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This advanced volume of lecture notes presents an open problem at the frontier of research into operator algebra theory, based on the author's university lecture courses and written in a widely accessible style for researchers and Ph.D. students with little experience in the area.

Martingales arise in many areas of probability theory. This book focuses on their applications to the geometry of Banach spaces and discusses the interplay of Banach space valued martingales with various other areas of analysis. It is accessible to graduates with a basic knowledge of real and complex analysis.