Quasi-equivalent representations

Two representations $\pi_i:A\to\mathcal B(H_i)$ of the same $C^*$ -algebra are quasi-equivalent if the assignment

\pi_1(a)\longmapsto\pi_2(a)

extends to a $*$ -isomorphism

\Phi:\pi_1(A)''\longrightarrow\pi_2(A)''

between their generated von Neumann algebras, and $\Phi$ is a normal $*$ -homomorphism with normal inverse. In particular, the two representations must have the same kernel. Normality is essential: it requires the isomorphism to preserve the weak-limit structure supplied by the two concrete representations, not only their norm-closed images.

Relation to unitary equivalence

Unitary equivalence implies quasi-equivalence by $\Phi(x)=UxU^*$ . The converse fails because quasi-equivalence forgets Hilbert-space multiplicity. For example, a nonzero representation $\pi$ and its amplification $a\mapsto\pi(a)\otimes I_K$ are quasi-equivalent: $x\mapsto x\otimes I_K$ normally identifies the generated von Neumann algebras. They need not be unitarily equivalent when the multiplicities differ.

Normal representation content

Quasi-equivalence says that the two weak closures carry the same normal representation theory while retaining the distinguished copy of $A$ . It preserves factoriality and the type of the generated von Neumann algebra. For nondegenerate representations it is also characterized by equality of the sets of positive functionals on $A$ that are normal relative to the representations.

Conventions and near misses

References

Jacques Dixmier, C-Algebras*, North-Holland, 1977. Publisher record. Relevant: Chapter 3 on equivalence and quasi-equivalence of representations.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter III, §2 on normal extensions and quasi-equivalence.