16571

14720

كارشناسي ارشد

رياضي محض

اصفهان : دانشگاه صنعتي اصفهان

1399

هشت،[ 70]ص. : مصور، جدول، نمودار

مهدي نعمتي

رسول نصر اصفهاني

واژه نامه

محمود منجگاني، فريد بهرامي

1400/06/03

كتابنامه

رياضي

رياضي

1400/06/13

2698693

در اين پايان نامه، ارتباط نظريه ي طيفي با مسئله ي پيوستگي خودكار همريختي ها براي جبر هاي شركت ناپذير را مطالعه مي كنيم. ضمن آن به معرفي انواع طيف هاي عملگري روي فضا هاي نرم دار مي پردازيم و برخي ويژگي هاي آن ها را بيان مي كنيم. همچنين طيف ها و ويژگي هاي آن ها در فضاي دوگان و كامل شده ي فضا هاي نرم دار را بررسي مي كنيم. در ادامه با معرفي عملگر هاي نادر به وسيله ي يك خاصيت طيفي خاص به بررسي بعضي از خصوصيات آن ها مي پردازيم، كه از اين ويژگي ها براي به دست آوردن نتايجي در پيوستگي خودكار همريختي ها استفاده مي كنيم. بر اين اساس مفهوم يك عنصر نادر در يك جبر نرم دار شركت ناپذير را معرفي مي كنيم و نشان مي دهيم وجود يك عنصر نادر، تنها مانع پيوستگي خودكار است. همچنين مسئله ي پيوستگي خودكار همريختي هاي با برد چگال را در حالتي بررسي مي كنيم كه تصوير همريختي در يك جبر شركت پذير قوي قرار دارد و به علاوه پيوستگي خودكار همريختي هاي پوشا روي جبر هاي شركت ناپذيري را مورد مطالعه قرار مي دهيم.

This M.Sc. thesis is based on the following papers • C-H. Cඁඎ ൺඇൽ M. V. Vൾඅൺඌർඈ., Automatic continuity of homomorphisms in non-associative Banach algebras, Canad. J. Math. Vol. 65 (5), 2013 pp. 989–1004 • J. Cൺඋඅඈඌ Mൺඋർඈඌ ൺඇൽ M. Vංർඍඈඋංൺ Vൾඅൺඌർඈ., Continuity of homomorphisms into power-associative complete normed algebras, Forum Math. 25 (2013), 1109 – 1125 Let X be a normed vector space. In this case L(X) denote the normed algebra of all bounded linear operators from X to itself. For each operator T ∈ L(X), we denote the spectrum of T by σ(T) = {λ ∈ C : T − λI is not invertible in L(X)}. and σs(T) is the surjective spectrum of T defined by σs(T) = {λ ∈ C : T − λI is not surjective}. We always denote by Xˆ the completion of X and by Tˆ ∈ L(Xˆ) the unique continuous extension of T to Xˆ. A linear operator T ∈ L(X) is called rare if σs(T) ∩ σs(Tˆ) = ∅. Let A be a normed algebra. An element a ∈ A is called rare if the left multiplication operator La : x ∈ A −→ ax ∈ A and right multiplication operator Ra : x ∈ A −→ xa ∈ A are rare operators on A. A normed algebra A is called simple if it’s product it nonzero and there is no nonzero proper ideal in A. A normed algebra A is said to have a simple completion if it’s completion Aˆ is a simple algebra. An associative algebra A is strongly semisimple whenever it’s strong radical (that is intersection of the modular maximal ideals of A) is zero. The basic automatic continuity problem is to give algebraic conditions on A and B which ensure the topological property that every homomorphism θ : A −→ B is necessarily continuous. In the study of associative Banach algebras, the theory of automatic continuity of homomorphisms and uniqueness of complete algebra norm is fundamental and has been well-established since the seminal works of Rickart [26] and Johnson [12]. We refer to [5, 19] for details. For non-associative algebras, the question of automatic continuity has been investigated by several authors; where the background and connections to spectral theory have been explained succinctly. We show that the existence of a rare element is the only obstruction to automatic continuity. This thesis is organized as follows: In chapter1, we recall relevant definitions and introduce some notation. In chapter2, we introduce the types of operating spectra on normed spaces and some of their features which we need in the next chapters. Then we study spectra and their features in dual speces and completed normed spaces. In chapter3, we introduce rare liner operators and we will examine some of their features that we use these features to get results in automatic continuity. In the following introduces the left and right multiplication operators and based on them, the cocept of a rare element in a non-associative normed algebra will be expressed. In chapter4, we check the problem of automatic continuity of in the dense-range homomorphism case where the homomorphism image is in a power-associative algebra and also automatic continuity of homomorphisms in nonassociative banach algebras

مهدي نعمتي

رسول نصر اصفهاني

محمود منجگاني، فريد بهرامي