Minimal Dynamical Systems on the Product of the Cantor Set and the Circle

The realization problem that we will state here can be divided into the following three problems A, B and C. Where I/O is the set of input/output maps that may be the input/output relation of a given black-box. CD is the category of dynamical systems which may have the same behavior (equivalently, input/output relation) as the black-box. A. The existence and uniqueness in algebraic sense. For any input/output map a I/O , find out at least one dynamical system CD such that the behavior of it is a . Also prove that any two dynamical systems that have the same behavior a are isomorphic in the sense of the category CD. B. The finite dimensionality of the dynamical systems. Clarify when a dynamical system CD is finite dimensional. Because finite dimensional dynamical systems are actually appearing by linear (or non-linear) circuits or computer programs, it is very important that these conditions become clear. C. Deriving the dynamical systems from finite data. Partial realization problems are to find the minimal dynamical system fit to a given finite input/output’s data and to clarify when the minimal dynamical systems are isomorphic.

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <∣X∣, the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory. 1991 Mathematics Subject Classification 20B35, 20B27.

In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group [Formula: see text] of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.

We say that a unital C ∗ \mathrm {C}^{*} -algebra A A has the approximate positive factorization property (APFP) if every element of A A is a norm limit of products of positive elements of A A . (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of C ∗ \mathrm {C}^{*} -algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial K 1 K_{1} , and stable rank 1. In the unital case, the K 0 K_{0} group separates the tracial states. The APFP passes to matrix algebras, and if I I is an ideal in A A such that I I and A / I A/I have the APFP, then so does A A . We also give some new examples of C ∗ \mathrm {C}^{*} -algebras with the APFP, including type I I 1 \mathrm {II}_{1} factors and infinite-dimensional simple unital direct limits of homogeneous C ∗ \mathrm {C}^{*} -algebras with slow dimension growth, real rank zero, and trivial K 1 K_{1} group. Simple direct limits of homogeneous C ∗ \mathrm {C}^{*} -algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a C ∗ \mathrm {C}^{*} -algebra that may be of interest in the future.

Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.

This dissertation focuses on finite group actions with the tracial Rokhlin property and the structure of the corresponding crossed products. It consists of two major parts. For the first part, we study several different aspects of finite group actions with certain versions of the Rokhlin property. We are able to give an explicit characterization of product-type actions with the tracial Rokhlin property or strict Rokhlin property. We also show that, in good circumstances, the actions with the tracial Rokhlin property are generic. In the second portion of this dissertation, we introduce the weak tracial Rokhlin property for actions on non-simple C*-algebras. The main results are as follows. Let A be a unital non-simple C*-algebra and α be an action of G on A with the weak tracial Rokhlin property. Assume that the crossed product C*: G, A, α) is simple. Suppose A has either of the following property: tracial rankk, stable rank one, real rank zero. Then C*: G, A, α) has the same property.

Alternative characterisations of nuclear dimension and decomposition rank in terms of finite dimensional subalgebras and approximate partitions of unity are given for C*-algebras with real rank zero. These characterisations aid the understanding of the two concepts, as well as the difference between them, and lead to elementary proofs of known, important results.

Let be a C*-algebra with real rank zero that has the stable weak cancellation property. Let be an ideal of such that is stable and satisfies the corona factorization property. We prove thatis a full extension if and only if the extension is stenotic and K-lexicographic. As an immediate application, we extend the classification result for graph C*-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West, and the first named author, our result may also be used to give a purely K-theoretical description of when an essential extension of two simple and stable graph C*-algebras is again a graph C*- algebra.

Letbe a minimal homeomorphism (n ≥1). We show that the crossed producthas rational tracial rank at most one. Let Ω be a connected, compact, metric space with finite covering dimension and with. Suppose that,where Giis a finite abelian group, i = 0,1. Let β:Ω→Ωbe a minimal homeomorphism. We also show thathas rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on X✗Ω, where X is the Cantor set.

In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic homology of a group algebra. For an analogous spectral sequence for the Hochschild homology of a crossed product algebra, see Brylinski [2], [3]. In this article, we give a new derivation of this spectral sequence, and generalize it to negative and periodic cyclic homology HC• (A) and HP•(A). The method of proof is itself of interest, since it involves a natural generalization of the notion of a cyclic module, in which the condition that the morphism τ ∈ Λ(n,n) is cyclic of order n + 1 is relaxed to the condition that it be invertible. We call this category the paracyclic category. Given a paracyclic module P , we can define a chain complex C(P ), with differentials b and B, which respectively lower and raise degree. The condition that the module P is paracyclic translates to the condition on C(P ) that 1− (bB +Bb) is invertible. Our main result is to show that there is an analogue of the Eilenberg-Zilber theorem for bi-paracyclic modules. It is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of Feigin and Tsygan. If M is a module over a commutative ring k, we will denote by M (k) the iterated tensor product, defined by M (0) = k and M (k+1) =M (k) ⊗M . If M and N are graded modules, we will denote by M ⊗N their graded tensor product. We would like to thank J. Block, J.-L. Brylinski and C. Kassel for a number of interesting discussions on the results presented here.

This paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.We begin with an overview of these fixed point results, including concise self contained sketches of their proofs. Then we proceed with a discussion of the relations among the ten classes of self-maps with domains ⟨X, d⟩ having various topological properties that often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable-path connectedness, and d-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between these classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of ℝ with f′ ≡ 0, which constitutes also a minimal dynamical system. We finish by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.

Learning in living organisms is typically associated with networks of neurons. The use of large numbers of adjustable units has also been a crucial factor in the continued success of artificial neural networks. In light of the complexity of both living and artificial neural networks, it is surprising to see that very simple organisms - even unicellular organisms that do not possess a nervous system - are capable of certain forms of learning. Since in these cases learning may be implemented with much simpler structures than neural networks, it is natural to ask how simple the building blocks required for basic forms of learning may be. The purpose of this study is to discuss the simplest dynamical systems that model a fundamental form of non-associative learning, habituation, and to elucidate technical implementations of such systems, which may be used to implement non-associative learning in neuromorphic computing and related applications.

A minimal dynamical system which has completely positive entropy, and does not have uniformly positive entropy is constructed. This answers a question that has been open for a certain time.

In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space Hand G, a group and let <TEX>${\alpha}$</TEX> : G<TEX>${\rightarrow}$</TEX> AutM be an action of G on M, where AutM is the group of all automorphisms on M. Then the crossed product <TEX>$\mathbb{M}=M{\times}{\alpha}$</TEX> G of M and G with respect to <TEX>${\alpha}$</TEX> is a von Neumann algebra acting on <TEX>$H{\bigotimes}{\iota}^2(G)$</TEX>, generated by M and <TEX>$(u_g)_g{\in}G$</TEX>, where <TEX>$u_g$</TEX> is the unitary representation of g on <TEX>${\iota}^2(G)$</TEX>. We show that <TEX>$M{\times}{\alpha}(G_1\;*\;G_2)=(M\;{\times}{\alpha}\;G_1)\;*_M\;(M\;{\times}{\alpha}\;G_2)$</TEX>. We compute moments and cumulants of operators in <TEX>$\mathbb{M}$</TEX>. By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if <TEX>$F_N$</TEX> is the free group with N-generators, then the crossed product algebra <TEX>$L_M(F_n){\equiv}M\;{\times}{\alpha}\;F_n$</TEX> satisfies that <TEX>$$L_M(F_n)=L_M(F_{{\kappa}1})\;*_M\;L_M(F_{{\kappa}2})$$</TEX>, whenerver <TEX>$n={\kappa}_1+{\kappa}_2\;for\;n,\;{\kappa}_1,\;{\kappa}_2{\in}\mathbb{N}$</TEX>.

In this paper, we will define a graph von Neumann algebra \(\Bbb{M}_{G}\) over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra \(\Bbb{M}_{G}\) = M ×α\(\Bbb{G}\) , where \(\Bbb{G}\) is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from \(\Bbb{M}_{G}\) onto its M-diagonal subalgebra \(\Bbb{D}_{G},\) we can see that this crossed product algebra \(\Bbb{M}_{G}\) is *-isomorphic to an amalgamated free product \(\underset{e\in E(G)}{\,*_{\Bbb{D}_{G}}}\)\(\Bbb{M} _{e},\) where \(\Bbb{M}_{e}\) = vN(M ×α\(\Bbb{G}_{e},\)\(\Bbb{D}_{G}),\) where \(\Bbb{G}_{e}\) is the subset of \(\Bbb{G}\) consisting of all reduced words in {e, e–1} and M ×α\(\Bbb{G} _{e}\) is a W*-subalgebra of \(\Bbb{M}_{G},\) as a new graph von Neumann algebra induced by a graph Ge. Also, we will show that, as a Banach space, a graph von Neumann algebra \(\Bbb{M}_{G}\) is isomorphic to a Banach space \(\Bbb{D}_{G}\) ⊕ \((\underset{w^{*}\in E(G)_{r}^{*}}{\oplus }\)\(\Bbb{M}_{w^{*}}^{o}),\) where \(E{\left( G \right)}^{*}_{r}\) is a certain subset of the set E(G)* of all words in the edge set E(G) of G.