scholarly journals Topologically irreducible representations of the Banach -algebra associated with a dynamical system

2017 ◽  
Vol 38 (5) ◽  
pp. 1768-1790
Author(s):  
AKI KISHIMOTO ◽  
JUN TOMIYAMA
Keyword(s):  
Dynamical System ◽  
Tensor Product ◽  
Banach Algebra ◽  
Irreducible Representations ◽  
Type I ◽  
Topological Dynamical System ◽  
Infinite Dimensional ◽  
New Class ◽  
Mild Conditions ◽  
Ergodic Measures

We describe (infinite-dimensional) irreducible representations of the crossed product C$^{\ast }$-algebra associated with a topological dynamical system (based on $\mathbb{Z}$) and we show that their restrictions to the underlying $\ell ^{1}$-Banach $\ast$-algebra are not algebraically irreducible under mild conditions on the dynamical system. The above description of irreducible representations has two ingredients, ergodic measures on the space and ergodic extensions for the tensor product with type I factors, the latter of which may not have been explicitly taken up before but which will be explored by means of examples. A new class of ergodic measures is also constructed for irrational rotations on the circle.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

Relations between the homologies of C*-algebras and their commutative C*-subalgebras

2002 ◽  
Vol 132 (1) ◽  
pp. 155-168
Author(s):  
ZINAIDA A. LYKOVA
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Banach Algebra ◽  
Left Ideal ◽  
Separable Hilbert Space ◽  
Infinite Dimensional ◽  
C Algebras

The paper concerns the identification of projective closed ideals of C*-algebras. We prove that, if a C*-algebra has the property that every closed left ideal is projective, then the same is true for all its commutative C*-subalgebras. Further, we say a Banach algebra A is hereditarily projective if every closed left ideal of A is projective. As a corollary of the stated result we show that no infinite-dimensional AW*-algebra is hereditarily projective. We also prove that, for a commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space, the following conditions are equivalent: (i) A is separable; and (ii) the C*-tensor product A [otimes ]minA is hereditarily projective. Howerever, there is a non-separable, hereditarily projective, commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space.

scholarly journals On a Class of Differential Variational Inequalities in Infinite-Dimensional Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 266 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Optimal Control Theory ◽  
Infinite Dimensional ◽  
Nash Games ◽  
New Class ◽  
Set Valued Mappings ◽  
Differential Variational Inequalities ◽  
Theoretical Developments

A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in infinite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

Infinite-Dimensional Grassmann-Banach Algebra

2004 ◽  
pp. 201-201
Author(s):  
John Howie ◽  
Steven Duplij ◽  
Ali Mostafazadeh ◽  
Masaki Yasue ◽  
Vladimir Ivashchuk
Keyword(s):  
Banach Algebra ◽  
Infinite Dimensional

scholarly journals Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

2005 ◽  
Vol 2005 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Hilbert Space ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Dimensional Lattice ◽  
Reaction Diffusion Systems ◽  
Infinite Dimensional ◽  
Diffusion Systems ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert spacel2×l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

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