Advanced Courses Of Mathematical Analysis V - Proceedings Of The Fifth International School

Proceedings of the Fifth International School

Juan Carlos Navarro Pascual, El Amín Kaidi

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789814699709
• Topic: Matemáticas
• Subtopic: Investigación y estudio matemático

This volume contains recent papers by several specialists in different fields of mathematical analysis. It offers a reasonably wide perspective of the current state of research, and new trends, in areas related to measure theory, harmonic analysis, non-associative structures in functional analysis and summability in locally convex spaces.

Those interested in researching any areas of mathematical analysis will find here numerous suggestions on possible topics with an important impact today. Often, the contributions are presented in an expository nature and this makes the discussed topics accessible to a more general audience.

Contents:

• Measurability and Semi-Continuity of Multifunctions (B Cascales)
• Introduction to Interpolation Theory (F Cobos)
• Optimality of Function Spaces in Sobolev Embeddings (L Pick)
• Derivations and Projections on Jordan Triples: An introduction to Nonassociative Algebra, Continuous Cohomology, and Quantum Functional Analysis (B Russo)
• Weighted Inequalities and Extrapolation (J Duoandikoetxea)
• A Note on the Off-Diagonal Muckenhoupt–Wheeden Conjecture (D Cruz-Uribe, J M Martell and C Pérez)
• On the Interplay Between Nonlinear Partial Differential Equations and Game Theory (J D Rossi)
• The Radon–Nikodým Theorem for Vector Measures and Integral Representation of Operators on Banach Function Spaces (E A Sánchez Pérez)
• The Orlicz–Pettis Theorem for Multiplier Convergent Series (C Swartz)

Readership: Graduate students in mathematics and researchers in mathematical analysis.

Derivations and projections on Jordan triples: An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis

Bernard Russo

Department of Mathematics, University of California, Irvine,
525 Rowland Hall, Irvine, CA 92697-3875, USA
E-mail: [email protected]

This paper is an elaborated version of the material presented by the author in a three-hour minicourse at V International Course of Mathematical Analysis in Andalusia, at Almeria, Spain September 12-16, 2011. The author wishes to thank the scientific committee for the opportunity to present the course and to the organizing committee for their hospitality. The author also personally thanks Antonio Peralta for his collegiality and encouragement.

The minicourse on which this paper is based had its genesis in a series of talks the author had given to undergraduates at Fullerton College in California. I thank my former student Dana Clahane for his initiative in running the remarkable undergraduate research program at Fullerton College of which the seminar series is a part. With their knowledge only of the product rule for differentiation as a starting point, these enthusiastic students were introduced to some aspects of the esoteric subject of non associative algebra, including triple systems as well as algebras. Slides of these talks and of the minicourse lectures, as well as other related material, can be found at the author’s website (www.math.uci.edu/~brusso). Conversely, these undergraduate talks were motivated by the author’s past and recent joint works on derivations of Jordan triples (¹,²,³), which are among the many results discussed here.

Part I (Derivations) is devoted to an exposition of the properties of derivations on various algebras and triple systems in finite and infinite dimensions, the primary questions addressed being whether the derivation is automatically continuous and to what extent it is an inner derivation.

Part II (Cohomology) discusses cohomology theory of algebras and triple systems, in both finite and infinite dimensions. Although the cohomology of associative and Lie algebras is substantially developed, in both finite and infinite dimensions (⁴,⁵,⁶; see⁷ for a review of⁴), the same could not be said for Jordan algebras. Moreover, the cohomology of triple systems has a rather sparse literature which is essentially non-existent in infinite dimensions. Thus, one of the goals of this paper is to encourage the study of continuous cohomology of some Banach triple systems. Occasionally, an idea for a research project is mentioned (especially in subsection 6.2). Readers are invited to contact the author, at [email protected], if they share this interest.

Part III (Quantum Functional Analysis) begins with the subject of contractive projections, which plays an important role in the structure theory of Jordan triples. The remainder of Part III discusses three topics, two very recent, which involve the interplay between Jordan theory and operator space theory (quantum functional analysis). The first one, a joint work of the author⁸, discusses the structure theory of contractively complemented Hilbertian operator spaces, and is instrumental to the third topic, which is concerned with some recent work on enveloping TROs and K-theory for JB^∗-triples⁹,¹⁰,¹¹. The second topic presents some very recent joint work by the author concerning quantum operator algebras which leans very heavily on contractive projection theory¹².

Parts I ...