universal C* algebras

Question

Is there a standard reference which has a discussion on universal $C^*$-algebras ? (definition, properties, examples, etc) Searching on the internet has led me to tidbits of information but I would like to read the initial papers or some standard references which introduced these ideas.

Answer 1

Blackadar's "Operator Algebras : Theory of C* algebras and Von-Neumann Algebras" has a section on universal C* algebras. So does Blackadar's paper on "Shape Theory for C* algebras".

Answer 2

A good place to start is with C*-algebras generated by isometries: See Coburn's 1967 paper for the case of a single isometry. See Cuntz's 1977 paper for the case of several isometries whose range projections partition the identity.

An even simpler case is the universal C-algebra generated by a single unitary which is just $C(\mathbb{T})$, as follows from functional calculus. It is instructive to convince yourself that there does not exist, for instance, a universal C-algebra generated by a single self-adjoint.

Group C-algebras and, more generally, crossed products constitute important examples of universal C-algebras. A 1986 paper of Ian Raeburn makes the universality of crossed-products precise.

Semigroup crossed products are also popular, see the classical case of Toeplitz algebras. Even more generally, one has graph C*-algebras.

Answer 3

Depends what you want to know exactly. There's some relatively elementary (but still useful) information in Davidson's `C$^*$-algebra's by Example' in Chapter 7 on Group C$^*$-algebras. It also goes into the related reduced C$^*$-algebras.

As I say it depends on your interest's I suspect. I had to learn about them for C$^*$-algebraic quantum groups for which Timmerman's book `An Invitation to Quantum Groups and Duality' has various bits of information scattered plus some brief (though again useful) information in the appendix.