Download On E19 Etale Groupoids - University of Hawaii Mathematics

Renault’s Equivalence Theorem for C ∗-Algebras of
Étale Groupoids
Ryan Felix
Abstract
The purpose of this paper is to prove directly that if two locally compact Hausdorff
étale groupoids are Morita equivalent, then their reduced groupoid C ∗ -algebras are
Morita equivalent.
Introduction
The connection between group theory and C ∗ -algebra theory begins with representation
theory. Group C ∗ -algebras were constructed with the purpose to better understand the
unitary representations of the group. The corresponding ∗-representations of the group C ∗ algebras allowed the study to progress in the setting of C ∗ -algebra theory. Similarly, in
studying an interesting generalization of groups, known as groupoids, Renault explains the
construction of groupoid C ∗ -algebras in [8] and proves the correspondence of representations
[8, 2.1.21].
Morita equivalence for C ∗ -algebras was introduced by Rieffel in [10] as a way to study
representations of C ∗ -algebras. A nice exposition of the modern view on Morita equivalence
of C ∗ -algebras is given in [7, Chapter 3].
Renault’s equivalence theorem was introduced at the very end of [9] with no details
provided. So later, along with Muhly and Williams, Renault proved the equivalence theorem
for locally compact Hausdorff, second countable groupoids with Haar systems in [6, Theorem
2.8]. More recently, Sims and Williams proved that Morita equivalent groupoids with the
same assumptions give rise to Morita equivalent reduced groupoid C ∗ -algebras [12, Theorem
13].
In this paper, a direct proof of an analogous result is given for an interesting class of locally compact Hausdorff groupoids known as étale groupoids. Note that it suffices to assume
the groupoid is étale in place of second countability. Definitions for Morita equivalence of
groupoids and C ∗ -algebras in this paper are not classical, but rather, are equivalent versions
which are more useful in the given setting.
1
C ∗ -Algebra Theory
This section introduces definitions of C ∗ -algebras with a main focus on proving the
Gelfand Theorem and the Continuous Functional Calculus. The majority of the material in
this section was taken or modified from an introductory C ∗ -algebra course taught by Erik
Guentner at the University of Hawai’i at Manoa.
Definition 1.1. A ∗-algebra is a complex algebra A which satisfies the following, for all
a, b ∈ A and λ ∈ C,
(a + b)∗ = a∗ + b∗
(λa)∗ = λa∗
a∗∗ = a
(ab)∗ = b∗ a∗
Definition 1.2. A Banach ∗-algebra is a complex normed ∗-algebra A which is complete
and satisfies ||ab|| ≤ ||a||||b|| for all a, b ∈ A.
Definition 1.3. A C ∗ -algebra is a Banach ∗-algebra A with ||a∗ a|| = ||a||2 for all a ∈ A.
From now on, let A denote a C ∗ -algebra.
Example 1.4.
1. C with the usual operations and norm is a C ∗ -algebra.
2. Mn (C) with the usual operations and operator norm is a C ∗ -algebra for any n ∈ N.
3. For any locally compact, Hausdorff space X,
C0 (X) := {f ∈ C(X) : for every > 0 the set {x : |f (x)| ≥ } is compact}
with pointwise addition, multiplication, and complex conjugation and the supremum
norm is a C ∗ -algebra.
Definition 1.5. For a unital C ∗ -algebra A (with unit denoted 1), an element a ∈ A is
invertible if there exists b ∈ A such that ab = 1 = ba.
Definition 1.6. The spectrum of an element a ∈ A is the set of complex numbers, λ, such
that a − λ1 is not invertible, denoted SpA (a). The spectral radius of an element a ∈ A is
sup {|λ| : λ ∈ SpA (a)}, denoted SprA (a).
Theorem 1.7. (Spectral Radius Formula): [11, 18.9] Let A be a unital complex Banach
algebra. If a ∈ A, then
1
SprA (a) = lim ||an || n .
n→∞
Definition 1.8. An element a in a C ∗ -algebra is self adjoint if a = a∗ ; a is normal if
aa∗ = a∗ a; a is positive if it is self adjoint and Sp(a) ⊆ [0, ∞).
2
Proposition 1.9. a ∈ A is positive if and only if there exists b ∈ A such that a = b∗ b.
Theorem 1.10. (Gelfand Theorem): Let A be commutative and unital. Then A ∼
= C(Â)
where  is the compact Hausdorff space of non-zero multiplicative linear functionals on A.
Notation until the end of the proof of the Gelfand Theorem,
(1) A will be assumed to be commutative and unital.
(2) For a ∈ A, Define â : Â → C by â(α) = α(a).
(3) Let Φ : A → C(Â) denote the Gelfand transform which is defined by Φ(a) = â.
