scholarly journals The multiplicative spectrum and the uniqueness of the complete norm topology

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 473-485 ◽  
Author(s):  
J.C. Marcos ◽  
M.V. Velasco
Keyword(s):  
Multiplication Operators ◽  
Automatic Continuity ◽  
Normed Algebra ◽  
Associative Algebras ◽  
Norm Topology ◽  
Dense Range ◽  
Theoretical Support ◽  
Evolution Algebra ◽  
Semisimple Algebras ◽  
Multiplicative Spectrum

We define the spectrum of an element a in a non-associative algebra A according to a classical notion of invertibility (a is invertible if the multiplication operators La and Ra are bijective). Around this notion of spectrum, we develop a basic theoretical support for a non-associative spectral theory. Thus we prove some classical theorems of automatic continuity free of the requirement of associativity. In particular, we show the uniqueness of the complete norm topology of m-semisimple algebras, obtaining as a corollary of this result a well-known theorem of Barry E. Johnson (1967). The celebrated result of C.E. Rickart (1960) about the continuity of dense-range homomorphisms is also studied in the non-associative framework. Finally, because non-associative algebras are very suitable models in genetics, we provide here a hint of how to apply this approach in that context, by showing that every homomorphism from a complete normed algebra onto a particular type of evolution algebra is automatically continuous.

Automatic Continuity of Homomorphisms in Non-associative Banach Algebras

2013 ◽  
Vol 65 (5) ◽  
pp. 989-1004
Author(s):  
C-H. Chu ◽  
M. V. Velasco
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Surjective Homomorphism ◽  
Automatic Continuity ◽  
Normed Algebra ◽  
Associative Algebras ◽  
Rare Element ◽  
Rare Elements

AbstractWe introduce the concept of a rare element in a non-associative normed algebra and show that the existence of such an element is the only obstruction to continuity of a surjective homomorphism from a non-associative Banach algebra to a unital normed algebra with simple completion. Unital associative algebras do not admit any rare elements, and hence automatic continuity holds.

scholarly journals The connection between evolution algebras, random walks and graphs

2019 ◽  
Vol 19 (02) ◽  
pp. 2050023 ◽  
Author(s):  
Paula Cadavid ◽  
Mary Luz Rodiño Montoya ◽  
Pablo M. Rodriguez
Keyword(s):  
Probability Theory ◽  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Associative Algebras ◽  
Biological Phenomena ◽  
Evolution Algebra ◽  
New Type ◽  
Markov Evolution

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras.

scholarly journals Algebras With Vanishing n-Cohomology Groups

1954 ◽  
Vol 7 ◽  
pp. 115-131 ◽  
Author(s):  
Masatoshi Ikeda ◽  
Hiroshi Nagao ◽  
Tadashi Nakayama
Keyword(s):  
Associative Algebra ◽  
Finite Rank ◽  
Unit Element ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Associative Algebras ◽  
Semisimple Algebras

Cohomology theory for (associative) algebras was first established in general higher dimensionalities by G. Hochschild [3], [4], [5]. Algebras with vanishing 1-cohomology groups are separable semisimple algebras ([3], Theorem 4.1). On extending and refining our recent results [6], [8], [12], we establish in the present paper the following:Let n ≧ 2. Let A be an (associative) algebra (of finite rank) possessing a unit element 1 over a field Ω, and N be its radical.

Automatic continuity with application to C*-algebras

1990 ◽  
Vol 107 (2) ◽  
pp. 345-347 ◽  
Author(s):  
Angel Rodriguez Palacios
Keyword(s):  
Banach Algebras ◽  
Direct Consequence ◽  
Automatic Continuity ◽  
Associative Algebras ◽  
C Algebras ◽  
Complete Algebra ◽  
Usual Norm

The fact proved by Cleveland [4], that the topology of any (non-complete) algebra norm on a C*-algebra is stronger than the topology of the usual norm, is reencountered as a direct consequence of a theorem, which we prove in this note, stating that homomorphisms from certain non-complete normed (associative) algebras onto some semisimple Banach algebras are automatically continuous.

scholarly journals New Condition of Automatic Continuity of Dense Range Homomorphisms on Jordan – Banach Algebras

2018 ◽  
Vol 27 (4) ◽  
pp. 146-150
Author(s):  
RUQAYAH N. BALO ◽  
NADIA A. ABDULRAZAQ ◽  
DUAA F. ABDULLAH
Keyword(s):  
Banach Algebras ◽  
Automatic Continuity ◽  
Dense Range

scholarly journals Properties of Nilpotent Evolution Algebras with no Maximal Nilindex

2021 ◽  
Vol 14 (1) ◽  
pp. 278-300
Author(s):  
Ahmad Alarfeen ◽  
Izzat Qaralleh ◽  
Azhana Ahmad
Keyword(s):  
Dynamical Systems ◽  
Lie Algebras ◽  
Structure Theorem ◽  
Deep Structure ◽  
Abstract Algebra ◽  
Classification Theorem ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Evolution Algebra

As a system of abstract algebra, evolution algebras are commutative and non-associative algebras. There is no deep structure theorem for general non-associative algebras. However, there are deep structure theorem and classification theorem for evolution algebras because it has been introduced concepts of dynamical systems to evolution algebras. Recently, in [25], it has been studied some properties of nilpotent evolution algebra with maximal index (dim E2 = dim E − 1). This paper is devoted to studying nilpotent finite-dimensional evolution algebras E with dim E2 =dim E − 2. We describe Lie algebras related to the evolution of algebras. Moreover, this result allowed us to characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras are found.

scholarly journals Determining When an Algebra Is an Evolution Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1349 ◽  
Author(s):  
Miguel D. Bustamante ◽  
Pauline Mellon ◽  
M. Victoria Velasco
Keyword(s):  
Asexual Reproduction ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Associative Algebras ◽  
Dimensional Algebra ◽  
Small Perturbations ◽  
Classical Genetic ◽  
Evolution Algebra ◽  
Necessary And Sufficient

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n-dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n×n matrices {M1,…,Mn} describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.

scholarly journals AUTOMATIC CONTINUITY OF n-HOMOMORPHISMS BETWEEN TOPOLOGICAL ALGEBRAS

2011 ◽  
Vol 83 (3) ◽  
pp. 389-400 ◽  
Author(s):  
TAHER G. HONARY ◽  
H. SHAYANPOUR
Keyword(s):  
Banach Algebras ◽  
Automatic Continuity ◽  
Topological Algebras ◽  
Dense Range

AbstractA map θ:A→B between algebras A and B is called n-multiplicative if θ(a1a2⋯an)=θ(a1) θ(a2)⋯θ(an) for all elements a1,a2,…,an∈A. If θ is also linear then it is called an n-homomorphism. This notion is an extension of a homomorphism. We obtain some results on automatic continuity of n-homomorphisms between certain topological algebras, as well as Banach algebras. The main results are extensions of Johnson’s theorem to surjective n-homomorphisms on topological algebras, a theorem due to C. E. Rickart in 1950 to dense range n-homomorphisms on topological algebras and two theorems due to E. Park and J. Trout in 2009 to * -preserving n-homomorphisms on lmc * -algebras.

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