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 Automatic continuity for Banach algebras with finite-dimensional radical

2006 ◽  
Vol 81 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Hung Le Pham
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity ◽  
Algebraic Condition ◽  
Finite Dimensional ◽  
Complete Algebra

AbstractThe paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

scholarly journals Topologically simple Banach algebras with derivation

1999 ◽  
Vol 60 (1) ◽  
pp. 153-161
Author(s):  
El Hossein Illoussamen ◽  
Volker Runde
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Pivotal Role ◽  
Automatic Continuity

It is not known if a commutative, topologically simple, radical Banach algebra exists. If, however, every derivation on such an algebra is continuous, this yields the automatic continuity of all derivations on commutative, semiprime Banach algebras. Utilising techniques used by Thomas in his proof of the Singer-Wermer conjecture, we show that, if A is a commutative, topologically simple Banach algebra with a non-zero derivation on it, then a quotient of a certain localisation of A has a power series structure. A pivotal role is played by what we call ample sets of denominators.

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 Stability of -Jordan Homomorphisms from a Normed Algebra to a Banach Algebra

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yang-Hi Lee
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Ring Homomorphism ◽  
Normed Algebra ◽  
Jordan Homomorphism ◽  
Commutative Banach Algebras ◽  
Jordan Homomorphisms

We establish the hyperstability of -Jordan homomorphisms from a normed algebra to a Banach algebra, and also we show that an -Jordan homomorphism between two commutative Banach algebras is an -ring homomorphism.

scholarly journals Automatic continuity of positive functionals on topological involution algebras

1981 ◽  
Vol 23 (2) ◽  
pp. 265-281 ◽  
Author(s):  
P.G. Dixon
Keyword(s):  
Banach Algebra ◽  
Spectral Theory ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Automatic Continuity ◽  
Topological Algebras ◽  
Positive Functional ◽  
Open Questions ◽  
Positive Functionals ◽  
Locally Convex

This paper surveys the known results on automatic continuity of positive functionals on topological *-algebras and then shows how two theorems on Banach *-algebras extend to complete metrizable topological *-algebras. The two theorems concerned are Loy's theorem on separable Banach *-algebras A with centre Z such that AZ is of countable codimension and Varopoulos' result on Banach *-algebras with bounded approximate identity. Both theorems have the conclusion that all positive functionals on such algebras are continuous. The extension of the second theorem requires the algebra to be locally convex and the approximate identity to be ‘uniformly bounded’. Neither extension requires the algebra to be LMC. This means that the proof of the first theorem is quite different from the corresponding Banach algebra result (which used spectral theory). The proof of the second is closer to the previously known LMC version, but actually neater by being more general. It is also shown that the well-known estimate of |f(a*ba)| for a positive functional f on a Banach *-algebra may be obtained without the usual use of spectral theory. The paper concludes with a list of open questions.

AUTOMATIC CONTINUITY OF 3-HOMOMORPHISMS ON TERNARY BANACH ALGEBRAS

2013 ◽  
Vol 10 (10) ◽  
pp. 1320013 ◽  
Author(s):  
MADJID ESHAGHI GORDJI ◽  
ALI JABBARI ◽  
ALI EBADIAN ◽  
SAEID OSTADBASHI
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity

In this paper we consider automatic continuity of 3-homomorphism and anti-3-homomorphism between non-unital ternary Banach algebras. We show that every surjective 3-homomorphism (anti-3-homomorphism) from ternary Banach algebra A into semisimple ternary Banach algebra B is continuous.

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.

scholarly journals Some Properties of Cone Inner Product Spaces over Banach Algebra

2021 ◽  
pp. 29-37
Author(s):  
Anas Yusuf ◽  
Abor Isa Garba
Keyword(s):  
Banach Algebra ◽  
Product Space ◽  
Banach Algebras ◽  
Schwarz Inequality ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Normed Algebra ◽  
Pythagoras Theorem ◽  
Ordered Banach Algebra

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

2021 ◽  
pp. 1-12
Author(s):  
PRAKASH A. DABHI ◽  
DARSHANA B. LIKHADA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Discrete Groups ◽  
Discrete Semigroups ◽  
Beurling Algebras

Abstract Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0\lt p\leq 1$ . We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$ . We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

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