APPENDIX B
Hilbert modules
Hilbert modules are a generalization of Hilbert spaces in which the complex scalars
are replaced with a C*-algebra or von Neumann algebra N , so that we talk about a left
or right N -module E with an N -valued inner product such that E is complete in the associated norm. For the structure theory of p.m.p. actions studied in Chapter 2, the key
example is the Hilbert module L2 (X|Y ) over the commutative von Neumann algebra
L∞ (Y ) which arises from a measure-preserving quotient map X → Y between probability spaces. In this case L2 (X|Y ) is the completion of L2 (X) under the norm associated
to the L∞ (Y )-valued inner product which we define using the conditional expectation
L∞ (X) → L∞ (Y ) (see Example B.6).
In this appendix we concentrate on modules over L∞ (Y ) where (Y, ν) is a standard
probability space, with the action on the left. Except for Lemma B.16 and Proposition B.18, everything works equally well, modulo minor notational changes, if L∞ (Y )
is replaced by a general von Neumann algebra, equipped with a faithful normal tracial
state where appropriate. We have simply focused our terminology and notation with a
view towards the applications in Chapter 2. For those interested in the general noncommutative case we have paid attention to the order of products, especially in conjunction
with the involution, which in L∞ (Y ) is conjugation. Note that, contrary to our left convention, the standard operator algebra practice is to use right modules.
We begin in Section B.1 by defining L∞ (Y )-valued inner products and Hilbert L∞ (Y )modules. Orthonormality and conditional Hilbert-Schmidt operators are then treated in
Sections B.2 and B.3, respectively. Almost none of what we do requires completeness,
and so for the most part we will be talking about an L∞ (Y )-module with L∞ (Y )-valued
inner product without passing to the Hilbert module completion. In Section B.2, for example, we approximate by finite orthonormal expansions without asking for an aymptotic
expansion as in the Hilbert space setting.
B.1. L∞ (Y )-valued inner products and Hilbert L∞ (Y )-modules
Throughout E is a (left) L∞ (Y )-module.
D EFINITION B.1. An L∞ (Y )-valued inner product on E is a map h·, ·iY : E × E →
L∞ (Y ) such that
215
216
B. HILBERT MODULES
(i) haf + g, hiY = ahf, hiY + hg, hiY for all f, g, h ∈ E and a ∈ L∞ (Y ),
(ii) hf, giY = hg, f iY for all f, g ∈ E,
(iii) hf, f iY ≥ 0 for all f ∈ E, with equality if and only if f = 0.
Note that (i) and (ii) together imply conjugate L∞ (Y )-linearity in the second variable.
We henceforth consider E to be equipped with an L∞ (Y )-valued inner product h·, ·iY .
P ROPOSITION B.2 (Cauchy-Schwarz inequality). For all f, g ∈ E we have
|hf, giY |2 ≤ khf, f iY khg, giY .
P ROOF. The inequality is obvious if f = 0, and so we may assume that hf, f iY 6= 0.
For every a ∈ L∞ (Y ) we have
0 ≤ haf + g, af + giY = ahf, f iY ā + 2 re(ahf, giY ) + hg, giY
≤ |ā|2 khf, f iY k + 2 re(ahf, giY ) + hg, giY .
Taking a = −hf, giY /khf, f iY k we obtain the desired inequality.
P ROPOSITION B.3. Setting kf k := khf, f iY k1/2 defines a norm on E, and khf, giY k ≤
kf kkgk and kaf k ≤ kakkf k for all f, g ∈ E and a ∈ L∞ (Y ).
P ROOF. For all f, g ∈ E, Proposition B.2 yields
khf, giY k2 = k|hf, giY |2 k ≤ khf, f iY khg, giY = kf k2 kgk2
and hence
kf + gk2 = khf + g, f + giY k
≤ khf, f iY k + khf, giY k + khg, f iY k + khg, giY k
≤ kf k2 + kf kkgk + kgkkf k + kgk2 = (kf k + kgk)2 .
It follows easily that k · k is a norm on E.
Finally, for all f ∈ E and a ∈ L∞ (Y ) we have
kaf k2 = khaf, af iY k = kahf, f iY āk ≤ kaāk · khf, f iY k = kak2 kf k2 .
D EFINITION B.4. An L∞ (Y )-module with an L∞ (Y )-valued inner product is a Hilbert
L∞ (Y )-module if it is complete under the norm in Proposition B.3.
Using Proposition B.3 one can easily verify the following, just as one does to show
that the completion of an ordinary inner product space under its associated norm is a
Hilbert space with inner product extending the original one.
