Noncommutative Analysis

Category: teaching

Introduction to von Neumann algebras, Lecture 5 (comparison of projections and classification into types of von Neumann algebras)

In the previous lecture we discussed the group von Neumann algebras, and we saw that they can never be isomorphic to B(H). There is something fundamentally different about these algebras, and this was manifested by the existence of a trace. von Neumann algebras with traces are special, and the existence or non-existence of a trace can be used to classify von Neumann algebras, into rather broad “types”. In this lecture we will study the theory of Murray and von Neumann on the comparison of projections and the use of this theory to classify von Neumann algebras into “types”. We will also see how traces (or generalized traces) fit in. (For preparing these notes, I used Takesaki (Vol I) and Kadison-Ringrose (Vol. II).)

Most of the time we will stick to the assumption that all Hilbert spaces appearing are separable. This will only be needed at one or two spots (can you spot them?).

In addition to “Exercises”, I will start suggesting “Projects”. These projects might require investing a significant amount of time (a student is not expected to choose more than one project).

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The preface to “A First Course in Functional Analysis”

I am not yet done being excited about my new book, A First Course in Functional Analysis. I will use my blog to advertise my book, one last time. This post is for all the people who might wonder: “why did you think that anybody needs a new book on functional analysis?” Good question! The answer is contained in the preface to the book, which is pasted below the fold.

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Introduction to von Neumann algebras, Lecture 4 (group von Neumann algebras)

As the main reference for this lecture we use (more-or-less) Section 1.3 in the notes by Anantharaman and Popa (here is a link to the notes on Popa’s homepage).

As for exercises:  Read the rest of this entry »

Our new baby book

Finally, after a long delay, a package arrived containing some hard copies of my book.

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A First Course in Functional Analysis (my book)

She’hechiyanu Ve’kiyemanu!

My book, A First Course in Functional Analysis, to be published with Chapman and Hall/CRC, will soon be out. There is already a cover, check it out on the CRC Press website.

This book is written to accompany an undergraduate course in functional analysis, where the course I had in mind is precisely the course that we give here at the Technion, with the same constraints. Constraint number 1: a course in measure theory is not mandatory in our undergraduate program. So how can one seriously teach functional analysis with significant applications? Well, one can, and I hope that this book proves that one can. I already wrote before, measure theory is not a must. Of course anyone going for a graduate degree in math should study measure theory (and get an A), but I’d like the students to be able to study functional analysis before that (so that they can do a masters degree in operator theory with me).

I believe that the readers will find many other original organizational contributions to the presentation of functional analysis in this book, but I leave them for you to discover. Instructors can request an e-copy for inspection (in the link to the publisher website above), friends and direct students can get a copy from me, and I hope that the rest of the world will recommend this book to their library (or wait for the libgen version).

The dominated convergence theorem for the Riemann and the improper Riemann integral (Measure theory is a must – part II)

(Hello students of Infi 2 – this post is for you).

In this post I will describe the dominated convergence theorem (DCT) for the Riemann and improper Riemann integrals. The previous post can serve as an introduction (a slanted one, beware) to this one. My goal is to convince that the important and useful convergence theorems in integration theory can (and therefore, needless to say, should) be taught in a first course on Riemannian integration.

The bounded convergence theorem for the Riemann integral is also known as Arzela’s Theorem, and this post does not contain anything new. In preparing this post I used as reference the short note “A truly elementary approach to the bounded convergence theorem”, J. W. Lewin, The American Mathematical Monthly. This post can be considered as a destreamlinization of that note. I think my presentation is even more “truly elementary”, since I avoid introducing inner measure. Warning: this post will really truly be at a very elementary level. Read the rest of this entry »

Measure theory is a must

[This post started out as an introduction to a post I was planning to write on convergence theorems for the Riemann integral. The introduction kind of got out hand, so I decided to post it separately. Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break (but in any case before we need them for the course I am teaching this term, Calculus 2)].

 

Mathematicians love to argue about subjective opinions. One of the most tiresome and depressing subjects of debate is “What should an undergraduate math major curriculum contain?”

 

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