Theorem 4.5. (Weyl’s Trick). Let G be a compact group and E a G-module which is, at the same time, a Hilbert space. Then there is a scalar product relative to which all operatorsπ(g) are unitary.
Specifically, if (• | •) is the given scalar product onE, then
(1) hx|yi=
Z
G
(gx|gy)dg defines a scalar product such that
(2) M−2(x|x)≤ hx|xi ≤M2(x|x) with
(3) M = sup{p
(gx|gx)|g∈G,(x|x)≤1}, and that
(4) hgx|gyi=hx|yi for all x, y∈E, g∈G.
Proof . For each x, y ∈ E the integral on the right side of (1) is well-defined, is linear in x and conjugate linear in y. Since Haar measure is positive, the information (gx|gx)≥0 yieldshx|xi ≥0. The positive numberM in (3) is well-defined sinceπ(G) is compact in Hom(E, E). Then hx|xi ≤R
GM2(x|x)dg = M2(x|x) since γ is positive and normalized. Also, (x|x) = (g−1gx|g−1gx)≤ M2(gx |gx), whence hx|xi ≥R
GM−2(x|x)dg=M−2(x|x). This proves (2) and thus also the fact thath• | •i is positive definite, that is, a scalar product.
Finally, leth∈G; thenhhx| hyi=R
G(ghx| ghy)dg =R
G(gx |gy)dg=hx| yi
by the invariance ofγ. ut
The idea of the construction is that for each g ∈ G we obtain a scalar product (x, y) 7→ (gx | gy). The invariant scalar product is the “average” or
“expectation” of this family with respect to the probability measureγ.
Definition 4.6. IfGis a topological group, then aHilbertG-moduleis a Hilbert space E and aG-module such that all operators π(g) are unitary, that is, such that
(gx|gy) = (x|y) for all x, y∈E, g∈G. ut The following Lemma will show that in every HilbertG-module of a locally compact group the action(g, x)7→gx:G×E→E is continuous.
Lemma 4.7. Assume that E is a G-module for a topological group G and that π:G→ Lp(E)is the associated representation (see2.2(ii)). IfE is a Baire space, then for every compact subspace K of G the setπ(K) ⊆Hom(E, E) is equicon-tinuous at0, that is, for any neighborhoodV of 0 inE there is a neighborhoodU of 0 such that KU ⊆V.
As a consequence, ifGis locally compact, the function (g, x)7→gx:G×E→E
is continuous.
Proof. First step: GivenV we find a closed 0-neighborhoodW withW−W ⊆V and [0,1]·W ⊆W. Notice that also the interior, InteriorW, of W is star-shaped, that is, satisfies [0,1]·InteriorW = InteriorW. Next we consider
C= \
g∈K
g−1W.
Since K is compact, Kx is compact for any x ∈ E and thus, as Kx ⊆ E = S
n∈Nn·InteriorW and then·W form an ascending family, we find ann∈N with K·x⊆n·W, that is, with x∈ T
g∈Kn·g−1W. Hence for each x∈E there is a natural numbernsuch that x∈n·C. Therefore
E= [
n∈N
n·C,
where all setsn·C are closed. ButE is a Baire space, and so for somen∈N, the setn·Chas interior points, and sincex7→n·xis a homeomorphism ofE, the setC itself has an interior pointc. Now for eachg∈Kwe findg(C−c)⊆W−W ⊆V. ButU =C−c is a neighborhood of 0, asKU ⊆V, our first claim is proved.
Second step: For a proof of the continuity of the functionα= (g, x)7→gx : G×E → E, it suffices to show the continuity of α at the point (1,0). To see this it suffices to note that for fixed h ∈ G and fixed y ∈ E the d ifference α(g, x)−α(h, y) = gx−hy = h h−1g(x−y) + (h−1gy−y)
= π(h) α(h−1g, x−y) + (h−1gy−y)
falls into any given neighborhood of 0 as soon as h−1g is close enough to 1and the difference x−y is close enough to zero, because αis
continuous at (1,0), becauseπ(h) is continuous and becausek 7→ ky:G →E is continuous by by the definition of the topology of pointwise convergence.
Third step: We now assume that G is locally compact and show that α is continuous at (1,0). For this purpose it suffices to know that for a compact neighborhoodKof1inGthe setπ(K)⊆Hom(E, E) is equicontinuous; for then any neighborhood V of 0 yields a neighborhood U of 0 in E with α(K×U) = π(K)(U)⊆V. This completes the proof of the second claim. ut According to the above theorem, ifG is a compact group, and E is a G-module which is at the same time a Banach space, the compact groupGacts on E; that is, (g, x)7→gx:G×E→E is continuous.
