scholarly journals Multipliers of Uniform Topological Algebras

2017 ◽  
Vol 31 (1) ◽  
pp. 71-81
Author(s):  
Mohammed El Azhari
Keyword(s):  
Topological Algebra ◽  
Topological Algebras ◽  
Algebra Isomorphism

Abstract Let E be a complete uniform topological algebra with Arens-Michael normed factors within an algebra isomorphism ϕ. If each factor Eα is complete, then every multiplier of E is continuous and ϕ is a topological algebra isomorphism where M(E) is endowed with its seminorm topology.

Generalized Frobenius Algebras and Hopf Algebras

2014 ◽  
Vol 66 (1) ◽  
pp. 205-240 ◽  
Author(s):  
Miodrag Cristian Iovanov
Keyword(s):  
Hopf Algebras ◽  
Topological Algebra ◽  
Homological Algebra ◽  
General Concept ◽  
Frobenius Algebra ◽  
Theoretic Approach ◽  
Topological Algebras ◽  
Infinite Dimensional ◽  
Finite Case ◽  
Frobenius Algebras

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

Local analytic structure in certain commutative topological algebras

1972 ◽  
Vol 6 (2) ◽  
pp. 161-167 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Analytic Structure ◽  
Frechet Space ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power ◽  
Locally Convex Topology ◽  
Locally Convex

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

scholarly journals On totally bounded universal algebras

2012 ◽  
Vol 21 (2) ◽  
pp. 151-165
Author(s):  
MITROFAN M. CHOBAN ◽  
◽  
INA D. CIOBANU ◽  
Keyword(s):  
Topological Algebra ◽  
Continuous Extension ◽  
Topological Algebras ◽  
Universal Algebras ◽  
Totally Bounded ◽  
General Properties

In the class of topological algebras of a given signature the notions of totally boundedness and a-pseudocompactness are introduced. A topological algebra is totally bounded if it is a subalgebra of a compact algebra. The general properties of totally bounded algebras are studied. The compactifications of topological algebras are investigated too. In particular, the problem of the continuous extension of the operation on the Stone-Cech compactification is studied.

scholarly journals On Uniform Topological Algebras

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
M. El Azhari
Keyword(s):  
Banach Algebra ◽  
Topological Algebra ◽  
Uniform Norm ◽  
Commutative Banach Algebra ◽  
Normed Algebra ◽  
Topological Algebras

Abstract The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M(A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.

Infrasequential Topological Algebras

1979 ◽  
Vol 22 (4) ◽  
pp. 413-418 ◽  
Author(s):  
T. Husain
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Convex ◽  
Locally Convex Algebra

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

scholarly journals Continuity of derivations on topological algebras of power series

1969 ◽  
Vol 1 (3) ◽  
pp. 419-424 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Frechet Space ◽  
Complex Field ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power

Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.

scholarly journals Multiplicative functionals and a class of topological algebras

1977 ◽  
Vol 17 (3) ◽  
pp. 391-399 ◽  
Author(s):  
Gerard A. Joseph
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Null Sequence ◽  
Topological Algebras ◽  
Multiplicative Functionals ◽  
Locally Convex ◽  
Locally Convex Algebra

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.

scholarly journals A New Proof of the Pythagorean Theorem and Its Application to Element Decompositions in Topological Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Fred Greensite
Keyword(s):  
Vector Space ◽  
Algebraic Structure ◽  
Geometric Mean ◽  
Topological Algebra ◽  
Space Structure ◽  
Pythagorean Theorem ◽  
Euclidean Norm ◽  
Topological Algebras

We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” (addressing unital algebraic structure rather than simply vector space structure). One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of different algebras.

scholarly journals Joint Spectra of Generators in Topological Algebras

2007 ◽  
Vol 2007 ◽  
pp. 1-7
Author(s):  
Antoni Wawrzyńczyk
Keyword(s):  
Topological Algebra ◽  
Topological Algebras ◽  
Joint Spectrum ◽  
Multiplicative Functionals

Let𝒜be a complex topological algebra with unit 1 and𝒰a family of proper closed ideals in𝒜. For an arbitraryS⊂𝒜we define a globally defined joint spectrumσ𝒰(S)={(λs)s∈S∈ℂS  |  ∃ I  ∈𝒰(s−λs)∈I ∀s∈S}. We prove that forSgenerating𝒜the spectrumσ𝒰(S)can be identified with the set𝔐𝒰of continuous multiplicative functionalsfon𝒜such that kerf∈𝒰. The relation is given by the formulaσ𝒰(S)={(f(s))s∈S  |   f∈𝔐𝒰}. If𝒜is aQ-algebra, the setσ𝒰(S)is rationally convex.

HYPERGROUP ALGEBRAS AS TOPOLOGICAL ALGEBRAS

2014 ◽  
Vol 90 (3) ◽  
pp. 486-493
Author(s):  
S. MAGHSOUDI ◽  
J. B. SEOANE-SEPÚLVEDA
Keyword(s):  
Haar Measure ◽  
Lebesgue Space ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Compact ◽  
Continuous Functionals ◽  
Locally Convex Topology ◽  
Compact Hypergroup ◽  
Locally Convex ◽  
Hypergroup Algebras

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.

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