nLab unitarization

Idea

In representation theory, unitarization refers to isomorphisms from a given linear representation on a Hermitian inner product space (Hilbert space) to that underlying a unitary representation.

Properties

Existence for compact groups

For $G$ a compact topological group then every linear representation $\rho \,\colon\,G \to Aut(\mathscr{H})$ on a (complex separable) Hilbert space may be unitarized by re-defining the inner product of $\mathscr{H}$ to be the “group averaging” of the original one, given by the integral

(1)$$\langle v, w \rangle_{new} \;\coloneqq\; \int_{g \in G} \big\langle \rho(g)(v) ,\, \rho(g)(w) \big\rangle_{old} \, \mathrm{d}g$$

against the Haar measure on $G$.

(cf. Knapp 1996, Prop. 4.6)

Remark

(Terminology) Some authors refer to (1) as Weyl’s unitarian trick or similar, cf. Gomez 2009 p 5. But the original “unitarian trick” of Weyl 1939 pp 137 (cf. Knapp96, Prop. 7.15) is really referring to a another construction which just uses the above unitarization as an intermediate step (cf. Knapp96, p 445).

Example

In the case that $G$ is a finite group, formula (1) reduces to the sum

$$\langle v, w \rangle_{new} \;\coloneqq\; \tfrac{1}{{\vert G \vert}} \sum_{g \in G} \big\langle \rho(g)(v) ,\, \rho(g)(w) \big\rangle_{old} \,,$$

and it is elementary to see that this new inner product is indeed preserved:

$$\begin{array}{l} \big\langle \rho(h)(v) ,\, \rho(h)(w) \big\rangle_{new} \\ \;=\; \tfrac{1}{{\vert G \vert}} \, \sum_{g \in G} \Big\langle \rho(h)\big(\rho(g)(v)\big) ,\, \rho(h)\big(\rho(g)(w)\big) \Big\rangle_{old} \\ \;=\; \tfrac{1}{{\vert G \vert}} \, \sum_{g \in G} \big\langle \rho(h g)(v) ,\, \rho(h g)(w) \big\rangle_{old} \\ \;=\; \tfrac{1}{{\vert G \vert}} \, \sum_{g' \in G} \big\langle \rho(g')(v) ,\, \rho(g')(w) \big\rangle_{old} \\ \;=\; \tfrac{1}{{\vert G \vert}} \, \big\langle \rho(h)(v) ,\, \rho(h)(w) \big\rangle_{new} \mathrlap{\,.} \end{array}$$

Existence for locally compact amenable groups

Provided that one assumes enough choice to ensure weak-star compactness in the unit ball $B(\mathscr{H})$ (for $\mathscr{H}$ the representing Hilbert space): every bounded strong-operator-continuous representation of a locally compact amenable group on a Hilbert space $\mathscr{H}$ is unitarizable. See the Wikipedia page for Uniformly bounded representation [version: 2025-12-08] for some of the historical references.

References

The terminology “unitarian trick” goes back to:

• Hermann Weyl: The Classical Groups – Their Invariants and Representations, Princeton University Press (1939) [jstor:j.ctv3hh48t, pdf, Wikipedia entry]

  “We therefore take refuge in what might be called the unitarian trick: each group is replaced by the subgroup of those elements that are unitary transformations.”

Textbook account:

• Anthony W. Knapp: Lie Groups Beyond an Introduction, Progress in Mathematics 140 (1996, 2002) [ISBN:9780817642594]

Lecture notes:

• Raul Gomez, p 5 of: Introduction to Representation Theory of Lie groups (2009) [pdf]

See also:

• Wikipedia: Unitarian trick