Lemma 1.11. For α ∈ Â,
1. α(1) = 1,
2. if a ∈ A is invertible, then α(a) 6= 0, and
3. ||α|| ≤ 1.
Proof. Since α is non-zero, there exists b ∈ A such that α(b) 6= 0. Since α is multiplicative,
α(b) = α(b · 1) = α(b)α(1). Hence, α(1) = 1.
Now, if a is invertible, 1 = α(a) = α(aa−1 ) = α(a)α(a−1 ). So, α(a) 6= 0.
assume that ||a||A ≤ 1 and λ ∈ C such that |λ| > ||a||.PSince || λa || < 1, the series
P∞Finally,
∞ an
an
−1
n=0 λ converges, and so, λ − a is invertible with inverse λ
n=0 λ . By (2) and the
linearity of α, 0 6= α(λ − a) = λ − α(a) which implies λ 6= α(a). Hence, ||α|| ≤ 1
Lemma 1.12. If a is not invertible, then there exists α ∈ â such that α(a) = 0.
Proof. Since a is not invertible, the ideal aA is a proper ideal of A, and thus contained in a
maximal proper ideal, denoted M . Then the quotient Banach algebra A/M is isomorphic to
C by the Gelfand Mazur theorem 7.71. Therefore, the projection p : A → A/M is contained
in â and p(a) = 0.
Proposition 1.13. Â is a compact Hausdorff space.
Proof. By 1.11, can embed  into Πa∈A Da by the correspondence α ↔ (α(a))a . Notice that
Πa∈A Da is compact and Hausdorff under the product topology and so it remains to show
that  is a closed subspace. This follows from the fact that the product topology on Â
coincides with the topology of pointwise convergence. So, limit points in  are clearly linear,
multiplicative, and since α(1) = 1 for all α, the limit points are also non-zero.
The proposition above shows that C(Â) is a C ∗ -algebra.
Lemma 1.14. SpA (a) = ran(â).
3
Proof.
λ ∈ SpA (a) ⇐⇒
⇐⇒
⇐⇒
⇐⇒
a − λ is not invertible.
there exists α ∈ â such that α(a − λ) = 0
there exists α ∈ â such that α(a) = λ.
λ ∈ ran(â).
by 1.11 and 1.12.
Proof. (Gelfand Theorem):
First, note that â is continuous for all a ∈ A since the topology on  coincides with that
of pointwise convergence; that is,
αλ → α ⇐⇒ αλ (a) → α(a) for all a ∈ A.
Now to check that the Gelfand map is indeed an isometric ∗-isomorphism.
(1) Linear and Multiplicative: This is straightforward and follows from the linearity and
multiplicative nature of the elements of  and the definition of each â.
(2) Continuity: In fact, the Gelfand map is a contraction.
||â|| = sup |â(α)| = sup |α(a)| ≤ ||a||.
α∈Â
α∈Â
The inequality follow from 1.11.
(3) Involutive: This will first be shown for self adjoint a ∈ A. The first claim is that
α(a) ∈ R for all α ∈ Â. Fix x = a + it, t ∈ R and write α(a) = λ + iµ. Then,
|α(x)|2 = |α(a + it)|2
= |α(a) + it|2
= |λ + iµ + it|2
= λ2 + (µ + t)2 = λ2 + µ2 + t2 + 2µt.
And,
|α(x)|2 ≤ ||x||2 = ||x∗ x||
= ||(a − it)(a + it)|| = ||a2 + t2 ||
≤ ||a2 || + t2 = ||a||2 + t2 .
So,
4
λ2 + µ2 + 2µt ≤ ||a||2 for all t ∈ R.
This can only be true when µ = 0 proving the first claim. Therefore, in the case that a
is self adjoint we have, for any α,
Φ(a∗ )(α) = α(a∗ ) = α(a) = α(a) = Φ(a)(α).
If a is not self adjoint, write a as
a + a∗
a − a∗
+i
:= b + ic.
2
2i
Note that b = b∗ and c = c∗ . For any α,
Φ(a∗ )(α) = α((b + ic)∗ ) = α(b∗ − ic∗ )
= α(b∗ ) − iα(c∗ ) = α(b) − iα(c)
= α(b) + iα(c) = α(b + ic) = α(a)
= (Φ(a)(α))∗
(4) Isometric (consequently injective): Again, this will be shown for self adjoint a first.
||Φ(a)|| = sup |α(a)|
α∈Â
= sup {|λ| : λ ∈ ran(â)}
= sup {|λ| : λ ∈ SpA (a)}
= SprA (a)
= ||a||
by 1.14
by 1.7, C ∗ -identity, and since a is self adjoint.
