A Note on Commutative Gelfand Theory for Real Banach Algebras

Pfaffenberger and Phillips [2] consider a real and unital case of the classical commutative Gelfand theorem and obtain two representation theorems. One is to represent a unital real commutative Banach algebra A as an algebra of continuous functions on the unital homomorphism space &#934;A. The other is to represent A as an algebra of continuous sections on the maximal ideal space MA. In this note, we point out that similar theorems for non-unital case hold and show that two representation theorems are essentially identical.</p

A perturbation of ring derivations on Banach algebras

AbstractSuppose A is a Banach algebra and suppose f:A→A is an approximate ring derivation in the sense of Hyers–Ulam–Rassias. This stability phenomenon was introduced for the first time in the subject of functional equations by Th.M. Rassias [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300]. If A has an approximate identity, or if A is semisimple and commutative, then we prove that f is an exact ring derivation

A Classification of Two-dimensional Endo-commutative Algebras over F_2

We introduce a new class of algebras called endo-commutative algebras in which the square mapping preserves multiplication, and provide a complete classification of endo-commutative algebras of dimension 2 over the field F_2 of two elements. We list all multiplication tables of the algebras up to isomorphism. This clarifies the difference between commutativity and endo-commutativity of algebras.Comment: 19 page

A classification of 2-dimensional endo-commutative straight algebras of rank 1 over a non-trivial field

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of 2-dimensional endo-commutative straight algebras of rank one over an arbitrary non-trivial field, where a straight algebra of dimension 2 satisfies the condition that there exists an element $x$ such that $x$ and $x^2$ are linearly independent. We list all multiplication tables of the algebras up to isomorphism.Comment: 10 pages. arXiv admin note: text overlap with arXiv:2304.1251

An extension of Helson-Edwards theorem to Banach Modules

An extension of the Helson-Edwards theorem for the group algebras to Banach modules over commutative Banach algebras is given. This extension can be viewed as a generalization of Liu-Rooij-Wang's result for Banach modules over the group algebras

Classification of Zeropotent Algebras of Dimension 3 over mathbbRmathbb{R} (Logic, Language, Algebraic system and Related Areas in Computer Science)

This is a summary of our results [Y. Kobayashi, K. Shirayanagi, S.-E. Takahasi and M. Tsukada, Classification of three-dimensional zeropotent algebras over an algebraically closed field, Comm. Algebra, Vol. 45 (12), 5037-5052, 2017.] and [K. Shirayanagi, S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of three-dimensional zeropotent algebras over the real number field, Comm. Algebra, Vol. 46 (11), 4665-4681, 2018.]

A note on finite codimensional linear isometries of C

Let (X,Y) be a pair of compact Hausdorff spaces. It is shown that a certain property of the class of continuous maps of Y onto X is equivalent to the non-existence of linear isometry of C(X) into C(Y) whose range has finite codimension >0

Continuously differentiable means

The Eurozone crisis and the refugee crisis are showcases of the problems associated with the EU's shift from market integration to the integration of core state powers. The integration of core state powers responds to similar demand factors as market integration (interdependence, externalities and spillover) but its supply is more tightly constrained by a high propensity for zero‐sum conflict, a functional requirement for centralized fiscal, coercive and administrative capacities, and high political salience. We show how these constraints structured the initial design of Economic and Monetary Union and of Schengen, made them vulnerable to crisis, and shaped policy options during the crises: they made horizontal differentiation unattractive, re‐regulation ineffective, centralized risk and burden‐sharing unfeasible, and the externalization of adjustment burdens to non‐EU actors necessary by default. In conclusion, we explore possible escape routes from the trap

Significance and Teaching Methods of Integration Constants

多くの教科書では不定積分の公式を掲げるが，紙面の都合の為か積分定数を省略する事が多い。これが為か学生達は積分定数について曖昧だったり，軽視する傾向にある。本論は具体的な例を挙げ，積分定数の意義と大切さ，その指導法について述べる。指導法・実践報