MAT 307: Noncommutative Functional Analysis
13 — 17 September 2004
- Lecture Notes for MAT 307 (Adobe Acrobat PDF)
| Instructor: | David P. Blecher Full Professor Department of Mathematics University of Houston Houston, Texas |
Dr. Blecher has some experience in successfully navigating undergraduates through advanced topics. He co-ran a very successful Research Experience for Undergraduates (REU) in Houston in the Summer of 2001. The students who worked with him then have papers published, or in the process of publication. Partly on the basis of this, one of these students (Kay Kirkpatrick) was a co-winner of the prestigious national award, the Alice T. Schafer Prize for Excellence in Mathematics by an Undergraduate Woman from the Association for Women in Mathematics, and has gone on to graduate school at UC Berkeley. The notes that Blecher produced for his REU students, will form the basis of the first half of this Lecture Mini-Series, and is a manageable crash course in basic functional analysis for undergraduates, including the beginnings of operator theory (from a matrix perspective). The ideas are a blend of linear algebra, analysis, more general algebra, and the basics of functional analysis. The students will be expected to reread these notes between lectures (we have designed the lectures so they do not rely on the student doing this, although we hope that most students will). Blecher is also in the process of writing a book on the subject of noncommutative functional analysis, which is to appear in the Oxford University Press in September 2004. Parts of Chapter 1 of this book will form the basis for the second half of this Lecture Mini-Series.
In the first half we also cover some ideas from “noncommutative mathematics”, to acquaint and familiarize students with key concepts. The central idea is that of “quantization”. We will show that studying some of the common spaces in mathematics (such as topological spaces, measure spaces, groups, and so on) is the same as studying ‘rings’ of appropriate functions on these spaces. Typically, these rings are commutative in most classical settings. These commutative rings can then be replaced by noncommutative rings having the same formal properties. There are now important theories of noncommutative topology, noncommutative probability, noncommutative differential geometry, quantum groups, noncommutative functional analysis, and so on. We will only spend a few minutes briefly surveying each of these theories before narrowing our interest down to the last one: the theory of operator spaces. Operator spaces may loosely be defined to be vector spaces whose elements are ‘operators on Hilbert space’.
A few selected features of operator space theory will be discussed in detail. We will discuss some connections with ordinary functional analysis. The items we need from these theories will be stated of course, and examples given of their use. For example, we hope to treat something like the following sequence of topics:
(1) The Hahn-Banach theorem, and a discussion of its operator space version, and of the duality of operator spaces.Prerequisites: Calculus I — III and Linear Algebra.
(2) A discussion of Banach-Stone type theorems and their noncommutative analogues.
(3) A discussion of injective operator spaces, culminating with some easy to understand recent results due to Blecher with Effros and Zarikian, which cut to the heart of ‘operator multiplication’.
(4) Application of the last item to the abstract characterization of operator algebras.
(5) A presentation of some key points of the recent solution by Pisier of the famous “Halmos similarity problem”.
Biographical Information: Prof. Blecher is a great communicator of mathematics. A captivating speaker, Blecher is also a prolific researcher of mathematics. Blecher studied in Cambridge, England, and Edinburgh, Scotland, where he received his Ph.D. in 1998. Since then he has been at the University of Houston, with the exception of visiting posts at the University of California, Berkeley, and the University of Missouri.
Prof. Blecher’s research interests include Operator Algebras and Functional Analysis. Blecher is a world-leading figure in the exciting new field of Operator Spaces. Recently his work has focused on Hilbert C*-modules (which are a noncommutative generalization of a vector bundle), and on injectivity and extremal representations of Operator Spaces. Blecher has published numerous deep, field influencing, papers. He is also the coauthor of two memoirs, and a book entitled “Operator algebras and their modules” (Oxford University Press, 2004).