For arbitrary a ∈ A, consider a∗ a which is self adjoint.
||a||2 = ||a∗ a|| = ||Φ(a∗ a)|| = ||Φ(a)∗ Φ(a)|| = ||Φ(a)||2 .
(5) Surjective: From (1), (3), (4), Φ(A) is a norm closed subalgebra which is closed under
involution. Also, Φ(1A ) = 1C(Â) by an easy computation and 1.11. It is also clear
that Φ(A) separates points by the definition of the Gelfand map. Hence, by the StoneWeierstrass theorem 7.72, Φ(A) = C(Â).
5
Lemma 1.15. (Spectral Permanence): If A0 is a C ∗ -subalgebra of A containing the unit
and x ∈ A, then
SpA0 (x) = SpA (x).
Theorem 1.16. (Functional Calculus): Let A be unital and a a normal element in A. Then
there is a unique isometric ∗-homomorphism φ : C(SpA (a)) → A such that φ(1) = 1 and
φ(z) = a, where z represents the identity function on SpA (a). Further, φ(SpA (a)) is the
C ∗ -subalgebra of A generated by 1 and a, denoted C ∗ (a, 1).
Proof. Let A0 := C ∗ (a, 1) ⊂ A. Since a is normal, A0 is commutative. By the Gelfand
theorem 1.10, C(Â0 ) ∼
= A0 with the correspondence x̂ ↔ x for all x ∈ A0 .
Now, show that Â0 ∼
= SpA (a). By spectral permanence 1.15, this reduces to showing
that Â0 ∼
= SpA0 (a). The element â ∈ C(Â0 ) will provide the desired isomorphism, as follows.
Recall that â(α) = α(a) for any α ∈ Â0 .
(1) Surjective: Given λ ∈ SpA0 (a), the element a − λI is not invertible in A0 . By 1.12,
there is a α ∈ Â0 such that α(a) = λ. Hence â(α) = λ.
(2) Injective: This follows from the fact that a generatesAP
0 . Assume α(a)
= β(a) and
ki ∗ li
take arbitrary b ∈ A0 . Then b is the limit of a sequence
i a (a ) n , ki , li ∈ N. So
α(b) = β(b), since α and β are linear, multiplicative, ∗-preserving, and continuous.
(3) Homeomorphism: By (1),(2), and the open mapping theorem 7.73, â is a homeomorphism. Note that (2) is required to invoke the open mapping theorem.
Ψ
Thus, C(SpA (a)) ∼
= C(Â0 ) under the map f 7→ f ◦ â. The map φ as described in the
theorem is given by the composition Φ−1 ◦Ψ (where Φ is the Gelfand transform). Uniqueness
follows from the continuity of φ and the properties φ(1) = 1 and φ(z) = a since the ∗polynomials are dense in C(SpA (a)).
Remark 1.17. A note on non unital C ∗ -algebras. In the non unital case, it is always
possible to adjoin a unit, called the unitization, and identify the non unital C ∗ -algebra with
a C ∗ -subalgebra of the unitization. The following gives a framework for adjoining a unit to
a non unital C ∗ -algebra, A.
Given A, define A+ = {(a, λ) : a ∈ A, λ ∈ C}. Endow A+ with the vector space operations coming from A⊕C so that A+ ∼
= A⊕C as vector spaces. However, define multiplication
as follows, for any a, b ∈ A and λ, µ ∈ C,
(a, λ)(b, µ) = (ab + λb + µa, λµ),
so that A+ ∼
6= A ⊕ C as algebras. Define the involution on A+ by
(a, λ)∗ = (a∗ , λ).
Finally, topologize A+ with the C ∗ -algebra norm defined by,
||(a, λ)|| = sup ||ab + λb||A .
b∈A
||b||A ≤1
With all this, identify a ∈ A with (a, 0) ∈ A+ , and note that ||a||A = ||(a, 0)||A+ .
6
Definition 1.18. An approximate identity for a C ∗ -algebra, A, is a net {ui } in A such that
||ui || ≤ 1, 0 ≤ ui for every i, ui ≤ uj when i ≤ j, and satisfies
lim ui a = lim aui = a
i
i
for all a ∈ A.
Theorem 1.19. [3, I.4.8] Every C ∗ -algebra has an approximate identity.
Proposition 1.20. (Automatic Continuity): Let A and B be C ∗ -algebras. If π : A → B is
a nonzero ∗-homomorphism, then ||π(a)|| ≤ ||a|| for all a ∈ A.
Proof. It is straightforward to show that π extends to a unital ∗-homomorphism π̃ : A+ → B+
with π̃(a, λ) = (π(a), λ). It will be shown that π̃ is contractive and then the desired result
will follow using the fact that ||a||A = ||(a, 0)||A+ , as in the remark above 1.17.
