Matej Brešar - Zero Product Determined Algebras (Frontiers in Mathematics)

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An algebra A is said to be zero product determined (zpd for short) if every bilinear functional on A with the property that (x, y)=0 whenever xy=0 is of the form (x, y)=(xy) for some linear functional . If A is a Banach algebra, then we also require that and are continuous.

Perhaps this definition does not tell much. The intuition behind it is that many properties of A are determined by pairs of elements whose product is zero. This may still sound vague. The point, however, is that zpd (Banach) algebras form a rather wide class of not necessarily associative (Banach) algebras in which problems of different kinds can be solved. Their theory has reached a certain level of maturity. The purpose of this short book is to present it in a unified and concise manner, accessible to researchers and students of different backgrounds. The material is mostly taken from numerous papers published over the last 15 years, but some new results and new proofs of known results are also included.

The concept of a zpd algebra arose from two unrelated papers, [53] and [4]. The first one, written jointly with P. emrl, studies commutativity preserving linear maps on central simple algebras, and the second one, written jointly with J. Alaminos, J. Extremera, and A. Villena, studies local derivations and related maps on C-algebras. Their common feature is that the results on certain linear maps were derived from the consideration of bilinear maps that vanish on pairs of elements whose product is zero. This eventually led to the introduction of the concept of a zpd algebra and its systematic study, which evolved in two parallel directions: the algebraic and the (in many ways richer) analytic. Some terminological misfortunes occurred on the way: the analytic branch mostly uses the term Banach algebra with property (introduced in [6], in conjunction with the related property ) for the concept that is similar to that of a zero product determined algebra used in the algebraic branch. One of the aims of the book is to unify both branches, and along the way we will make some terminological adjustments. Property will therefore play only an auxiliary role.

The book is divided into three parts. Part I considers the algebraic theory and Part II the analytic theory. Part III is devoted to applications, that is, it examines problems from different areas of mathematics for which the zpd concept has proved useful. Some results in this last part are stated without detailed proofs. This is because the topics treated are very diverse and so many of them would need their own technical introduction. In the course of writing I realized that it is better to refer to original sources for details and focus primarily on ideas that illustrate the usefulness of the concepts studied in this book.

Parts of the book present results obtained jointly with my colleagues Jeronimo Alaminos, Jose Extremera, and Armando Villena from the University of Granada. I would like to thank them for the fruitful and pleasant long-term collaboration (and for being wonderful hosts during my frequent, enjoyable visits to Granada). The idea for the book arose from discussions with Armando, who was the leading force in developing the analytic branch of the theory. Although he did not join me as a coauthor, his influence can be felt as he has been giving me continuous help in the process of writing. My special thanks to him!

Finally, I would also like to thank an Bajuk for pointing out some errors in an earlier version, and the referees for useful comments.