Universal C*-algebra

In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable as algebras of bounded operators on a Hilbert space by the Gelfand-Naimark-Segal construction and the relations must prescribe a uniform bound on the norm of each generator. This means that depending on the generators and relations, a universal C*-algebra may not exist. In particular, free C*-algebras do not exist.

Definitions

A relation is a pair consisting of a *-polynomial p in n non-commuting variables and a non-negative number ε. This definition can be generalized to include continuous functions using continuous functional calculus. A representation on a set of relations R in a C*-algebra A is a *-homomorphism φ from the free algebra (not a C*-algebra) generated by n elements: x₁, x₂, ..., x_n such that ||(φ(p(x₁, ..., x_n))|| ≤ ε for each relation (p,ε) in R.

A set of relations R is called bounded if a representation on R exists for some C*-algebra and ||ρ(x_i)|| is uniformly bounded for all i and all representations ρ.

Examples

• The universal C*-algebra generated by a unitary element u has presentation <u | u*u = uu* = 1>. By the functional calculus, this C*-algebra is the continuous functions on the unit circle in the complex plane. Any C*-algebra generated by a unitary element is the homomorphic image of this universal C*-algebra.