Tensor product of unitary representations - Imported Knowlpedia Content

Definition

Let $G$ be a topological group, and let $\pi:G\to\mathcal U(\mathcal H)$ and $\sigma:G\to\mathcal U(\mathcal K)$ be strongly continuous unitary representations on complex Hilbert spaces. Their tensor product representation is the homomorphism

\pi\otimes\sigma:G\longrightarrow \mathcal U(\mathcal H\widehat\otimes_2\mathcal K)

specified on elementary tensors by

(\pi\otimes\sigma)(g)(\xi\otimes\eta) =\pi(g)\xi\otimes\sigma(g)\eta.

Here $\mathcal H\widehat\otimes_2\mathcal K$ is the completed Hilbert tensor product. The formula preserves inner products, extends uniquely to a unitary operator for each $g$ , and defines a strongly continuous representation.

Continuity and coefficients

Strong continuity is immediate on finite sums of elementary tensors and extends to the completion because all operators have norm one. Matrix coefficients multiply:

\langle(\pi\otimes\sigma)(g)(\xi\otimes\eta), \xi'\otimes\eta'\rangle =\langle\pi(g)\xi,\xi'\rangle \langle\sigma(g)\eta,\eta'\rangle.

This identity makes tensor products fundamental in studying positive-definite functions, weak containment, and representations possessing invariant vectors.

Exterior tensor products

If $\pi$ represents $G$ and $\sigma$ represents another group $H$ , their exterior tensor product is the representation of $G\times H$ defined by

(\pi\boxtimes\sigma)(g,h)=\pi(g)\otimes\sigma(h).

Restricting this exterior product along the diagonal map $G\to G\times G$ recovers the tensor product in the core. Thus the two constructions are related but are not synonyms.

Conventions and scope

The completion is essential: the algebraic tensor product is dense but usually not complete. The construction here is for unitary Hilbert-space representations, not projective tensor products of Banach representations. In particular, “tensor product representation” does not mean a direct sum or a direct integral.

References

Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan's Property (T), Cambridge University Press, 2008. Appendix A DOI record. Relevant: Appendix A on tensor products of unitary representations.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 3 on unitary representations.