C*-algebra by generators and relations - Imported Knowlpedia Content

Definition

A $C^*$ -algebra by generators and relations consists of generators $\mathcal G$ , relations $\mathcal R$ , and a universal $C^*$ -algebra $C^*(\mathcal G\mid\mathcal R)$ containing a distinguished realization of them. Every admissible realization of $(\mathcal G,\mathcal R)$ in a $C^*$ -algebra $B$ must induce a unique $*$ -homomorphism $C^*(\mathcal G\mid\mathcal R)\to B$ carrying universal generators to the chosen elements. Admissibility includes the unital or nonunital convention and enough uniform norm control that every $*$ -polynomial in the generators has finite supremum norm over all realizations.

Construction and existence

Form the algebraic involutive algebra on $\mathcal G$ , impose $\mathcal R$ , and define the norm of a class $x$ as the supremum of $\|\pi(x)\|$ over all admissible realizations $\pi$ . One then quotients by the zero-norm elements and completes. This produces the universal algebra when every supremum is finite. Algebraic consistency alone is insufficient: relations can admit representations while failing to bound a generator, so no universal $C^*$ -norm exists Loring, Chapter 3.

Standard examples

The relation $u^*u=uu^*=1$ defines a universal unitary and yields $C(\mathbb T)$ . Requiring an isometry $v^*v=1$ yields the Toeplitz algebra, while the Cuntz relations define the Cuntz algebras. In each case, the relations supply norm bounds. The presentation determines the algebra only together with its admissible representation class and its convention about units.

Relation to algebraic presentations

An algebraic presentation uses a quotient of a free involutive algebra. A $C^*$ -presentation additionally uses analytic completion in a maximal representation seminorm. It may therefore identify more elements through the common kernel of all bounded realizations. The phrase “generated by” can also describe a particular $C^*$ -subalgebra inside an ambient algebra; that relative construction does not by itself assert this universal mapping property.

References

Bruce Blackadar, Operator Algebras: Theory of $C^*$ -Algebras and von Neumann Algebras, Springer, 2006. DOI record. Relevant: §II.8 on universal constructions and generators and relations.
Terry A. Loring, Lifting Solutions to Perturbing Problems in $C^*$ -Algebras, American Mathematical Society, 1997. AMS DOI record. Relevant: Chapter 3 on admissible $C^*$ -relations and universal algebras.