scholarly journals Self-Similar Inverse Semigroups from Wieler Solenoids

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 266
Author(s):  
Inhyeop Yi
Keyword(s):  
Inverse Limit ◽  
Inverse Semigroups ◽  
Stable Sets ◽  
Limit System ◽  
Tracial State ◽  
C Algebras ◽  
Self Similar ◽  
Smale Space ◽  
Factor Maps ◽  
Totally Disconnected

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs and the tight groupoids of these inverse semigroups are equivalent to the unstable groupoids of Wieler solenoids. We also show that the C ∗ -algebras of the groupoids of germs have a unique tracial state.

The stable algebra of a Wieler solenoid: inductive limits and -theory

2019 ◽  
Vol 40 (10) ◽  
pp. 2734-2768 ◽  
Author(s):  
ROBIN J. DEELEY ◽  
ALLAN YASHINSKI
Keyword(s):  
Inverse Limit ◽  
Inductive Limit ◽  
Stable Sets ◽  
Inductive Limits ◽  
One Dimensional ◽  
Stable Algebra ◽  
Smale Space ◽  
Totally Disconnected ◽  
Compact Spectrum

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable $C^{\ast }$-algebra is the stationary inductive limit of a $C^{\ast }$-stable Fell algebra that has a compact spectrum and trivial Dixmier–Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to, in principle, compute the $K$-theory of the stable $C^{\ast }$-algebra. A specific one-dimensional Smale space (the $aab/ab$-solenoid) is considered as an illustrative running example throughout.

scholarly journals SELF-SIMILAR INVERSE SEMIGROUPS AND SMALE SPACES

2006 ◽  
Vol 16 (05) ◽  
pp. 849-874 ◽  
Author(s):  
VOLODYMYR NEKRASHEVYCH
Keyword(s):  
Markov Partition ◽  
Inverse Semigroups ◽  
Automata Theory ◽  
Markov Partitions ◽  
Self Similar ◽  
Smale Space ◽  
Smale Spaces

Self-similar inverse semigroups are defined using automata theory. Adjacency semigroups of s-resolved Markov partitions of Smale spaces are introduced. It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup. The notions of the limit solenoid and a contracting semigroup is described.

scholarly journals Self-similar graph $C^*$-algebras and partial crossed products

2016 ◽  
Vol 75 (2) ◽  
pp. 299-317 ◽  
Author(s):  
Ruy Exel ◽  
Starling Starling
Keyword(s):  
Crossed Products ◽  
C Algebras ◽  
Self Similar

scholarly journals Examples of non connective C *-algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Anna Gąsior ◽  
Andrzej Szczepański
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Blow Up ◽  
Fractional Diffusion ◽  
Uniqueness Of Solutions ◽  
C Algebras ◽  
Space Fractional Diffusion Equation ◽  
Self Similar ◽  
Similar Solutions

Abstract This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder’s and Banach’s fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.

Iterated Limits of Lattices

1974 ◽  
Vol 26 (6) ◽  
pp. 1301-1320 ◽  
Author(s):  
Craig Platt
Keyword(s):  
Inverse Limit ◽  
Limit System ◽  
Universal Algebras

In this paper the results of [5] are extended to classes of lattices. We assume familiarity with [5], but we recall for convenience the principal definitions and notations. If is a category and if is a direct [resp., inverse] limit system in , then is the direct [resp., inverse] limit of (determined only up to isomorphism in ). If is an inverse limit system of sets or universal algebras, let denote the canonical construction of inverse limit described for example in [1, Chapter 3].

scholarly journals Continuous Orbit Equivalence on Self-Similar Graph Actions

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 990
Author(s):  
Inhyeop Yi
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Orbit Equivalence ◽  
Semigroup Actions ◽  
Infinite Path ◽  
Self Similar

For self-similar graph actions, we show that isomorphic inverse semigroups associated to a self-similar graph action are a complete invariant for the continuous orbit equivalence of inverse semigroup actions on infinite path spaces.

scholarly journals The C*-algebras of some inverse semigroups

1990 ◽  
Vol 42 (2) ◽  
pp. 335-348 ◽  
Author(s):  
Rachel Hancock ◽  
Iain Raeburn
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Bicyclic Semigroup ◽  
C Algebras

We discuss the structure of some inverse semigroups and the associated C* algebras. In particular, we study the bicyclic semigroup and the free monogenic inverse semigroup, following earlier work of Conway, Duncan and Paterson. We then associate to each zero-one matrix A an inverse semigroup CA, and show that the C*-algebra OA of Cuntz and Krieger is closely related to the semigroup algebra C*(CA).

Free group C*-algebras associated with ℓp

2014 ◽  
Vol 25 (07) ◽  
pp. 1450065 ◽  
Author(s):  
Rui Okayasu
Keyword(s):  
Free Group ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Linear Functionals ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Positive Linear Functionals ◽  
The Ideal

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

scholarly journals Crossed products of Hilbert pro-C-bimodules and associated pro-C-algebras

2016 ◽  
Vol 32 (2) ◽  
pp. 195-201
Author(s):  
MARIA JOITA ◽  
◽  
RADU-B. MUNTEANU ◽  
Keyword(s):  
Compact Group ◽  
Inverse Limit ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Locally Compact ◽  
C Algebras

An action (γ, α) of a locally compact group G on a Hilbert pro-C∗-bimodule (X, A) induces an action γ × α of G on A ×X Z the crossed product of A by X. We show that if (γ, α) is an inverse limit action, then the crossed product of A ×α G by X ×γ G respectively of A ×α,r G by X ×γ,r G is isomorphic to the full crossed product of A ×X Z by γ × α respectively the reduced crossed product of A ×X Z by γ × α.

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