Жалпылама C* алгебралары: Генераторлар мен қатынастар арқылы сипаттама

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.

Alternatively, one can use a more concrete characterization of universal C* algebras that more closely resembles the construction in abstract algebra. Unfortunately, this restricts the types of relations that are possible. Given a set G, a relation on G is a set R consisting of pairs (p, η) where p is a * polynomial on X and η is a non negative real number. A representation of (G, R) on a Hilbert space H is a function ρ from X to the algebra of bounded operators on H such that for all (p, η) in R. The pair (G, R) is called admissible if a representation exists and the direct sum of representations is also a representation. Then

is finite and defines a seminorm satisfying the C* norm condition on the free algebra on X. The completion of the quotient of the free algebra by the ideal is called the universal C* algebra of (G,R).

The noncommutative torus can be defined as a universal C* algebra generated by two unitaries with a commutation relation. The Cuntz algebras, graph C* algebras and k graph C* algebras are universal C* algebras generated by partial isometries. The universal C* algebra generated by a unitary element u has presentation By continuous functional calculus, this C* algebra is the algebra of continuous functions on the unit circle in the complex plane. Any C* algebra generated by a unitary element is isomorphic to a quotient of this universal C* algebra.