Disjoint representations - Imported Knowlpedia Content

Definition

Let $A$ be a complex $C^*$ -algebra, and let $\pi_i:A\to B(H_i)$ be nondegenerate representations. They are disjoint if they have no nonzero subrepresentations that are unitarily equivalent. Equivalently, the only bounded operator $T:H_1\to H_2$ satisfying

T\pi_1(a)=\pi_2(a)T\qquad(a\in A)

is $T=0$ . Nondegeneracy removes irrelevant zero summands on which every operator would intertwine. Disjointness is a relation between the represented copies of $A$ , not merely a statement that the Hilbert spaces or the norm-closed image algebras are nonisomorphic.

Equivalent characterizations

If a nonzero intertwiner $T$ exists, its polar decomposition supplies a nonzero partial isometry intertwining the two representations. Its initial and final spaces then carry unitarily equivalent subrepresentations. This proves the equivalence between the two formulations in the core Dixmier, Chapter 5, §5.2.

After passing to the universal representation of $A$ , each $\pi_i$ has a normal extension to a central summand of $A^{**}$ . The representations are disjoint exactly when the corresponding central support projections are orthogonal. This is why disjointness is suited to central and direct-integral decompositions.

Relation to other equivalence notions

Unitary equivalence is the opposite extreme: it gives a unitary intertwiner, so two nonzero unitarily equivalent representations are never disjoint. Quasi-equivalence also precludes disjointness for nonzero representations, but it permits different Hilbert-space multiplicities. Factor representations exhibit the basic dichotomy: two factor representations are either quasi-equivalent or disjoint Takesaki, Chapter III, §2.

Examples and scope

For a commutative algebra $C_0(X)$ , point-evaluation representations at distinct points are disjoint: their one-dimensional subrepresentations are inequivalent. Repeating one evaluation with any nonzero multiplicity does not make it disjoint from the original evaluation, because a one-dimensional common subrepresentation remains.

References

Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: Chapter 5, §5.2 on disjointness and quasi-equivalence of representations.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. DOI record. Relevant: Chapter III, §2 on normal extensions, quasi-equivalence, and disjointness.