Universal C-star algebras

Definition#

Suppose that a set \(\mathcal{G} = \{x_i : i\in\Omega\}\) of generators and a set \(\mathcal{R}\) of relations are given the relations can be of a very general nature, but are usually algebraic relations among the generators and their adjoints, or more generally of the form

\[|| p(x_{i_1},\ldots, x_{i_n}, x^\ast_{i_1},\ldots,x^\ast_{i_n}|| \leq \eta\]

where  \(p\) is a polynomial in \(2n\) noncommuting variables with complex coefficients and \(\eta\geq 0\) - the only restriction being that they must be realizable among operators on a Hilbert space and must at least implicitly place an upper bound on the norm of each generator when realized as an operator.

A representation of \((\mathcal{G}|\mathcal{R})\) has the obvious meaning: a set \(\{T_i : i\in\Omega\}\) of bounded operators on a Hilbert space \(\mathcal{H}\) satisfying the relations. A representation of \((\mathcal{G}|\mathcal{R})\) defines a \(\ast\)-representation of the free \(\ast\)-algebra \(\mathcal{A}\) on the set \(\mathcal{G}\). For \(x\in\mathcal{A}\), let

\[||x|| = \mathrm{sup}\{||\pi(x)||:\pi\text{ a representation of }(\mathcal{G}|\mathcal{R})\}.\]

If for all \(x\in\mathcal{A}\) this supremum is finite, it defines a \(C^\ast\)-seminorm on \(\mathcal{A}\), and the completion with elements of seminorm \(0\) divided out is called the universal \(C^\ast\)-algebra on \((\mathcal{G}|\mathcal{R})\), denoted \(C^\ast(\mathcal{G}|\mathcal{R})\).

More Info#

The construction is extremely general: every \(C^\ast\)-algebra can be written as the universal \(C^\ast\)-algebra on a suitable set of generators and relations, albeit it might be uninterseting.