First, it will be shown that, for self adjoint a in an arbitrary C ∗ -algebra, ||a|| = Spr(a).
n
n
By the C ∗ -identity, ||a||2 = ||a2 ||. This implies, by induction, that ||a||2 = ||a2 ||. Hence,
n
−n
||a|| = ||a2 ||2 .
Taking a limit as n → ∞, the spectral radius formula 1.7 implies ||a|| = Spr(a).
Next, it will be shown, for any a ∈ A+ , that SpB+ (π̃(a)) ⊆ SpA+ (a). This is easily seen
assuming there is some λ ∈ SpB+ (π̃(a)) \ SpA+ (a). Then a − λIA+ is invertible in A+ . Use
the fact that π̃ is a unital homomorphism to show that π̃(a) − λIB+ must also be invertible.
Now, use the C ∗ -identity and the previous two results to show π̃ is contractive. Let
a ∈ A+ . Note that a∗ a is self adjoint.
||π̃(a)||2 = ||π̃(a∗ a)|| = SprB+ (π̃(a∗ a)) ≤ SprA+ (a∗ a) = ||a∗ a|| = ||a||2
Finally, use this result for the case when π is not unital. Let a ∈ A.
||π(a)||B = ||(π(a), 0)||B+ = ||π̃(a, 0)||B+ ≤ ||(a, 0)||A+ = ||a||A .
Proposition 1.21. (Quotient C ∗ -Algebra): [3, I.5.4] If I is a norm closed two sided ideal
of A, then the quotient algebra A/I, with adjoint defined by (a + I)∗ = (a∗ + I) and quotient
norm ||(a + I)|| = inf x∈I ||a − x||, is a C ∗ -algebra.
Theorem 1.22. [3, I.5.5] Let A and B be C ∗ -algebras. If π : A → B is a nonzero ∗homomorphism, then ||π|| = 1 and π(A) is a C ∗ -subalgebra of B. If π is injective, then it is
isometric. So in general, A/ ker(π) ∼
= π(A).
Note that the automatic continuity of any ∗-homomorphism 1.20 is important so that
ker(π) is a norm closed two sided ideal of A. And so, the quotient C ∗ -algebra A/ ker(π)
makes sense as in 1.21.
7
Étale Groupoids
This section introduces groupoids with an emphasis on étale groupoids. Most of the
material is based on §5.6 from Brown and Ozawa’s C ∗ -algebras and Finite-Dimensional Approximations [2, §5.6].
Definition 2.23. A groupoid consists of a set G of morphisms and a distinguished subset
G(0) ⊆ G of objects, called units, together with source and range maps s, r : G → G(0) , and a
composition map G(2) = {(α, β) ∈ G × G : s(α) = r(β)} → G defined by (α, β) 7→ αβ such
that
(1) s(αβ) = s(β) and r(αβ) = r(α) for every (α, β) ∈ G(2) ,
(2) s(x) = x = r(x) for every x ∈ G(0) ,
(3) γs(γ) = γ = r(γ)γ for every γ ∈ G,
(4) (αβ)γ = α(βγ), and
(5) every γ has an inverse γ −1 , with γγ −1 = r(γ) and γ −1 γ = s(γ).
A topological groupoid G is a groupoid with a topology on G such that all structure maps
are continuous (only need continuity of inverse and composition as continuity of source and
range follow, see remark). Note that G(2) inherits the relative topology from G × G.
Remark 2.24. Continuity of inverse and composition implies continuity of range and source.
Proof. Show that the range map is continuous (the argument for the source map will be
similar). Denote the composition map by f and the inverse map by g. Define φ : G → G(2)
by φ(γ) = (γ, γ −1 ). Then
r ≡ f ◦ φ.
So, continuity of the range map depends on the continuity of φ, as continuity of f is assumed.
Take arbitrary open W ⊆ G(2) . Without loss of generality, it has the form π1−1 (U )∩π2−1 (V )∩
G(2) where πi is the projection of G×G onto the i-th coordinate and U, V are open in G since
these sets form a basis for the topology. Let γ ∈ φ−1 (W ). Since g is continuous, g −1 (V ) ∩ U
is an open neighborhood of γ contained in φ−1 (W ).
Definition 2.25. A topological groupoid G is said to be étale if s and r are local homeomorphisms.
Remark 2.26. If the source map (or range map) is a local homeomorphism, then so is the
range map (source map).
Proof. Assume that the source map is a local homeomorphism. Given γ ∈ G, claim that
there is an open neighborhood V such that r|V is a homeomorphism. Since s is a local
homeomorphism, there is an open neighborhood U of γ −1 such that s|U is a homeomorphism.
