An Invitation to C*-Algebras (Graduate Texts in Mathematics 39) 1st Edition by William Arveson (PDF)

Description

This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-algebras become GCR. This practice probably creates an impression that nothing of value is known about other C*-algebras. Of course that is not true. But insofar as representations are con­ cerned, we can point to the empirical fact that to this day no one has given a concrete parametric description of even the irreducible representations of any C*-algebra which is not GCR. Indeed, there is metamathematical evidence which strongly suggests that no one ever will (see the discussion at the end of Section 3. 4). Occasionally, when the idea behind the proof of a general theorem is exposed very clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. 1 and 2. 2. (it would be desirable but not necessary to include Section 1. 1 as well), whereas someone interested in Borel structures could read Chapter 3 separately. Chapter I could be used as a bare-bones introduction to C*-algebras. Sections 2.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐”Operator algebras! Oh, right! That is Arveson’s book, isn’t it?”–From a typical conversation I would have with a colleague from the other end of campus.– Or, an impatient student, I meet in the elevator, wants to quickly get an idea of what it is all about,– and have fun reading about the material, at the same time: That would be Arveson’s lovely little book!– Sure, there *are* lots of great books in the subject; but they haven’t had this same kind of wide impact. The central ideas are very attractively presented: It *is* an invitation! The other books in C*-algebra theory will typically be thicker, and they might be more narrowly focused, –more for the specialists, if you like. The first edition of Arveson’s book is from the seventies; but still,– now many years later, everyone knows “An invitation”, and reads it. Authors keep immitating its approach and its style: You see immitators,–authors in other specialties of math writing books entitled “An invitation to …”. But none of the immitations seem to have quite the charm of the original. You can’t very well **plan to** write a charming book in math. But when one arrives, we all know it. The subject of Arveson’s book started with quantum theory, Hilbert space, spectral theory, and representations of groups and algebras. And, in the half century plus, since its inception, the subject has found applications in a surprisingly wide range of other fields: geometry, K-theory, fundamental physics, symbolic dynamics, and tiling theory, to mention just a few.

⭐This text appeared near the end of my Ph.D program on ideal theory in C*-Algebras.Although brief (compared to, say, Dixmier), it cuts to the chase. The first paragraph of the preface well summarizes my comments: For convenience Hilbert Spaces are separable and C*-Algebras are GCR. I would like to emphasize one point:In certain cases, specific results are proven, and the more general cases are left as exercises. Here is an invitation to do all the exercises in the text! If you succeed, you are well on your way. I have used, and continue to use, this fine text in a variety of ways.

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