These are a minimally polished set of lecture notes for a course
on C∗-algebras which I’ve given several times at Dartmouth College.
The
LATEX version here was produced during the Spring 2011 instance of the course.
I used Murphy’s book [Mur90] as the primary source and no doubt some of
the arguments here are his.
It seem unlikely that these notes will ever be published. In addition to
Murphy’s book, there are the classics by Dixmier [Dix77, Dix81], Takesaki
[Tak79] and Pedersen [Ped82], and a host of newer books such as Davidson
[Dav96] and Kadison-Ringrose [KR83].
There is even an Encyclopaedia of
Mathematical Sciences volume by Blackadar [Bla06].
Nevertheless, I am providing these to future graduate students as a quick
introduction. Anyone if free to use them as they see fit. However, I would
be very grateful to hear about typos, corrections and especially suggestions to
improve the exposition. In particular, I will continue to tinker with them so
the page and theorem numbers will certainly change over time.
Thanks to James Ronan and Alexander Kazantzidhs for providing valuable
feedback after reading through an earlier draft.
Author(s): Dana P. Williams
Year: 2020
Introduction
Algebras
Getting Started
Exercises
The Spectrum
Exercises
The Gelfand Transform
Exercises
Examples: Abelian Harmonic Analysis
Getting it Right: the C*-Norm Identity
Banach *-algebras
Exercises
The Functional Calculus
Exercises
Positivity
Exercises
Approximate Identities
Exercises
Homomorphisms
Exercises
Representations
Representations
Exercises
Representations and Ideals
Exercises
Compact Operators
Exercises
Representations of the Compacts
Exercises
Classes of C*-Algebras
Exercises
The Gelfand-Naimark Theorem
The GNS Construction
Exercises
Pure States
Exercises
Simple Examples
Direct Limits of C*-Algebras
Exercises
Projections and Traces
Exercises
UHF Algebras
Exercises
AF Algebras
Exercises
K-Theory
Matrix Algebras
Exercises
The K0 Group
Exercises
The Positive Cone
Exercises
Projections in Direct Limits
Elliot's Theorem
Elliot's Theorem for Unital AF-Algebras
Back to Glimm Algebras
Exercises
Answers to Some of the Exercises
Notation and Symbol Index
Index
Bibliography