Let I denote the inverse map on G. Then I −1 (U ) is an open neighborhood of γ, and
r(I −1 (U )) = s(U ) which is homeomorphic to U via s. As I is a homeomorphism, U is
homeomorphic to I −1 (U ), and so r|I −1 (U ) is a homeomorphism. In general, the composition
of a homeomorphism and a local homeomorphism is again a local homeomorphism. In this
case, r = s ◦ I.
8
Proposition 2.27. If G is a locally compact, Hausdorff, étale groupoid, then the subspace
topology on Gx = {γ ∈ G : s(γ) = x} (Gx = {γ ∈ G : r(γ) = x}) is equivalent to the discrete topology for all x ∈ G(0) .
Proof. Assume for a contradiction that there exists some g ∈ Gx such that {g} is not open in
the subspace topology. Then any open set, U , of g must contain at least one other element
in Gx . Hence, for any open U of g, s|U is not injective and thus not a homeomorphism.
Proposition 2.28. If G is a locally compact, Hausdorff, étale groupoid, then G(0) ⊆ G is a
clopen subset.
Proof. To show that G(0) is closed, it will be shown that the complement is open. Take
γ ∈ (G(0) )c . By the Hausdorff property, there are disjoint, open sets U, V in G such that
s(γ) ∈ U and r(γ) ∈ V . Since s, r are continuous, s−1 (U ) ∩ r−1 (V ) is an open set in (G(0) )c
which contains γ.
To show that G(0) is open, let x ∈ G(0) . Since G is étale, there is an open U containing
x such that s is homeomorphic on U . By the continuity of s, s−1 (U ) is also open in G.
Then V := U ∩ s−1 (U ) is an open set containing x which is contained in G(0) . To see that
V ⊂ G(0) , take γ ∈ V and note that s(γ) = s(s(γ)). Since γ and s(γ) are contained in U ,
on which s is injective, the note implies that γ = s(γ). Hence γ ∈ G(0) .
Example 2.29. Let Γ be a group and X a locally compact Hausdorff space. Then
(1) G(0) = {e} ⊆ G = Γ is a groupoid where e is the identity in Γ.
(2) G(0) = G = X is a groupoid. If a space, Y , is considered as a groupoid in this way, then
it will be denoted Y with the trivial groupoid structure. Further, if Y only contains one
element, then it will be called the trivial groupoid.
(3) If X is a discrete set, define the pair groupoid to be X × X := {(x, y) : x, y ∈ X}
with composition given by (x, y)(y, z) = (x, z), G(0) = {(x, x) : x ∈ X}, s(x, y) =
(y, y), r(x, y) = (x, x), and endowed with the product topology.
(4) If Γ acts on X by homeomorphisms, then X oΓ = {(x, g, y) ∈ X × Γ × X : x = g · y} is
a groupoid with s(x, g, y) = (y, e, y), r(x, g, y) = (x, e, x) and (x, g, y)(y, h, z) = (x, gh, z).
Note that (x, g, y) is uniquely determined by (x, g). Since X o Γ = X × Γ as sets, can
endow it with the product topology where Γ is discrete. With this topology (note that the
action being by homeomorphisms is necessary here), X o Γ is an étale, locally compact,
Hausdorff groupoid, called the transformation groupoid.
(5) Let G be a groupoid and U = {Ui : i ∈ I} be an open covering of G(0) . The localization
of G over U is the groupoid
GU = {(i, γ, j) ∈ I × G × I : s(γ) ∈ Uj and r(γ) ∈ Ui }
(0)
with GU = {(i, x, i) : i ∈ I, x ∈ Ui } and with
9
s(i, γ, j) = (j, s(γ), j), r(i, γ, j) = (i, r(γ), i) and (i, α, j)(j, β, k) = (i, αβ, k).
The relative product topology, where I is discrete, makes GU into a locally compact,
Hausdorff groupoid, and if G is étale so is GU .
Example 2.30. For a discrete group Γ, the transformation groupoid Γ o Γ is isomorphic to
the pair groupoid Γ × Γ. Consider the map φ : Γ × Γ → Γ o Γ defined by φ(g, h) = (g, gh−1 )
and note that the subgroup {gh−1 : g, h ∈ Γ} of Γ is in fact equal to Γ.
Definition 2.31. Two étale, locally compact, Hausdorff groupoids G and G0 are said to be
Morita equivalent if there are localizations GU and G0U 0 , respectively, which are isomorphic
as topological groupoids.
Example 2.32.
(1) If X := {xi }i∈I where I is a discrete set, then the pair groupoid X × X is Morita
equivalent to the trivial groupoid, denoted by {x}. Let U = {U } where U = X × X and
V = {Vi }i∈I where Vi = {x} for all i. Then [X × X]U ∼
= {x}V under the correspondence
(xi , xj ) ↔ (i, x, j).