Example 4.8. Let G be a compact group. Set E = C(G,K); then E is a Banach space with respect to the sup-norm given by kfk = supt∈G|f(t)|. We definegf =π(g)(f) by gf(t) =f(tg). Then π:G→ Lp(E) is a faithful (that is, injective) representation, andGacts onE.
Proof . We note |f1(tg1)−f2(tg2)| ≤ |f1(tg1)−f1(tg2)|+|f1(tg2)−f2(tg2)| ≤
|f1(tg1)−f1(tg2)|+kf1−f2k. Since G is compact, f1 is uniformly continuous.
Hence the first summand is small if g1 and g2 are close. The second summand is small if f1 and f2 are close in E. This shows that (g, f) 7→ gf:G×E → E is continuous. It is straightforward to verify that this is a linear action. Finally π(g) = idE is tantamount tof(tg) =f(t) for allt∈Gand allf ∈C(G,K). Since the continuous functions separate the points, takingt= 1 we concludeg= 1. ut In Example 4.8, under the special hypotheses present, we have verified the conclusion of Lemma 4.7 directly.
By Weyl’s Trick 4.5, for compactG, it is never any true loss of generality to assume for aG-module on a Hilbert space that E is a Hilbert module. Every finite dimensional K-vector space is a Hilbert space (in many ways). Thus, in particular, every representation of a compact group on a finite dimensional K -vector space may be assumed to be unitary.
Hilbert modules are the crucial type ofG-modules for compact groupsGas we shall see presently. For the moment, let us observe, that every compact group Ghas at least one faithful Hilbert module.
Example 4.9. Let G be a compact group and H0 the vector space C(G,K) equipped with the scalar product
(f1|f2) =γ(f1f2) = Z
G
f1(g)f2(g)dg.
Indeed the function (f1, f2)7→(f1 |f2) is linear in the first argument, conjugate linear in the second, and (f | f) =γ(f f)≥0 sinceγ is positive. Also, if f 6= 0, then there is ag∈Gwithf(g)6= 0. Then the open setU ={t∈G|(f f)(t)>0}
containsg, hence is nonempty. The relation (f | f) = 0 would therefore imply
that U does not meet the support ofγ, which isG—an impossibility. Hence the scalar product is positive definite andH0is a pre-Hilbert space. Its completion is a Hilbert spaceH, also called L2(G,K).
The translation operators π(g) given by π(g)(f) =gf are unitary since (π(g)f |π(g)f) =R
Gf(tg)f(tg)dt=R
gf(t)f(t)dt= (f |f) by invariance. Every unitary operator on a pre-Hilbert spaceH0extends uniquely to a unitary operator on its completionH, and we may denote this extension with the same symbolπ(g).
The spaceL(H) of bounded operators on the Hilbert spaceHis a C∗-algebra andU(H) =U L(H)
denotes its unitary group. Thenπ:G→ U(H) is a morphism of groups. We claim that it is continuous with respect to the strong operator topology, that is,g7→gf:G→ His continuous for eachf ∈ H. Letε >0 and let f0∈C(G,K) be such thatkf−f0k2< εwherekfk22= (f |f). Thenkgf−hfk2≤ kgf−gf0k2+kgf0−hf0k2+khf0−hfk2=kgf0−hf0k2+2kf−f0k2<kgf0−hf0k2+2εin view of the fact thatπ(g) is unitary. Butkgf0−hf0k2≤ kgf0−hf0k∞wherekf0k∞ is the sup-norm supg∈G|f0(g)|for a continuous functionf0. By Example 4.8 the functiong 7→gf is continuous with respect to the sup-norm; hencekgf0−hf0k∞
can be made less thanε forgclose enough toh. For these gand hwe then have kgf −hfk2 < 3ε. This shows the desired continuity. Since π(g) = idH implies π(g)|H0 = idH0 and this latter relation already implies g = 1 by Example 4.8, the representationπis injective. Thus L2(G,K)is a faithful Hilbert module. It is called the regularG-module and the unitary representationπ:G→ U L2(G,K)
is called theregular representation. ut
For the record we write:
Remark 4.10. Every compact group possesses faithful unitary representations
and faithful Hilbert modules. ut