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.
Asked
Active
Viewed 2,042 times
3 Answers
5
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".
Prahlad Vaidyanathan
- 31,554
-
In both the sources when talking about the set of relations "We could allow the relations in $\mathcal{R}$ to be any kind of relations which could be formulated for operators on a Hilbert space or for elements of a $C^*$-algebra ..." Is $||a|| \ge ||b||$ a valid kind of relation, where $a$, $b$ are members of the generating set ? – nsoum Aug 18 '13 at 10:25
-
No, I believe the only valid relationships are of the form $|p(x_1,x_2, \ldots, x_n, x_1^{\ast}, x_2^{\ast}, \ldots, x_n^{\ast})| \leq \alpha$ for some polynomial $p$ with comlex coefficients in $2n$ commuting variables, $x_1, x_2, \ldots, x_n$ in the generating set, and $\alpha \geq 0$ – Prahlad Vaidyanathan Aug 18 '13 at 12:10
-
The author says "for specificity" we consider the relations of the form you wrote. I thought the wording was a bit vague and did not rule out possibilities like the one I mentioned. What is the reason for only considering relations of the above form ? – nsoum Aug 18 '13 at 16:42
-
I quote Blackadar : "The only restriction on the relations is that they must be realizable among operators on a Hilbert space and they must (at least implicitly) place an upper bound on the norm of each generator when realized as an operator." So, I think conditions like $|a| \leq |b|$ might be permissible, provided the universal C* algebra so constructed has a representation! – Prahlad Vaidyanathan Aug 18 '13 at 16:54
4
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.
Glorfindel
- 3,955
Mike F
- 22,196
-
1Interestingly, the universal Cstar algebra generated by a partial isometry is apparently quite horrible: http://www.ams.org/journals/proc/2012-140-01/S0002-9939-2011-10988-2/S0002-9939-2011-10988-2.pdf – Rasmus Jun 02 '15 at 11:05
-
If we work in the category of unital $C^*$-algebras and unital maps, do you know whether it is equally horrible? – Ulrik Feb 10 '16 at 20:09
2
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.
Strottos
- 168