(2) Let Z act on R by translation. The transformation groupoid R o Z is Morita equivalent
to R/Z with the trivial groupoid structure. Let
(
U=
(
Ui =
i
2
i
2
− 14 , 2i + 14 if i is even
− 25 , 2i + 25 if i is odd
)
.
i∈Z
Let
(
V=
(
[[0], 14 ) ∪ ( 43 , [1]) if i is even
Vi = 1 9 ( 10 , 10 ) if i is odd
)
.
i∈Z
Then (R o Z)U ∼
= (R/Z)V under the correspondence (i, γ, j) ↔ (i, [s(γ)], j).
Hilbert C ∗ -Modules
This section introduces Hilbert C ∗ -modules. While learning about Hilbert C ∗ -modules,
keep in mind that they are very similar to Hilbert spaces in regards to their algebraic structure but are not similar in regards to their geometric structure. Most of the material is based
on Chapter 1 from Lance’s Hilbert C ∗ -modules [5, Chapter 1].
Definition 3.33. A right A-module is a set M together with a binary operation, +, which
makes M into an abelian group (associativity, invertibility, existence of identity) and an
action of A on M which satisfies, for all a, b ∈ A, m, n ∈ M,
(1) m(a + b) = ma + mb
10
(2) m(ab) = (ma)b
(3) (m + n)a = ma + na.
Definition 3.34. An inner product A-module is a linear space E which is a right A-module
where scalar multiplication is compatible (i.e. α(xa) = (αx)a = x(αa) for α ∈ C, x ∈ E,
and a ∈ A), with a map (x, y) 7→ hx, yi : E × E → A such that, for all x, y, z ∈ E, α, β ∈ C,
and a ∈ A,
(1) hx, αy + βzi = α hx, yi + β hx, zi
(2) hx, yai = hx, yi a
(3) hy, xi = hx, yi∗
(4) hx, xi ≥ 0; if hx, xi = 0 then x = 0.
Proposition 3.35. If E is an inner product A-module and x, y ∈ E then
hy, xi hx, yi ≤ || hx, xi || hy, yi .
Proof. Begin with the case when || hx, xi || = 1. First, it will be shown that || hx, xi || ≥
hx, xi. Consider (hx, xi , 0) ∈ A+ as in 1.17. Note that (hx, xi , 0) is self adjoint. By
the functional calculus 1.16, the C ∗ -subalgebra C ∗ ((hx, xi , 0), (0, 1)) of A+ is isomorphic
to C(SpA+ (hx, xi , 0)) as C ∗ -algebras and (hx, xi , 0) corresponds to the identity function.
Hence, ||(hx, xi , 0)||A+ ≥ (hx, xi , 0), and so || hx, xi ||A ≥ hx, xi.
Now, by 1.9, there exists a ∈ A such that || hx, xi || − hx, xi = a∗ a. So it follows from
the computation below and 1.9, that for any arbitrary b ∈ A, b∗ || hx, xi ||b − b∗ hx, xi b is
positive (*):
b∗ || hx, xi ||b − b∗ hx, xi b = b∗ (|| hx, xi || − hx, xi)b = b∗ (a∗ a)b = (ab)∗ ab.
Now,
0 ≤ hx hx, yi − y, x hx, yi − yi
= hx, yi∗ hx, xi hx, yi − hy, xi hx, yi − hx, yi∗ hx, yi + hy, yi
= hx, yi∗ hx, xi hx, yi + hy, yi − 2 hy, xi hx, yi
≤ hx, yi∗ hx, yi + hy, yi − 2 hy, xi hx, yi by (*).
Therefore,
hy, xi hx, yi ≤ hy, yi = || hx, xi || hy, yi .
If || hx, xi || 6= 1, then consider the element √
x
||hx,xi||
and apply the above result.
Remark 3.36. From the above result, the following are obvious consequences.
(1) The inner product is continuous.
11
(2) By the C ∗ identity, || hx, yi || ≤ ||x||||y|| (3.1).
(3) Using (3.1), || · || : E → [0, ∞) defined by
1
||x|| = || hx, xi || 2
is a norm on any inner product A-module.
(4) Also using (3.1), ||x|| = sup {|| hx, yi || : ||y|| ≤ 1}.
Definition 3.37. An inner product A-module which is complete with respect to its norm is
called a Hilbert A-module.
Example 3.38.
(1) A is itself a Hilbert A-module with the obvious action and inner product given by
ha, bi = a∗ b. Denote this Hilbert A-module by EA .
(2) If H is a Hilbert space and X is a compact hausdorff space, then C(X, H) is a Hilbert
C(X)-module with pointwise multiplication as the action and inner product given by
hf, gi (x) = hf (x), g(x)iH .
