APPENDIX D
Gaussian actions
Gaussian Hilbert spaces provide a mechanism for transforming the additive structure
of a real Hilbert space H into the multiplicative structure of a σ-algebra. This is done via a
kind of exponentiation process that converts direct sums into independent σ-subalgebras.
As explained in Section D.4, for a group G this construction enables us to convert an
orthogonal representation π : G → B(H) into a p.m.p. action G y (X, µ) whose orthogonal Koopman representation is the “second quantization” of π. By a realification
and complexification procedure, one can also start instead with a unitary representation
π : G → B(H), and in fact it is in this framework that we will formulate the main
theorem (Theorem D.14) since our applications in the main part of the book all involve
unitary representations. The requisite background on realification, complexification, and
symmetric Fock space can be found in Sections D.1 and D.2, while Gaussian Hilbert
spaces are introduced in Section D.3.
Our presentation is mostly modeled on [8]. The main difference is our interest in the
unitary case, which is more in line with [83]. Standard references on Gaussian Hilbert
spaces are [110, 74, 67].
D.1. Realification and complexification of representations
Realification. Let H be a complex Hilbert space with inner product h·, ·i. The realification HR of H is the real Hilbert space obtained by viewing H as a real vector space
under restriction of scalars and defining on it the inner product hξ, ζiHR = rehξ, ζi. Note
that the norms k·kHR and k·kH coincide.
For a unitary operator u on H, we write uR for the orthogonal operator on HR which
is formally identical to u. Given a unitary representation π : G → B(H) of a group, we
define the orthogonal representation πR : G → B(HR ) by s 7→ π(s)R , and call it the
realification of π.
Complexification. Let H be a real Hilbert space with inner product h·, ·i. The complexification HC of H is the complex Hilbert space consisting of all pairs (ξ, ζ) ∈ H × H,
written as formal sums ξ + iζ, equipped with the addition
(ξ1 + iζ1 ) + (ξ2 + iζ2 ) = (ξ1 + ξ2 ) + i(ζ1 + ζ2 ),
235
236
D. GAUSSIAN ACTIONS
scalar multiplication
(a + ib)(ξ + iζ) = (aξ − bζ) + i(bξ + aζ),
and inner product
hξ1 + iζ1 , ξ2 + iζ2 iHC = hξ1 , ξ2 i + hζ1 , ζ2 i + ihζ1 , ξ2 i − ihξ1 , ζ2 i.
The positive definiteness of the inner product follows from the fact that
kξ + iζk2HC = hξ + iζ, ξ + iζiHC = hξ, ξi + hζ, ζi = kξk2 + kζk2 .
If u is an orthogonal operator on H, then we can define a unitary operator on HC by
ξ + iζ 7→ uξ + iuζ. We denote this operator by uC . Given an orthogonal representation
π : G → B(H) of a group, we define a unitary representation πR : G → B(HC ) by
s 7→ π(s)C , and call it the complexification of π.
Realification + complexification. Let H be a complex Hilbert space with inner product h·, ·i. Our goal is to give a simple description of the unitary representation (πR )C :
G → B((HR )C ) that arises from given a unitary representation π : G → B(H). Define
the maps V, W : H → (HR )C by
1
1
V ξ = √ (ξ + i(−iξ)), W ξ = √ (ξ + i(iξ)).
2
2
Then V is linear and W is conjugate-linear, and for all ξ, ζ ∈ H one computes using the
equations
hξ, −iζiHR = hiξ, ζiHR = rehiξ, ζi = re ihξ, ζi = −imhξ, ζi
that
hV ξ, V ζi(HR )C = hξ, ζi,
hW ξ, W ζi(HR )C = hξ, ζi,
hV ξ, W ζi(HR )C = 0.
Thus V is a Hilbert space embedding, W can be viewed as a Hilbert space embedding of
the conjugate H via the identification of H with H as a set, and the images of V and W
are orthogonal. Note also that if ξ + iζ is any vector in (HR )C then we can express it as
1
1
√ V (ξ + iζ) + √ W (ξ − iζ).
2
2
We furthermore observe that if U is a unitary operator on H then we have V U = (UR )C V
and W Ū = (UR )C W where U is the unitary operator on H which is formally identical to
U . Putting everything together, we conclude the following.
