---

Written as a hybrid between a research monograph and a textbook the first half of this book is concerned with basic concepts for the study of Banach algebras that, in a sense, are not too far from being commutative. Essentially, the algebra under consideration either has a sufficiently large center or is subject to a higher order commutator property (an algebra with a so-called polynomial identity or in short: Pl-algebra). In the second half of the book, a number of selected examples are used to demonstrate how this theory can be successfully applied to problems in operator theory and numerical analysis.

Distinguished by the consequent use of local principles (non-commutative Gelfand theories), PI-algebras, Mellin techniques and limit operator techniques, each one of the applications presented in chapters 4, 5 and 6 forms a theory that is up to modern standards and interesting in its own right.

Written in a way that can be worked through by the reader with fundamental knowledge of analysis, functional analysis and algebra, this book will be accessible to 4th year students of mathematics or physics whilst also being of interest to researchers in the areas of operator theory, numerical analysis, and the general theory of Banach algebras.

---

Banach algebras.- Local principles.- Banach algebras generated by idempotents.- Singular integral operators.- Convolution operators.- Algebras of operator sequences.

---

From the reviews:

"This book consists of two parts, the first half of which can be thought of as a textbook suitable for a course on Banach algebras. ... The book is remarkably well written, and puts together in a systematic manner the material that was only available before in the form of journal publications ... . it is both accessible to graduate (and even advanced undergraduate) students and of interest to seasoned researchers working on various aspects of operator theory and numerical analysis." (I. Spitkovsky, Mathematical Reviews, Issue 2012 e)

"The theme of this book is local principles, which are formulated in the language of Banach algebras and may be regarded as noncommutative Gelfand theories. ... The book shall be useful for students and researchers working in operator theory, functional analysis and numerical analysis." (Mohammad Sal Moslehian, Zentralblatt MATH, Vol. 1209, 2011)

---