Definition 3.39. Let E and F be Hilbert A-modules. An adjointable operator from E to
F is a map T : E → F for which there is a map T ∗ : E → F such that, for x ∈ E, y ∈ F ,
hT x, yi = hx, T ∗ yi .
Denote by L(E) the set of all adjointable operators from E to itself.
Proposition 3.40. If T ∈ L(E), then T is bounded on E; that is, there exists c < ∞ such
that ||T x|| ≤ c||x|| for all x ∈ E.
Proof. Let T ∈ L(E). It will be shown that the graph of T is closed, so by the closed graph
theorem 7.74, T will be bounded. Assume xn → x in E and T (xn ) → y in F . Then
hy, yi = lim hT (xn ), yi
Dn
E
∗
= lim xn , T y
by the continuity of h·, ·i.
n
= hT x, yi
= hT x, T xi
by a similar argument as above.
To summarize (although all equivalences were not shown) hy, yi = hT x, yi = hT x, T xi =
hy, T xi. And so,
||T (x) − y||2 = || hT (x) − y, T (x) − yi || = 0
which implies, T (x) = y. Hence, the graph of T is closed and therefore T is bounded on E.
12
Example 3.41. (Bounded operator that is not adjointable) Consider example 3.38 (2)
with H = C and X = [0, 1]. Define E = {f ∈ C([0, 1]) : f (0) = 0}. The inclusion operator
i : E → C([0, 1]) is clearly bounded. Assume that i is adjointable, with adjoint i∗ . Let g
be the characteristic function of [0, 1]. Consider i∗ (g), by the definition of E, there exists an
x ∈ (0, 1] such that i∗ (g)(x) 6= 1. Fix this x and let f ∈ E such that f (x) = g(x) − i∗ (g)(x).
Then,
hi(f ), gi = hf, i∗ (g)i
=⇒ 0 = hf, g − i∗ (g)i
=⇒ 0 = hf (x), g(x) − i∗ (g)(x)iC
The last equality is a contradiction, and so, i is not adjointable.
Any adjointable operator on an A-module is A-linear; that is, for x, y ∈ E and a ∈ A,
T (x + y) = T (x) + T (y)
T (xa) = T (x)a.
With everything above, it is straightforward to show that L(E) is a C ∗ -algebra for any
Hilbert C ∗ -module E. The following introduces an important closed, two sided ideal of L(E).
Definition 3.42. For x, y ∈ E, define θx,y : E → E by θx,y (z) = x hy, zi, such operators will
be known as rank one operators.
The space of compact operators, K(E), in L(E) is the closed linear span of all rank one
operators. The fact that K(E) is a closed, two sided ideal follows from properties of rank
one operators, for details see [5, Ch. 1].
Construction of C ∗ -algebras Associated to Étale Groupoids
This section outlines the construction of C ∗ -algebras from étale groupoids, gives concrete descriptions of specific examples, and contains some general theory of constructions of
transformation groupoids. The construction is taken from [2, §5.6] with added detail. The
general theory on transformation groupoids was outlined in notes by Rufus Willett.
Lemma 4.43. Let E0 be a dense subset of a normed linear space E. Let F be a Banach
space and T : E0 → F be an bounded linear operator. There exists a unique linear operator
T̃ : E → F extending T which is also bounded. Moreover, if E = F is a Hilbert C ∗ -module
and T is a projection, then T̃ is also a projection.
Proof. Assume x ∈ E \ E0 . Since E0 is dense in E, there exists a sequence {xn } in E0 such
that lim xn = x. Claim that {T (xn )} is a Cauchy sequence in F and so lim T (xn ) exists.
Choose N such that for n, m ≥ N , ||xn − xm || < . Using the bound on T ,
||T (xn ) − T (xm )|| = ||T (xn − xm )|| ≤ c||xn − xm || < c.
13
So {T (xn )} is indeed a Cauchy sequence. Now, define T̃ : E → F by
(
T (x)
if x ∈ E0
T̃ (x) =
lim T (xn ) if x ∈ E \ E0 and xn → x
At first glance this definition may not seem well defined, however it is easy to check that
lim T (xn ) = lim T (yn ) for any sequences {xn }, {yn } in E0 converging to the same x ∈ E.
Indeed,
||T (xn ) − T (yn )|| ≤ c||xn − yn || ≤ c(||xn − x|| + ||x − yn ||) → 0
It is left to show that T̃ is linear and bounded. Linearity follows easily from the linearity
of T and the definition of T̃ . Boundedness follows from the continuity of the norm.
If E = F , the fact that the extension preserves projections is clear from direct computations using the definition of the extension given above.