P ROPOSITION D.1. The map ϕ : H⊕H → (HR )C defined by ϕ(η1 , η2 ) = V η1 +W η2
is a unitary isomorphism, and if π : G → B(H) is a unitary representation then ϕ
conjugates π ⊕ π to (πR )C .
D.2. SYMMETRIC FOCK SPACE
237
D.2. Symmetric Fock space
We describe here the contruction of the symmetric Fock space S(H) of a real or
complex Hilbert space H, along with the orthogonal/unitary operators and group representations on S(H) that are induced from those on H. For simplicity we will assume
in the following three paragraphs that H is real. The complex case is formally identical
but with “orthogonal” replaced everywhere by “unitary” and taking H⊙0 below to be C
instead of R.
Let n ∈ N. We define an orthogonal representation σ 7→ Uσ of the symmetric group
Sn on the nth tensor power H⊗n by declaring
U
σ
ξ1 ⊗ · · · ⊗ ξn 7−→
ξσ(1) ⊗ · · · ⊗ ξσ(n)
on elementary tensors. The nth symmetric power of H is the closed subspace
H⊙n := {ξ ∈ H⊗n : Uσ ξ = ξ for all σ ∈ Sn }
P
of H⊗n . The orthogonal projection of H⊗n onto H⊙n is equal to (1/n!) σ∈Sn Uσ , and
we write f1 ⊙ · · · ⊙ fn for the image of an elementary tensor f1 ⊗ · · · ⊗ fn under this
projection. Letting H⊙0 denote the one-dimensional Hilbert space R, we define symmetric
L
⊙n
Fock space S(H) to be the Hilbert space direct sum ∞
.
n=0 H
Given an orthogonal operator U on H, the assignment
ξ 1 ⊙ · · · ⊙ ξ n 7→ U ξ 1 ⊙ · · · ⊙ U ξ n
on elementary tensors defines an orthogonal operator on H⊙n , denoted by U ⊙n . Then
L∞
⊙n
is an orthogonal operator on S(H), which we write as S(U ).
n=0 U
For each n ≥ 0 the map U 7→ U ⊙n from the orthogonal group of H to the orthogonal
group of H⊙n is a homomorphism, so that every orthogonal representation π : G →
B(H) gives rise to an orthogonal representation s 7→ π(s)⊙n of G on H⊙n , which we
L
⊙n
denote by π ⊙n . We then define S(π) to be the orthogonal representation ∞
of G
n=0 π
on S(H).
P ROPOSITION D.2. Let H be a real Hilbert space. Then there is a canonical unitary
isomorphism S(HC ) → S(H)C which conjugates S(πC ) to S(π)C for every orthogonal
representation π of a group on H.
P ROOF. For each n ∈ N we define a map from (HC )⊗n to (H⊗n )C by
(x1,0 + ix1,1 ) ⊗ · · · ⊗ (xn,0 + ixn,1 )
X −1
X −1
7→
i|ω (1)|−1 x1,ω(1) ⊗ · · · ⊗ xn,ω(n)
i|ω (1)| x1,ω(1) ⊗ · · · ⊗ xn,ω(n) + i
ω∈Ω
ω∈Ω′
238
D. GAUSSIAN ACTIONS
where Ω is the set of all functions in {0, 1}{1,...,n} taking value one an even number of
times and Ω′ is the set of all functions in {0, 1}{1,...,n} taking value one an odd number
of times. This is a unitary isomorphism which maps (HC )⊙n to (H⊙n )C , and piecing
together the resulting isomorphisms (HC )⊙n → (H⊙n )C over all n ∈ N along with the
identity operator C → C in the case n = 0, we get a unitary isomorphism S(HC ) →
S(H)C , which is readily seen to satisfy the requirement in the proposition statement. D.3. Gaussian Hilbert spaces
A random variable on a probability space (X, µ) is a measurable function f : X → R.
The distribution of such an f is the measure µf on R which is the image of µ under f .
When f is integrable we define its expectation or mean
Z
Z
x dµf (x)
f (x) dµ(x) =
E(f ) :=
X
R
2
and when it is in L (X) we define its variance
Var(f ) := E((f − E(f ))2 ) = E(f 2 ) − E(f )2 .
We say that f is centred if E(f ) = 0.