P ROPOSITION B.5. The completion of E has a Hilbert L∞ (Y )-module structure extending that of E.
B.2. ORTHONORMAL SETS
217
E XAMPLE B.6. The key example for our purposes in Section 2.2 of Chapter 2.2 is
the L∞ (Y )-module arising from a measure-preserving factor map ϕ : (X, µ) → (Y, ν)
of probability spaces. This is explained in detail in Section 2.1 of Chapter 2.2. We view
L∞ (Y ) as a von Neumann subalgebra of L∞ (X) via the composition map f 7→ f ◦ ϕ.
Then L∞ (X) becomes an L∞ (Y )-module via the multiplication map (f, g) 7→ f g, and we
define on it the L∞ (Y )-valued inner product hf, giY := EY (f ḡ) where EY : L∞ (X) →
L∞ (Y ) is the conditional expectation. We write L2 (X|Y ) for the Hilbert L∞ (Y )-module
obtained by completing L∞ (X) according to Proposition B.5 with respect to the norm
defined by h·, ·iY . We have the natural inclusions L∞ (X) ⊆ L2 (X|Y ) ⊆ L2 (X).
B.2. Orthonormal sets
The notions of orthogonality and orthonormality generalize from the Hilbert space
setting as follows. As completeness is not needed for what we do here, we merely assume
that E is an L∞ (Y )-module with an L∞ (Y )-valued inner product h·, ·iY .
D EFINITION B.7. If f and g are elements of E satisfying hf, giY = 0 (equivalently,
hg, f iY = 0) then we say that f and g are orthogonal and write f ⊥ g. We will also speak
of the orthogonality of an element f ∈ E to a set Ω ⊆ E, meaning that f ⊥ g for all
g ∈ Ω.
D EFINITION B.8. A set Ω ⊆ E is said to be orthonormal if f ⊥ g for all distinct
f, g ∈ Ω and hf, f iY is a nonzero projection for every f ∈ Ω.
Our aim is to show that, up to an arbitrarily small error, we can expand finitely many
given elements of E with respect to a common finite orthonormal set, which can moreover
be taken to extend a prescribed finite orthonormal set (Proposition B.11).
L EMMA B.9. Let Ω be a finite orthonormal subset of E. Then for every f ∈ E one
P
has (f − g∈Ω hf, giY g) ⊥ h for all h ∈ Ω.
P ROOF. For every h ∈ Ω, since p := hh, hiY is a projection we see by expanding that
hh − ph, h − phiY = 0 and hence h = ph, so that
X
X
f−
hf, giY g, h
= hf, hiY −
hf, giY hg, hiY
g∈Ω
Y
g∈Ω
= hf, hiY − hf, hiY p
= hf, hiY − hf, phiY = 0.
L EMMA B.10. Let f be a nonzero element of E and ε > 0. Then there is a g ∈
L (Y )f such that hg, giY is a nonzero projection and kf − hf, giY gk < ε.
∞
218
B. HILBERT MODULES
P ROOF. Set a = hf, f iY . Since a 6= 0 we can find a nonzero projection p ∈ L∞ (Y )
and a b ∈ L∞ (Y ) with b̄ = b such that pb2 a = p and ka(1 − p)k < ε2 . Set g = bpf . Then
hg, giY is a nonzero projection, and since hf, giY g = pf we have
f − hf, giY g, f − hf, giY g Y = hf − pf, f − pf iY = a(1 − p)2 = a(1 − p)
so that kf − hf, giY gk = ka(1 − p)k1/2 < ε.
P ROPOSITION B.11. Let Ω be a finite orthonormal subset of E. Let f1 , . . . , fn ∈
E and ε > 0. Then there is a finite orthonormal set Ω′ ⊆ E containing Ω such that
P
kfj − g∈Ω′ hfj , giY gk < ε for every j = 1, . . . , n.
P ROOF. Let Ω1 and Ω2 be finite orthonormal subsets of E such that Ω1 ⊆ Ω2 , and let
P
f ∈ E. Writing ui for f − g∈Ωi hf, giY g, we have, using Lemma B.9,
X
X
hu1 , u1 iY = u2 +
hf, giY g, u2 +
hf, giY g
g∈Ω2 \Ω1
= hu2 , u2 iY +
g∈Ω2 \Ω1
= hu2 , u2 iY +
X
Y
g∈Ω2 \Ω1
X hf, giY g, hf, giY g
Y
hf, giY hf, giY hg, giY
g∈Ω2 \Ω1
≥ hu2 , u2 iY ,
and hence khu1 , u1 iY k ≥ khu2 , u2 iY k. By induction on n, we may therefore assume that
n = 1 and write f for f1 .
