arXiv subjects

Tsuyoshi Kajiwara

Publications and source records attributed to Tsuyoshi Kajiwara.

11 recordsLinked to original sources

arXiv2018

Dimension groups for self-similar maps and matrix representations of the core of the associated C*-algebras

We introduce a dimension group for a self-similar map as the ${\rm K}_0$-group of the core of the $C^*$-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty}\cong {\mathbb Z}[t]$ together with the unilatral shift, i.e. the multiplication map by $t$ as an abstract group. Thus the canonical endomorphisms on the ${\rm K}_0$-groups are not automorphisms in geneal. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

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arXiv2013

Ideals of the core of C*-algebras associated with self-similar maps

We give a complete classification of the ideals of the core of the C*-algebras associated with self-similar maps under a certain condition. Any ideal is completely determined by the intersection with the coefficient algebra C(K) of the self-similar set K. The corresponding closed subset of K is described by the singularity structure of the self-similar map. In particular the core is simple if and only if the self-similar map has no branch point. A matrix representation of the core is essentially used to prove the classification.

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arXiv2012

C*-algebras associated with complex dynamical systems and backward orbit structure

Let $R$ be a rational function. The iterations $(R^n)_n$ of $R$ gives a complex dynamical system on the Riemann sphere. We associate a $C^*$-algebra and study a relation between the $C^*$-algebra and the original complex dynamical system. In this short note, we recover the number of $n$-th backward orbits counted without multiplicity starting at branched points in terms of associated $C^*$-algebras with gauge actions. In particular, we can partially imagine how a branched point is moved to another branched point under the iteration of $R$. We use KMS states and a Perron-Frobenius type operator on the space of traces to show it.

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arXiv2012

Traces on cores of C*-algebras associated with self-similar maps

We completely classify the extreme tracial states onthe cores of the C*-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.

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arXiv2010

KMS states on finite-graph C*-algebras

We study KMS states on finite-graph C*-algebras with sinks and sources. We compare finite-graph C*-algebras with C*-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature $β$ is large, then the set of extreme $β$-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational funcition from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshevyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

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arXiv2008

$C^*$-algebras associated with algebraic correspondences on the Riemann sphere

Let $p(z,w)$ be a polynomial in two variables. We call the solution of the algebraic equation $p(z,w) = 0$ the algebraic correspondence. We regard it as the graph of the multivalued function $z \mapsto w$ defined implicitly by $p(z,w) = 0$. Algebraic correspondences on the Riemann sphere $\hat{\mathbb C}$ give a generalization of dynamical systems of Klein groups and rational functions. We introduce $C^*$-algebras associated with algebraic correspondences on the Riemann sphere. We show that if an algebraic correspondence is free and expansive on a closed $p$-invariant subset $J$ of $\hat{\mathbb C}$, then the associated $C^*$-algebra ${\mathcal O}_p(J)$ is simple and purely infinite.

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arXiv2006

KMS states and branched points

We completely classify the KMS states for the gauge action on a $C^*$-algebra associated with a rational function $R$ introduced in our previous work. The gauge action has a phase transition at $β= \log °R$. We can recover the degree of $R$, the number of branched points, the number of exceptional points and the orbits of exceptional points from the structure of the KMS states. We also classify the KMS states for $C^*$-algebras associated with some self-similar sets, including the full tent map and the Sierpinski gasket by a similar method.

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arXiv2004

KMS states on C^*-algebras associated with self-similar sets

In this paper, we study KMS states for the gauge actions on C${}^*$-algebras associated with self-similar sets whose branch points are finite. If the self-similar set does not contain any branch point, the Hutchinson measure gives the unique KMS state. But if the self-similar set dose contain a branch point, there sometimes appear other KMS states which come from branch points. For this purpose we construct explicitly a basis for a Hilbert C${}^*$-module associated with a self-similar set with finite branch condition. Using this we get condition for a Borel probability measure on K to be extended to a KMS state on the C${}^*$-algebra associated with the original self-similar set. We classify KMS states for the case of dynamics of unit interval and the case of Sierpinski gasket which is related with Complex dynamical system. KMS states for these examples are unique and given by the Hutchinson measure if $β$ is equal to $\log N$, where $N$ is the number of contractions. They are expressed as convex combinations of KMS states given by measures supported on the orbit of the branched points if $β> \log N$.

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arXiv2004

C^*-algebras associated with self-similar sets

Let $γ= (γ_1,...,γ_N)$, $N \geq 2$, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset $K$. We consider the union ${\mathcal G} = \cup_{i=1}^N \{(x,y) \in K^2 ; x = γ_i(y)\}$ of the cographs of γ_i$. Then $X = C({\mathcal G})$ is a Hilbert bimodule over $A = C(K)$. We associate a $C^*$-algebra ${\mathcal O}_γ(K)$ with them as a Cuntz-Pimsner algebra ${\mathcal O}_X$. We show that if a system of proper contractions satisfies the open set condition in $K$, then the $C^*$-algebra ${\mathcal O}_γ(K)$ is simple and purely infinite, which is not isomorphic to a Cuntz algebra in general.

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arXiv2003

C^*-algebras associated with complex dynamical systems

Iteration of a rational function $R$ gives a complex dynamical system on the Riemann sphere. We introduce a $C^*$-algebra ${\mathcal O}_R$ associated with $R$ as a Cuntz-Pimsner algebra of a Hilbert bimodule over the algebra $A = C(J_R)$ of continuous functions on the Julia set $J_R$ of $R$. The algebra ${\mathcal O}_R$ is a certain analog of the crossed product by a boundary action. We show that if the degree of $R$ is at least two, then $C^*$-algebra ${\mathcal O}_R$ is simple and purely infinite. For example if $R(z) = z^2 - 2$, then the Julia set $J_R = [-2,2]$ and the restriction $R : J_R \to J_R$ is topologically conjugate to the tent map on $[0,1]$. The algebra ${\mathcal O}_{z^2 - 2}$ is isomorphic to the Cuntz algebra ${\mathcal O}_{\infty}$. We also show that the Lyubich measure associated with $R$ gives a unique KMS state on the $C^*$-algebra ${\mathcal O}_R$ for the gauge action at inverse temperature $\log (°R)$ if the Julia set contains no critical points.

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arXiv1998

Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules

Pimsner introduced the C*-algebra O_X generated by a Hilbert bimodule X over a C*-algebra A. We look for additional conditions that X should satisfy in order to study simplicity and, more generally, the ideal structure of O_X when X is finite projective. We introduce two conditions: `(I)-freeness' and `(II)-freeness', stronger than the former, in analogy with [J. Cuntz, W. Krieger, Invent. Math. 56, 251-268] and [J. Cuntz, Invent. Math. 63, 25-40] respectively. (I)-freeness comprehend the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite dimension and the case of `Cuntz-Krieger bimodules'. If X satisfies this condition the C*-algebra O_X does not depend on the choice of the generators when A is faithfully represented. As a consequence, if X is (I)-free and A is X-simple, then O_X is simple. In the case of Cuntz-Krieger algebras, X-simplicity corresponds to irreducibility of the defining matrix. If A is simple and p.i. then O_X is p.i., if A is nonnuclear then O_X is nonnuclear. We therefore provide examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore if X is (II)-free, we determine the ideal structure of O_X.

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