A family {fi }i∈I of random variables is said to be independent if, for all finite sets
F ⊆ I and Borel sets Bi ⊆ R for i ∈ F , the collection of sets fi−1 (Bi ) for i ∈ F is
independent, i.e.,
\
Y
−1
µ
fi (Bi ) =
µ(fi−1 (Bi )).
i∈F
i∈F
D EFINITION D.3. A real-valued random variable f on (X, µ) is Gaussian if either it
is constant or its distribution µf is given by integration with respect to Lebesgue measure
against a density function of the form
1
2
2
e−(x−m) /2σ
x 7→ √
2πσ 2
for some real numbers m and σ > 0, in which case E(f ) = m and Var(f ) = σ 2 .
Note that, given the form of its density function, a Gaussian random variable on (X, µ)
lies in Lp (X) for every 1 ≤ p < ∞.
D EFINITION D.4. A Gaussian Hilbert space is a closed subspace H of L2R (X) for
some probability space (X, µ) such that the elements of H are all centred Gaussian random variables.
Let f be a random variable. Its characteristic function is the function on R defined
by t 7→ E(eitf ). This is the same thing as the Fourier transform but without the usual
D.3. GAUSSIAN HILBERT SPACES
239
normalization constant. Proofs of the following basic facts can be found in Sections 9.4
and 9.5 of [31] or Section 3.3.1 of [32].
(i) The characteristic function completely determines the random variable.
(ii) In the case that f is a Gaussian random variable with mean m and variance σ 2 ,
2 2
the characteristic function is t 7→ eimt−σ t /2 .
(iii) If a family {f1 , . . . , fn } of random variables is independent then
E(eitf1 · · · eitfn ) = E(eitf1 ) · · · E(eitfn )
for all t ∈ R.
We will tacitly use all of these facts in the proof of the following.
P ROPOSITION D.5. Let {f1 , . . . , fn } be an independent family of centred Gaussian
random variables. Then the linear span of {f1 , . . . , fn } is a Gaussian Hilbert space.
P ROOF. For c1 , . . . , cn ∈ R, the value of the characteristic function of the random
P
variable f = j cj fj at a given t ∈ R is
E(eitf ) = E(eitc1 f1 · · · eitcn fn ) = E(eitc1 f1 ) · · · E(eitcn fn )
2 2 2 /2
= e−c1 σ1 t
2
2 2 /2
· · · e−cn σn t
2 2
2
2
= e−(c1 σ1 +···+cn σn )t
2 /2
.
This shows that f is a centered Gaussian random variable with variance c21 σ12 + · · · + c2n σn2 ,
yielding the result.
Although we will not need it, we mention the fact that if {Hi }i∈I is a collection of
closed subspaces of a Gaussian Hilbert space K ⊆ L2R (X) then, denoting by BHi the
σ-algebra generated by Hi , the collection {BHi }i∈I is independent if and only if the
subspaces Hi are pairwise orthogonal [110, Prop. 2.4].
P ROPOSITION D.6. The set of all centred Gaussian random variables on (X, µ) is
closed in L2R (X).
P ROOF. Suppose that {fn } is a sequence of centred Gaussian random variables converging in L2R (X) to some function f . Then | E(fn − f )| ≤ kfn − f k1 ≤ kfn − f k2 → 0
and hence E(f ) = 0, so that f is centred random variable. Moreover, the quantities
σn := E(fn2 )1/2 = kfn k2 converge to σ := E(f 2 )1/2 = kf k2 , and so unless σ 6= 0 (in
which case f = 0) we have
Z
Z
1
1
2
2
2
−x2 /2σn
lim p
e−x /2σ h(x) dx
e
h(x) dx = √
n→∞
2πσn2
2πσ 2
for every compactly supported continuous function h on R. Since L2 convergence implies weak convergence of the distributions (“convergence in distribution”) [20, Thms.
240
D. GAUSSIAN ACTIONS
4.1.4 and 4.4.5], this shows that the distribution of f is Gaussian with associated density
2
2
function x 7→ (2πσ 2 )−1/2 e−x /2σ .
2
E XAMPLE D.7. Consider the measure ν on R which has density function x 7→ e−x /2
with respect to Lebesgue measure. Let I be a countable index set and consider RI
equipped with the product Borel σ-algebra and product measure ν I . For each i ∈ I
the projection fi : RI → R onto the ith factor is a centred Gaussian random variable.
Moreover, any finite collection of these random variables is obviously independent and so
any linear combination of them is again a centred Gaussian random variable by Proposition D.5. The closed linear span of the fi in L2R (RI ) is thus a Gaussian Hilbert space by
Proposition D.6.
