Universal C*-algebra - Imported Knowlpedia Content

Definition

Fix generators $\mathcal G$ , relations $\mathcal R$ , and a choice of unital or nonunital category of $C^*$ -algebras. A universal $C^*$ -algebra for this data is a $C^*$ -algebra $C^*(\mathcal G\mid\mathcal R)$ with distinguished generators satisfying $\mathcal R$ , such that every realization of the same relations in an object $B$ induces a unique $*$ -homomorphism $C^*(\mathcal G\mid\mathcal R)\to B$ carrying each universal generator to its realization. The chosen morphism class is part of the specification. When it exists, the universal algebra is determined up to a unique $*$ -isomorphism preserving the generators.

Existence and construction

Formal generators and relations first define an algebraic $*$ -algebra. One then takes the supremum of the operator seminorms arising from all admissible $*$ -representations and completes after quotienting by the common kernel. Existence requires these seminorms to be finite on every element; in generators-and-relations language this is typically ensured by admissibility or explicit norm bounds. Arbitrary algebraic relations need not define a nonzero, or even existent, universal $C^*$ -algebra Blackadar, §II.8.

Standard examples

The universal unital $C^*$ -algebra generated by one unitary is $C(\mathbb T)$ : a unitary $u$ in any unital $C^*$ -algebra determines a unique unital $*$ -homomorphism sending the coordinate function to $u$ . The full group $C^*$ -algebra is universal for unitary representations of a group, while the Cuntz algebras are universal for isometries satisfying the Cuntz relations. Each example depends on its stated morphism and unit conventions.

Conventions and scope

“Universal $C^*$ -algebra” describes a representing property, not one single algebra. In the unital category the comparison maps are required to preserve the identity; in a nonunital category they are not. Some sources reserve “universal enveloping $C^*$ -algebra” for the $C^*$ -completion of a $*$ -algebra. This notion is unrelated to universal algebra in the sense of algebraic signatures and varieties, and “universal algebra” is therefore not used here as an alias.

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. DOI record. Relevant: Chapter 3 on universal $C^*$ -algebras and admissible relations.