1,\infty)$. In the equivariant setting, if $A$, $B$ and $ψ$ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if $G$ is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.
Generalizing a result of MacDonald we give necessary and sufficient conditions for an arbitrary subshift to embed into an irreducible sofic shift factoring through a given cover by an irreducible subshift of finite type (SFT). We obtain also necessary and sufficient conditions for an arbitrary subshift to embed into an irreducible sofic shift factoring through \emph{some} sliding block code out of an irreducible SFT. We do that when the code is required to be surjective, and hence a factor code, and when it is required to be injective or almost invertible, or is allowed to be arbitrary. These results require concepts of the period of an irreducible sofic shift as well as a concept of a $p$-periodic subshift. Several equivalent formulations of the period are developed.
It is shown that the pairing of the K00 group of a C*-algebra with the densely defined traces of the algebra can be extended to a pairing with the densely defined weights. For traces the pairing can be extended to the K0 group without the semi-continuity assumption occurring in the work of Connes and Elliott.
We present methods to construct flows with varying sets of KMS infinity states on a given simple unital AF-algebra. It follows, for example, that for any pair D and E of non-empty compact metric spaces there is a flow on the CAR algebra whose set of KMS infinity states is homeomorphic to D while the set of KMS infinity states for the inverted flow is homeomorphic to E. Remarkably the flows that realize all such pairs D and E can be chosen to have isomorphic KMS bundles.
We obtain three results: 1) Every compact simplex bundle with exactly one point in the fiber over 0 is the KMS bundle of a periodic flow on the Jiang-Su algebra. 2) Let A be a separable unital C*-algebra with a unique trace state. Suppose that A tensorially absorbs the Jiang-Su algebra. The (weak) cocycle-conjugacy classes of flows that are not approximately inner are uncountable. 3) Let B be a separable, simple, unital, purely infinite and nuclear C*-algebra in the UCT class. Assume that the K1 group of B is torsion free. Every proper simplex bundle with empty fiber over 0 is the KMS bundle of a periodic flow on B.
It is shown that any bundle of KMS state spaces which can occur for a flow on a unital separable C*-algebra with a trace state can also be realized by a flow on any given unital infinite-dimensional simple AF algebra with a tracial state space affinely homeomorphic to the fiber in the bundle over 0.
It is shown that for any infinite dimensional simple unital AF algebra A and any closed lower bounded set K of real numbers containing zero there is a flow on A for which the set of possible inverse temperatures is K.
Let $φ:X\to X$ be a homeomorphism of a compact metric space $X$. For any continuous function $F:X\to \mathbb{R}$ there is a one-parameter group $α^{F}$ of automorphisms on the crossed product $C^*$-algebra $C(X)\rtimes_φ\mathbb{Z}$ defined such that $α^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and $U$ is the canonical unitary in the construction of the crossed product. In this paper we study the 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 $C(X) \rtimes_ϕ \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$.
We exhibit a unital simple mono-tracial AF algebra A with the property that for any compact set K of real numbers containing 0 there is a periodic flow on A such the set of possible inverse temperatures for that flow is K, and for each number in K the corresponding equilibrium state is unique.
We describe the structure of ground states and ceiling states for generalized gauge actions on an UHF algebra. It is shown that both sets are affinely homeomorphic to the state space of a unital AF algebra, and that any pair of unital AF algebras can occur in this way, independently of the field of KMS states. In addtion, we study the KMS-infinity states.
A classical construction associates to a transient random walk on a discrete group $Γ$ a compact $Γ$-space $\partial_M Γ$ known as the Martin boundary. The resulting crossed product $C^*$-algebra $C(\partial_M Γ) \rtimes_r Γ$ comes equipped with a one-parameter group of automorphisms given by the Martin kernels that define the Martin boundary. In this paper we study the KMS states for this flow and obtain a complete description when the Poisson boundary of the random walk is trivial and when $Γ$ is a torsion free non-elementary hyperbolic group. We also construct examples to show that the structure of the KMS states can be more complicated beyond these cases.
We provide a general description of the KMS states for flows whose fixed point algebra satisfies a certain regularity condition. This is the applied to crossed products by discrete groups, and in particular to certain flows on crossed products by discrete abelian groups where the methods can be combined with spectral analysis for abelian automrphism groups.
We calculate the S-invariant of Connes for the von Neumann algebra factors arising from KMS-weights of a generalized gauge action on a simple graph C*-algebra when the associated measure on the infinite path space of the graph is dissipative under the action of the shift.
The paper develops a method to construct one-parameter groups of automorphisms on the CAR C*-algebra with a prescribed field of KMS states.
The paper develops a series of tools for the study of KMS-weights on graph C*-algebras and KMS states on their corners. The approach adopts methods and ideas from graph theory, random walks and dynamical systems.
We determine the factor generated by the GNS-representation defined by a KMS-weight for a generalized gauge action on a simple graph C*-algebra when the corresponding measure on the path space of the graph is conservative for the shift. This is an improvment of Theorem 3.2 in arXives 1412.6762.