D EFINITION D.8. We say that a Gaussian Hilbert space H ⊆ L2R (X) generates the
σ-algebra if the σ-algebra of all measurable subsets of X is the smallest σ-algebra containing f −1 (B) for every f ∈ H and Borel set B ⊆ R (what we really mean here is to
quantify over all representatives of elements of H, consistent with our convention that the
σ-algebra contains all µ-null sets).
Since the Gaussian Hilbert space in Example D.7 clearly generates the σ-algebra, and
all atomless standard probability spaces are isomorphic, we conclude the following.
P ROPOSITION D.9. Let (X, µ) be an atomless standard probability space. Then there
is a Gaussian Hilbert space in L2R (X, µ) which generates the σ-algebra.
The final goal of this section is to show that if H ⊆ L2R (X) is a Gaussian Hilbert space
which generates the σ-algebra then there is a canonical isometric isomorphism S(H) →
L2R (X).
Let H be a real Hilbert space. For every ξ ∈ H we set
∞
X
1
√ ξ ⊙n
exp(ξ) =
n!
n=0
where ξ ⊙n = ξ ⊙ · · · ⊙ ξ ∈ H⊙n and ξ ⊙0 = 1. Viewing each H⊙n as a subspace of S(H)
in the canonical way, this defines an element of S(H), since
∞ ∞
X
X
1
1 ⊙n 2
kξk2n
√ ξ ⊙n =
H < ∞.
n!
H
n!
n=0
n=0
For ξ, ζ ∈ H we have
(26)
∞ D
∞
X
1 ⊙n 1 ⊙n E X 1
√ ξ ,√ ζ
hexp(ξ), exp(ζ)i =
hξ, ζin = ehξ,ζi .
=
n!
n!
n!
n=0
n=0
L EMMA D.10. The set {exp(ξ) : ξ ∈ H} is total in S(H).
D.3. GAUSSIAN HILBERT SPACES
241
P ROOF. Let n ∈ N and ξ1 , . . . , ξn ∈ H, and define on Rn the function f (t1 , . . . , tn ) =
exp(t1 ξ1 + · · · + tn ξn ). Then we compute that
∂
∂
···
f (0, . . . , 0) = n!ξ1 ⊙ · · · ⊙ ξn ,
∂t1
∂tn
which shows that ξ1 ⊙ · · · ⊙ ξn belongs to the closure of the linear span of {exp(ξ) : ξ ∈
H}. Since exp(0) and the vectors of the form ξ1 ⊙ · · · ⊙ ξn for some n ∈ N together span
a dense subspace of S(H), we obtain the result.
Let f ∈ L2R (X) be a centred Gaussian random variable with variance σ 2 . In the case
that f is not constant we have
Z
1
2
2
2
f
E(e ) = √
(27)
ex−x /2σ dx = eσ /2 .
σ 2π R
f
p
In particular, e lies in L (X) for all 1 ≤ p < ∞.
L EMMA D.11. Let (X, µ) be a probability space and H ⊆ L2R (X) a Gaussian Hilbert
space which generates the σ-algebra. Let 1 ≤ p < ∞. Then the set {ef : f ∈ H} is total
in LpR (X).
P ROOF. We have already observed above from (27) that ef belongs to LpR (X) for
every f ∈ H. Using the fact that Lp (X)∗ ∼
= Lq (X) where qR= (p − 1)/p, it is then
enough to show that, given an element g of LqR (X) satisfying X ef g dµ = 0 for every
f ∈ H, we have g = 0. Let f1 , . . . , fn ∈ H, and define the map F : X → Rn by
P
x 7→ (f1 (x), . . . , fn (x)). Let t1 , . . . , tn ∈ R and set f = j tj fj , which is an element
R
of H. Then one can verify that the function z 7→ X g(x)ezf (x) dµ(x) on C is analytic,
which means
is zero everywhere since it vanishes on R by our hypothesis on g. In
R that it−if
particular X g(x)e (x) dµ(x) = 0. Writing ν for the measure on Rn which is the push
forward of g dµ under F , it follows that the Fourier transform νb satisfies
Z
P
νb(t1 , . . . , tn ) =
e−i tj xj dν(x1 , . . . , xn )
n
ZR
P
g(x)e−i tj fj (x) dµ(x) = 0.
