Exploring the structure and properties of semisimple rings, this book delves into their classification and representation theory. It covers foundational concepts and advanced topics, providing a comprehensive framework for understanding how these rings operate within algebra. The text is aimed at both students and researchers, offering clear explanations and numerous examples to illustrate complex ideas. The emphasis on classical results and modern applications makes it a valuable resource for anyone interested in algebraic structures.
Many problems in operator theory lead to the consideration ofoperator equa tions, either directly or via some reformulation. More often than not, how ever, the underlying space is too 'small' to contain solutions of these equa tions and thus it has to be 'enlarged' in some way. The Berberian-Quigley enlargement of a Banach space, which allows one to convert approximate into genuine eigenvectors, serves as a classical example. In the theory of operator algebras, a C*-algebra A that turns out to be small in this sense tradition ally is enlarged to its (universal) enveloping von Neumann algebra A". This works well since von Neumann algebras are in many respects richer and, from the Banach space point of view, A" is nothing other than the second dual space of A. Among the numerous fruitful applications of this principle is the well-known Kadison-Sakai theorem ensuring that every derivation 8 on a C*-algebra A becomes inner in A", though 8 may not be inner in A. The transition from A to A" however is not an algebraic one (and cannot be since it is well known that the property of being a von Neumann algebra cannot be described purely algebraically). Hence, ifthe C*-algebra A is small in an algebraic sense, say simple, it may be inappropriate to move on to A". In such a situation, A is typically enlarged by its multiplier algebra M(A).
Mathieu's Lehrbuch, als Einführung in die Funktionalanalysis verfaßt, wendet sich an Studierende der Mathematik nach dem Vorexamen, also in der Regel ab dem 5. Fachsemester. Die Darstellung weicht von einer historisch gewachsenen, klassisch „deutschen“ ab und folgt - durch die vielfältige internationale Lehr- und Forschungstätigkeit des Autors gefördert - den neueren Entwicklungen der Funktionalanalysis, die mehr und mehr Hilfsmittel und Denkweisen der Algebra benötigen. Mathieu hat bewußt ein „Arbeitsbuch“ geschrieben, das den Studierenden zur Mitarbeit zwingt. Beispielsweise sind in den Text Aufgaben gestreut, deren Lösungen Teile der Beweisführungen enthalten und damit Bestandteile des Lehrstoffs werden. Ein fachlich und zugleich didaktisch höchst ungewöhnlicher, moderner Zugang zu einem wichtigen Teilgebiet der Mathematik.
