Universal representation of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a nonzero $C^*$ -algebra. For every state $\varphi\in S(A)$ , choose its GNS representation $(\pi_\varphi,H_\varphi,\xi_\varphi)$ . The universal representation is the direct sum

\pi_u=\bigoplus_{\varphi\in S(A)}\pi_\varphi \quad\text{on}\quad H_u=\bigoplus_{\varphi\in S(A)}H_\varphi.

It is a faithful nondegenerate representation. Its unitary-equivalence class does not depend on the chosen models of the GNS triples. “Universal” here refers to simultaneously containing the cyclic representations generated by states, not to an initial or terminal object among all representations.

Why it is faithful

States separate the positive elements of a $C^*$ -algebra: for every nonzero $a$ , some state is nonzero on $a^*a$ . The corresponding GNS summand then acts nontrivially on $a$ , so $\pi_u(a)\neq0$ . This also gives a direct GNS proof of the faithful-representation part of the Gelfand–Naimark theorem Murphy, §§3.3–3.4.

Enveloping von Neumann algebra

The bicommutant $\pi_u(A)''$ is canonically isomorphic, as a von Neumann algebra, to the bidual $A^{**}$ . Under this identification the canonical map $A\to A^{**}$ agrees with $\pi_u$ , and every bounded functional on $A$ extends uniquely to a normal functional on $A^{**}$ . This is the universal enveloping von Neumann algebra construction Takesaki, vol. I, Chapter III.

Scope of the sum

Some sources sum GNS representations over all positive functionals, or choose only enough states to separate points. Such sums can also be faithful, but their multiplicities—and even their unitary-equivalence classes—need not agree with $\pi_u$ . The definition here fixes the direct sum over all states. It should not be confused with a universal $C^*$ -algebra specified by generators and relations.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §§3.3–3.4 on GNS representations and the faithful representation theorem.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III on the universal representation and enveloping von Neumann algebra.