=
X
Since the Fourier transform determines the measure (see for example Theorem 9.5.1 of
[31], which is framed in the language of characteristic functions on Rn ), we deduce that ν
is the zero measure. Consequently the measure g dµ vanishes on the σ-algebra generated
by f1 , . . . , fn . Since g lies in L1 (X), we can apply the dominated convergence theorem
to verify that the algebra of measurable subsets of X on which g dµ vanishes is closed
under both increasing and decreasing countable unions, which implies by [82, Thm. 10.1]
that this algebra is a σ-algebra. By our hypothesis on H, it follows that g dµ is the zero
measure. Hence g = 0, which establishes the result.
242
D. GAUSSIAN ACTIONS
T HEOREM D.12. Let (X, µ) be a probability space and H ⊆ L2R (X) a Gaussian
Hilbert space which generates the σ-algebra. Then there is a unique isometric isomorphism ϕ : S(H) → L2R (X) such that
ϕ(exp(f )) = ef −E(f
2 )/2
for all f ∈ H.
P ROOF. For all f, g ∈ H we have, using (27) and (26),
hef −E(f
2 )/2
, eg−E(g
2 )/2
i = E(ef −E(f
2 )/2
eg−E(g
= E(ef +g )e−(E(f
= e(E((f +g)
2 )/2
)
2 )+E(g 2 ))/2
2 )−E(f 2 )−E(g 2 ))/2
= eE(f g) = ehf,gi = hexp(f ), exp(g)i.
It follows that there is a well-defined isometric linear map from the linear span of {exp(f ) :
P
P
2
f ∈ H} to L2R (X) given by i ci exp(fi ) 7→ i ci efi −E(fi )/2 on linear combinations.
Since {exp(f ) : f ∈ H} is total in S(H) by Lemma D.10, this map extends uniquely
to an isometric embedding ϕ : S(H) → L2R (X), which Lemma D.11 then shows to be
surjective.
D.4. Gaussian actions
T HEOREM D.13. Let (X, µ) and (Y, ν) be probability spaces and let H ⊆ L2R (X)
and K ⊆ L2R (Y ) be Gaussian Hilbert spaces which both generate the σ-algebra. Let
ϕ : S(H) → L2R (X) and ψ : S(K) → L2R (Y ) be the corresponding isomorphisms given
by Theorem D.12. Then for every isometric isomorphism U : H → K there is a measure
isomorphism T : X → Y such that ψ ◦ S(U ) ◦ ϕ−1 is the map f 7→ f ◦ T −1 .
P ROOF. Let U : H → K be an isometric isomorphism. Write V for the isometric
isomorphism ψ ◦ S(U ) ◦ ϕ−1 : L2R (X) → L2R (Y ). Given f, g ∈ H we set h = ϕ(exp(f ))
and k = ϕ(exp(g)) and observe that
(28)
hk = eE(f g) ef +g−E(f +g)
2 /2
= eE(f g) ϕ(exp(f + g)),
which shows that hk lies in L2R (X) and
V (hk) = eE(f g) ψ(exp(U (f + g)))
= eE(U (f )U (g)) eU (f +g)−E(U (f +g)
= eU (f )−E(U (f )
2 )/2
eU (g)−E(U (g)
2 )/2
2 )/2
= ψ(exp(U (f )))ψ(exp(U (g))) = V (h)V (k).
D.4. GAUSSIAN ACTIONS
243
A similar computation replacing U by U −1 yields V −1 (hk) = V −1 (h)V −1 (k) whenever
h = ψ(exp(f )) and k = ψ(exp(g)) for some f, g ∈ K.
We next argue that this multiplicativity also holds on indicator functions. So let A and
B be measurable subsets of X. Since the linear span D of the vectors of the form ef is
dense L4R (X) by Lemma D.11, there are sequences {fn } and {gn } in D such that fn → 1A
and gn → 1B in L4 norm. Then fn gn → 1A 1B in L2 norm, so that V (fn )V (gn ) =
V (fn gn ) → V (1A 1B ) in L2 norm. By passing to subsequences, we may assume V (fn ),
V (gn ), and V (fn gn ) each converge pointwise a.e., so that V (fn )V (gn ) → V (1A )(1B )
and V (fn gn ) → V (1A 1B ) pointwise a.e., and consequently V (1A 1B ) = V (1A )V (1B ).