P
Set h = f − g∈Ω hf, giY g. We may assume that h 6= 0, for otherwise we can take
Ω′ = Ω. By Lemma B.10 we can find an h′ ∈ L∞ (Y )h such that hh′ , h′ iY is a nonzero
projection and kh − hh, h′ iY h′ k < ε. Then Ω′ := Ω ∪ {h′ } is a finite orthonormal set and
X
′
′
f −
hf, giY g = kh − hh, h iY h k < ε.
g∈Ω′
B.3. Conditional Hilbert-Schmidt operators
Throughout E is an L∞ (Y )-module with L∞ (Y )-valued inner product h·, ·iY .
D EFINITION B.12. A normed L∞ (Y )-module is a (left) L∞ (Y )-module F with a
norm k · k such that kaf k ≤ kakkf k for all f ∈ F and a ∈ L∞ (Y ).
Examples of normed L∞ (Y )-modules include
(i) an L∞ (Y )-module with an L∞ (Y )-valued inner product (in particular, a Hilbert
L∞ (Y )-module), and
(ii) a Hilbert space H together with a representation of L∞ (Y ) on it.
B.3. CONDITIONAL HILBERT-SCHMIDT OPERATORS
219
For our applications in Section 2.2 of Chapter 2, we are interested in the special case of
(ii) that arises in the setting of Example B.6. As described there, a measure-preserving
factor map X → Y between probability spaces induces an inclusion L∞ (Y ) ⊆ L∞ (X).
The Hilbert space in question is then L2 (X), with L∞ (Y ) acting on it by multiplication
operators.
Henceforth F will denote a normed L∞ (Y )-module.
D EFINITION B.13. A bounded L∞ (Y )-linear operator T : E → F is conditionally
P
Hilbert-Schmidt if f ∈Ω kT f k2 < ∞ for every orthonormal set Ω ⊆ E.
L EMMA B.14. Suppose that E is nonzero. Let T : E → F be a bounded L∞ (Y )linear map. Then
kT k = sup{kT f k : f ∈ E and hf, f iY is a projection}.
P ROOF. Let f be a nonzero element of E with kf k ≤ 1, and let ε > 0. By Lemma B.10,
for every δ > 0 we can find a g ∈ L∞ (Y )f such that hg, giY is a projection and
kf − hf, giY gk < δ, in which case
kT f − T (hf, giY g)k ≤ kT kkf − hf, giY gk ≤ δkT k.
Taking δ small enough, we can ensure that kT (hf, giY g)k ≥ kT f k−ε. Since khf, giY k ≤
kf kkgk ≤ 1, we therefore get
kT gk ≥ khf, giY kkT gk ≥ khf, giY T gk = kT (hf, giY g)k ≥ kT f k − ε.
P ROPOSITION B.15. Let T : E → F be a bounded L∞ (Y )-linear conditionally
Hilbert-Schmidt operator. Then for every ε > 0 there exists a finite orthonormal set
Ω ⊆ E such that kT f k ≤ εkf k for all f ∈ E orthogonal to Ω.
P ROOF. By Lemma B.14, we can find an f1 ∈ E such that hf1 , f1 iY is a projection
and kT f1 k ≥ kT k − ε/2. Denote by E1 the submodule of E consisting of all elements
orthogonal to f1 . Now apply Lemma B.14 to T |E1 to obtain an f2 ∈ E1 such that hf2 , f2 iY
is a projection and kT f2 k ≥ kT |E1 k − ε/2. Proceeding recursively in this manner, we
construct an orthonormal sequence {fn } in E such that kT fn+1 k ≥ kT |En k − ε/2 and
fn+1 ∈ En where En denotes the submodule of E consisting of all elements orthogonal
P
2
to {f1 , . . . , fn }. Since T is conditionally Hilbert-Schmidt, we have ∞
n=1 kT fn k < ∞.
Then there is a particular n such that kT fn+1 k < ε/2, in which case kT |En k ≤ ε, so that
we may take Ω to be {f1 , . . . , fn }.
It is a standard and easy fact that a Hilbert-Schmidt operator H → K between ordinary Hilbert spaces is a compact operator, meaning that it maps the unit ball of H to a
precompact subset of K. We round out this section by establishing a conditional analogue
of this in Proposition B.18.
220
B. HILBERT MODULES
L EMMA B.16. Suppose that the norm on F is associated to an L∞ (Y )-valued inner
T : E → F be an L∞ (Y )-linear map. Let λ > 0 be such that
R product. Let 1/2
( Y hT g, T giY dν) ≤ λkgk for all f ∈ E. Then there is a set A ⊆ Y with ν(A) > 1−λ
such that hT g, T giY 1A ≤ kgk2 λ1A for all g ∈ E.
