BIPROJECTIVITY AND BIFLATNESS OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2014 ◽  
Vol 91 (1) ◽  
pp. 134-144 ◽  
Author(s):  
F. ABTAHI ◽  
A. GHAFARPANAH ◽  
A. REJALI
Keyword(s):  
Banach Algebra ◽  
Product Space ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Algebra Morphism ◽  
Cartesian Product Space

AbstractLet ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$.

scholarly journals ARENS REGULARITY AND AMENABILITY OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2012 ◽  
Vol 87 (2) ◽  
pp. 195-206 ◽  
Author(s):  
S. J. BHATT ◽  
P. A. DABHI
Keyword(s):  
Banach Algebra ◽  
Product Space ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Commutative Banach Algebra ◽  
Arens Regularity ◽  
Algebra Morphism ◽  
Cartesian Product Space

AbstractGiven a morphism T from a Banach algebra ℬ to a commutative Banach algebra 𝒜, a multiplication is defined on the Cartesian product space 𝒜×ℬ perturbing the coordinatewise product resulting in a new Banach algebra 𝒜×Tℬ. The Arens regularity as well as amenability (together with its various avatars) of 𝒜×Tℬ are shown to be stable with respect to T.

scholarly journals Tensor Products of Banach Algebras

1959 ◽  
Vol 11 ◽  
pp. 297-310 ◽  
Author(s):  
Bernard R. Gelbaum
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Commutative Banach Algebra ◽  
Maximal Ideal Space ◽  
Locally Compact Abelian Group ◽  
Ideal Space ◽  
Integrable Functions ◽  
Group A

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B1 = Bl(G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G, 3m1the maximal ideal space of A, m2the maximal ideal space of L1(G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G, m3the maximal ideal space of B1. Then m3and the Cartesian product m1 X m2are homeomorphic when the spaces mi, i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m3and m1 X m2is such as to permit a description of any epimorphism E3: B1 → B1/m3 in terms of related epimorphisms E1: A → A/M1 and E2:L1(G) → Ll(G)/M2, where M1 is in mi i = 1, 2, 3.

scholarly journals Iterated Functions Systems Composed of Generalized θ-Contractions

2021 ◽  
Vol 5 (3) ◽  
pp. 69
Author(s):  
Pasupathi Rajan ◽  
María A. Navascués ◽  
Arya Kumar Bedabrata Chand
Keyword(s):  
Product Space ◽  
Complete Metric Space ◽  
Cartesian Product ◽  
Iterated Function Systems ◽  
Theoretical Work ◽  
Finite Collection ◽  
Self Similarity ◽  
Iterated Function ◽  
Finite Cartesian Product ◽  
Cartesian Product Space

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS.

scholarly journals More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism

2018 ◽  
Vol 68 (1) ◽  
pp. 147-152
Author(s):  
Mohammad Ramezanpour
Keyword(s):  
Banach Algebra ◽  
Direct Product ◽  
Banach Algebras ◽  
Algebra Morphism

AbstractFor two Banach algebrasAandB, theT-Lau productA×TB, was recently introduced and studied for some bounded homomorphismT:B→Awith ∥T∥ ≤ 1. Here, we give general nessesary and sufficent conditions forA×TBto be (approximately) cyclic amenable. In particular, we extend some recent results on (approximate) cyclic amenability of direct productA⊕BandT-Lau productA×TBand answer a question on cyclic amenability ofA×TB.

On the uniform ergodicity of Markov processes of order 2

2003 ◽  
Vol 40 (2) ◽  
pp. 455-472 ◽  
Author(s):  
Ulrich Herkenrath
Keyword(s):  
Time Series ◽  
Markov Process ◽  
Markov Processes ◽  
Product Space ◽  
Transition Probability ◽  
Cartesian Product ◽  
Time Series Models ◽  
Uniform Ergodicity ◽  
General State Space ◽  
Cartesian Product Space

We study the uniform ergodicity of Markov processes (Zn, n ≥ 1) of order 2 with a general state space (Z, 𝒵). Markov processes of order higher than 1 were defined in the literature long ago, but scarcely treated in detail. We take as the basis for our considerations the natural transition probability Q of such a process. A Markov process of order 2 is transformed into one of order 1 by combining two consecutive variables Z2n–1 and Z2n into one variable Yn with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Yn, n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Zn, n ≥ 1) is defined in terms of the same property for (Yn, n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Zn, n ≥ 1). We apply the general results to study the uniform ergodicity of Markov processes of order 2 which arise in some nonlinear time series models and as sequences of smoothed values in sequential smoothing procedures of Markovian observations. As for the time series models, Markovian noise sequences are covered.

On the uniform ergodicity of Markov processes of order 2

2003 ◽  
Vol 40 (02) ◽  
pp. 455-472 ◽  
Author(s):  
Ulrich Herkenrath
Keyword(s):  
Time Series ◽  
Markov Process ◽  
Markov Processes ◽  
Product Space ◽  
Transition Probability ◽  
Cartesian Product ◽  
Time Series Models ◽  
Uniform Ergodicity ◽  
General State Space ◽  
Cartesian Product Space

We study the uniform ergodicity of Markov processes (Z n , n ≥ 1) of order 2 with a general state space (Z, 𝒵). Markov processes of order higher than 1 were defined in the literature long ago, but scarcely treated in detail. We take as the basis for our considerations the natural transition probability Q of such a process. A Markov process of order 2 is transformed into one of order 1 by combining two consecutive variables Z 2n–1 and Z 2n into one variable Y n with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Y n , n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Z n , n ≥ 1) is defined in terms of the same property for (Y n , n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Z n , n ≥ 1). We apply the general results to study the uniform ergodicity of Markov processes of order 2 which arise in some nonlinear time series models and as sequences of smoothed values in sequential smoothing procedures of Markovian observations. As for the time series models, Markovian noise sequences are covered.

A NOTE ON CYCLIC AMENABILITY OF THE LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2015 ◽  
Vol 92 (2) ◽  
pp. 282-289 ◽  
Author(s):  
F. ABTAHI ◽  
A. GHAFARPANAH
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Stability Result ◽  
Algebra Homomorphism ◽  
Arens Regularity ◽  
Algebra Morphism

Let $T$ be a Banach algebra homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$ with $\Vert T\Vert \leq 1$. Recently, Bhatt and Dabhi [‘Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc.87 (2013), 195–206] showed that cyclic amenability of ${\mathcal{A}}\times _{T}{\mathcal{B}}$ is stable with respect to $T$, for the case where ${\mathcal{A}}$ is commutative. In this note, we address a gap in the proof of this stability result and extend it to an arbitrary Banach algebra ${\mathcal{A}}$.

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