By the same argument, V −1 (1A 1B ) = V −1 (1A )V −1 (1B ).
We can now conclude by Theorem 0.5 that there is a measure isomorphism T : X →
Y such that V is the map f 7→ f ◦ T −1 .
T HEOREM D.14. Let π : G → B(H) be an unitary representation on a separable
Hilbert space. Then there is, up to measure conjugacy, a unique p.m.p. action G y (X, µ)
such that there exists an isometric embedding HR ֒→ L2R (X, µ) with the property that, if
we identify HR with its image under this embedding,
(i) HR is a Gaussian Hilbert space which generates the σ-algebra, and
(ii) the corresponding isometric isomorphism ϕ : S(HR ) → L2R (X, µ) given by
Theorem D.12 conjugates S(πR ) to the orthogonal Koopman representation of
the action.
P ROOF. By Proposition D.9 there exists a standard probability space (X, µ) such that
by means of some isometric embedding we may regard HR as a Gaussian Hilbert space
in L2R (X) which generates the σ-algebra. By Theorem D.13, taking both of the Gaussian
Hilbert spaces there to be HR ⊆ L2R (X), we see that there is a p.m.p. action G y (X, µ)
such that the isometric isomorphism ϕ : S(HR ) → L2R (X) given by Theorem D.12
conjugates S(πR ) to the orthogonal Koopman representation.
Now let G y (Y, ν) be another p.m.p. action such that we may also regard HR as
a subspace of L2R (Y ) via some isometric embedding so that the properties in the theorem statement are satisfied. Let ψ : S(HR ) → L2R (Y ) be the corresponding isometric isomorphism as given by Theorem D.12. Applying Theorem D.13 to the two
inclusions HR ⊆ L2R (X) and HR ⊆ L2R (Y ) and the identity operator on HR , we obtain measure isomorphism T : X → Y such that ψ ◦ ϕ−1 is the map f 7→ f ◦ T −1 .
For every s ∈ G, writing αs for the transformation x 7→ sx of X and βs for the
transformation y 7→ sy of Y , we observe that the two operators f 7→ f ◦ βs and
f 7→ f ◦(T ◦αs ◦T −1 ) on L2R (X) coincide, the first being equal to ψ ◦Sπ(s)◦ψ −1 and the
second to (ψ ◦ϕ−1 )◦(ϕ◦Sπ(s)◦ϕ−1 )◦(ψ ◦ϕ−1 )−1 . It follows that βs = T ◦αs ◦T −1 , and
244
D. GAUSSIAN ACTIONS
so we conclude that T conjugates G y (X, µ) to G y (Y, ν), yielding the uniqueness
part of the theorem.
D EFINITION D.15. The p.m.p. action in Theorem D.14 is called the Gaussian action
associated to π.
The following fact will be useful in Example D.17.
P ROPOSITION D.16.
E XAMPLE D.17. The Gaussian action associated to the left regular representation
λ : G → B(ℓ2 (G)), or any countable multiple of it, is the Bernoulli action G y (Y G , ν G )
where (Y, ν) is an atomless standard probability space (of which there is only one up to
isomorphism by ???). Indeed, given a countable index set J we can identify (ℓ2 (G)J )R =
ℓ2R (G){0,1}×J with the Gaussian Hilbert space L2 (R{0,1}×J×G ) in Example D.7 (taking the
index set there to be {0, 1} × J × G) via the embedding which sends the standard basis
vector supported at (i, j, s) ∈ {0, 1} × J × G to the projection R{0,1}×J×G → R onto the
coordinate at (i, j, s). The induced action G y R{0,1}×J×G = (R{0,1}×J )G is then clearly
the Bernoulli one, with base R{0,1}×J .
T HEOREM D.18. Let π : G → B(H) be a unitary representation. Then the Koopman
representation of the associated Gaussian action is unitarily equivalent to S(π ⊕ π).
P ROOF. By Proposition D.1 there is a unitary isomorphism conjugating (πR )C to
π ⊕ π. This then produces, in the obvious canonical way, a unitary isomorphism conjugating S((πR )C ) to S(π ⊕ π). Since S(πR )C and S((πR )C ) are unitarily equivalent by
Proposition D.2, and the (unitary) Koopman representation of the action is in the obvious way the complexification of its orthogonal Koopman representation, we can apply
Theorem D.14 to obtain the result.