P ROOF. Denote by BE the unit ball of E. We claim that the set {min(λ+1, hT g, T giY ) :
g ∈ BE }, where λ + 1 is interpreted as a constant function on Y , is directed under the
natural order. Let g, h ∈ BE . Then we can find a measurable set D ⊆ Y such that
hT g, T giY 1D ≥ hT h, T hiY 1D and hT h, T hiY 1Dc ≥ hT g, T giY 1Dc . We have
h1D g + 1Dc h, 1D g + 1Dc hiY = hg, giY 1D + hh, hiY 1Dc ≤ 1,
so that 1D g + 1Dc h ∈ BE . Then
min(λ + 1, hT (1D g + 1Dc h), T (1D g + 1Dc h)iY )
= min(λ + 1, hT g, T giY 1D + hT h, T hiY 1Dc )
= min(λ + 1, max(hT g, T giY , hT h, T hiY ))
= max(min(λ + 1, hT g, T giY ), min(λ + 1, hT h, T hiY )),
establishing our claim.
The net {min(λ + 1, hT g, T giY ) : g ∈ BE } is increasing and bounded above by λ + 1.
Thus it converges to some f ∈ L∞ (Y ) in the strong operator topology, and we have
Z
Z
Z
hT g, T giY dν ≤ λ2 .
min(λ + 1, hT g, T giY ) dν ≤ sup
f dν = sup
Y
g∈BE
Y
g∈BE
Y
Therefore we can find a set A ⊆ Y with measure greater than 1 − λ such that f 1A ≤ λ1A .
Then for any g ∈ BE we have
min((λ + 1)1A , hT g, T giY 1A ) = min(λ + 1, hT g, T giY )1A ≤ f 1A ≤ λ1A
and hence hT g, T giY 1A ≤ λ1A .
D EFINITION B.17. A subset of the normed L∞ (Y )-module E is called a finitely genP
erated module zonotope if it is of the form h∈Ω BL∞ (Y ) h where Ω is a finite subset of
E and BL∞ (Y ) denotes the unit ball of L∞ (Y ). A set K ⊆ E is said to be conditionally
precompact if for every ε > 0 there are a set D ⊆ Y with ν(D) > 1 − ε and a finitely
generated module zonotope Z in E such that 1D K ⊆ε Z.
R
On the L∞ (Y )-module E we define the inner product hf, gi = Y hf, giY dν and then
complete in the associated norm to obtain a Hilbert space, which we denote by L2 (E).
Note that the natural embedding E ֒→ L2 (E) is contractive. In the case of the Hilbert
module L2 (X|Y ) arising from an extension X → Y , the Hilbert space L2 (E) is simply
B.3. CONDITIONAL HILBERT-SCHMIDT OPERATORS
221
L2 (X). In fact, in our application in the proof of Lemma 2.11 the L∞ (Y )-module will be
L∞ (X) with the inner product h·, ·iY , in which case L2 (E) is again L2 (X).
P ROPOSITION B.18. Let T : E → E be an L∞ (Y )-linear map which as a map into
L (E) is bounded and conditionally Hilbert-Schmidt. Then the image of the unit ball of
E under T is conditionally precompact.
2
P ROOF. Let ε > 0. By Lemma B.15 we can find a finite orthonormal set Ω ⊆ E
such that kT f k2 ≤ ε2 kf k for all f in the submodule Ω⊥ of E consisting of all elements
P
orthogonal to Ω. Note that Z := h∈Ω BL∞ (Y ) T h is a finitely generated module zonotope
in E, where BL∞ (Y ) is the unit ball of L∞ (Y ). By Lemma B.16 we can find a set A ⊆ Y
with measure greater than 1 − ε2 such that hT f, T f iY 1A ≤ kf k2 ε2 1A for all f ∈ Ω⊥ .
Let f be an element of the unit ball of E. By Lemma B.9 we know that the element
P
g := f − h∈Ω hf, hiY h lies in Ω⊥ and kgk ≤ kf k ≤ 1. Thus h1A T g, 1A T giY =
hT g, T giY 1A ≤ ε2 1A and hence k1A T gk ≤ ε. We also have khf, hiY k∞ ≤ khk·kf k ≤ 1
P
for each h ∈ Ω, and hence h∈Ω hf, hiY T h ∈ Z. Therefore
X
1A T f = 1A T g +
1A hf, hiY T h ∈ε Z.
h∈Ω