Unitary equivalence of C*-representations - Imported Knowlpedia Content

Definition

Let $\pi:A\to\mathcal B(H_\pi)$ and $\rho:A\to\mathcal B(H_\rho)$ be $C^*$ -representations of the same algebra. They are unitarily equivalent if there is a unitary operator $U:H_\pi\to H_\rho$ such that

U\pi(a)=\rho(a)U\qquad(a\in A).

Equivalently, $\rho(a)=U\pi(a)U^*$ for every $a$ . The unitary $U$ is an intertwiner, and it identifies the representations together with their Hilbert-space inner products. Similarity by a merely bounded invertible operator is not the equivalence relation defined here.

Equivalence relation

Unitary equivalence is reflexive, symmetric, and transitive: use the identity, $U^*$ , and composition of implementing unitaries, respectively. It is therefore meaningful to classify representations by unitary-equivalence classes rather than by chosen Hilbert-space realizations.

Preserved structure

The intertwining relation carries invariant and cyclic subspaces from one representation to the other. It also preserves kernels, faithfulness, nondegeneracy, irreducibility, and multiplicities in direct-sum decompositions. Moreover, their commutants satisfy

U\pi(A)'U^*=\rho(A)',

so they are spatially isomorphic. These basic invariances are part of the standard representation theory developed in Pedersen, Chapter 3.

Pointed cyclic representations

For cyclic representations with pointed triples $(\pi,H,\xi)$ and $(\rho,K,\eta)$ , equivalence of the pointed triples additionally requires $U\xi=\eta$ . The uniqueness statement in the GNS construction is of this stronger pointed form. Two unpointed representations can be unitarily equivalent even when a chosen cyclic vector is not carried to another chosen vector.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: Chapter 3 on representations and intertwiners.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 3 on unitary equivalence and cyclic representations.