Weak containment of unitary representations - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group, and let $\pi$ and $\rho$ be strongly continuous unitary representations on Hilbert spaces $H_\pi$ and $H_\rho$ . The representation $\pi$ is weakly contained in $\rho$ , written $\pi\prec\rho$ , if for every $\xi\in H_\pi$ , compact set $K\subseteq G$ , and $\varepsilon>0$ , there exist $\eta_1,\ldots,\eta_n\in H_\rho$ such that

\left|\langle\pi(g)\xi,\xi\rangle- \sum_{k=1}^n\langle\rho(g)\eta_k,\eta_k\rangle\right|<\varepsilon

for every $g\in K$ . The integer $n$ and the approximating vectors may depend on $\xi$ , $K$ , and $\varepsilon$ . Thus the condition compares positive definite diagonal coefficients uniformly on compact subsets, allowing finite sums of coefficients of $\rho$ ; it does not require an isometric embedding of $H_\pi$ into $H_\rho$ .

Equivalent operator-norm criterion

For the integrated representations of $L^1(G)$ , weak containment is equivalent to

\lVert\pi(f)\rVert\leq\lVert\rho(f)\rVert \qquad\text{for every }f\in L^1(G).

Equivalently, after passing to the full group $C^*$ -algebra, $\ker\rho\subseteq\ker\pi$ . The direction of this inclusion is important: the containing representation detects at least as large a norm. These equivalences are part of Fell's theory of weak containment Fell, 1962.

Coefficients and direct sums

The approximation condition uses diagonal coefficient functions; polarization recovers approximation statements for general coefficients. A finite sum of diagonal coefficients of $\rho$ is a single diagonal coefficient of a finite direct sum of copies of $\rho$ . Consequently, $\pi\prec\rho$ and $\rho\prec\sigma$ imply $\pi\prec\sigma$ .

Conventions and scope

References

J. M. G. Fell, "Weak Containment and Induced Representations of Groups," Canadian Journal of Mathematics 14 (1962), 237–268. DOI record. Relevant: the definition and basic permanence properties of weak containment.
Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan's Property (T), Cambridge University Press, 2008. DOI record. Relevant: Appendix F on weak containment.