Let G be an étale, locally compact, Hausdorff groupoid. By 4.50, Gx and Gx are
discrete in G for every x ∈ G(0) . Let Cc (G) denote the the space of all continuous compactly
supported functions f : G → C with composition and ∗-operation given by
X
X
f (γβ −1 )g(β) and f ∗ (γ) = f (γ −1 ).
f (α)g(β) =
(f ∗ g)(γ) =
αβ=γ
β∈Gs(γ)
Proposition 4.44. The space Cc (G) with operations as above is a ∗-algebra.
Proof. It will only be shown that the convolution is well defined; that is, given f, g ∈
Cc (G), f ∗ g is also contained in Cc (G). From there, checking the axioms of a ∗-algebra
are straightforward.
Let f, g ∈ Cc (G). To prove that f ∗ g has compact support let supp(f ) · supp(g) denote
the set {γ ∈ G : γ = αβ, α ∈ supp(f ), β ∈ supp(g)}. Note that this set is compact as it is
the continuous image (image of the composition map) of a compact set (supp(f )×supp(g) ⊆
G × G). Claim that supp(f ∗ g) ⊆ supp(f ) · supp(g). Let γ ∈ supp(f ∗ g). Then
X
f (α)g(β) 6= 0
αβ=γ
=⇒ there are α, β such that αβ = γ and f (α)g(β) 6= 0
=⇒ f (α) =
6 0 and g(β) 6= 0
=⇒ α ∈ supp(f ) and β ∈ supp(g).
Hence, γ ∈ supp(f ) · supp(g). So, supp(f ∗ g) is a closed subset of a compact set, and thus
compact.
To prove that f ∗ g is continuous, it suffices to show that for any γ ∈ G there is an open
subset U of G containing γ such that (f ∗g)|U is continuous. Fix γ ∈ G and let K := supp(g).
14
Since K is compact, cover it with finitely many open sets, {Ui }n , such that the source map
is a homeomorphism on each Ui . So, for any α ∈ G,
|K ∩ Gs(α) | ≤ | [∪n Ui ] ∩ Gs(α) | ≤ n.
More generally, for future reference, given any compact K 0 ⊆ G,
sup |K 0 ∩ Gx | < ∞,
(4.1)
x
and similarly,
sup |K 0 ∩ Gx | < ∞.
(4.2)
x
Back to the problem at hand, enumerate the set K ∩ Gs(γ) = {γi }m where m ≤ n. Since
G is Hausdorff, there are disjoint open sets, {Ui }m , such that γi ∈ Ui for all i. It can be
assumed that
Sm s is a homeomorphism on these sets since s is a local homeomorphism. Since
E := K \[ i Ui ] is compact, continuity of s implies s(E) is compact. Note that s(γ) ∈
/ s(E),
(0)
and so there are disjoint sets U and V such that s(γ) ∈ U and s(E) ⊆ V since G is also
Hausdorff.
Let
"m
#
\
W := U ∩
s(Ui ) and Wi := Ui ∩ s−1 (W )
i
−1
so
Smthat s|Wi is a homeomorphism from Wi to W for all i and s (W ) ∩ K is contained in
1 Wi .
Let V now denote an open set containing γ such that s|V is a homeomorphism. It will
be shown that the restriction of f ∗ g to the open set V ∩ s−1 (W ) is continuous. Note that
s|Wi ∩s−1 (V ) is a local homeomorphism from Wi ∩ s−1 (V ) to s(V ) ∩ W for all i.
For each i, let hi : s(V )∩W → Wi ∩s−1 (V ) be the inverse homeomorphism of s|Wi ∩s−1 (V ) .
Define
φi : V ∩ s−1 (W ) → C, φi (α) = f (α(hi (s(α))−1 )g(hi (s(α)))
Each φi is continuous as it is the composition of continuous functions. Define
Φ : V ∩ s−1 (W ) → Cm ,
Φ(α) = (φ1 (α), . . . , φm (α)),
which is also continuous since each component function is continuous. Thus,
(f ∗ g)|V ∩s−1 (W ) ≡ F ◦ Φ
where F : Cm → C is component addition, another continuous function. So, it has been
shown that for any γ there is an open set U containing γ such that (f ∗ g)|U is continuous.
Therefore, f ∗ g is continuous.
Now, Cc (G) can be made into a right C0 (G(0) )-module with action and C0 (G(0) ) valued
inner product defined by
X
(f · ξ)(γ) = f (γ)ξ(s(γ)) and hf, gi (x) =
f (γ)g(γ)
γ∈Gx
15
for f, g ∈ Cc (G) and ξ ∈ C0 (G(0) ).
Let L2 (G) be the completion of Cc (G) as a C0 (G(0) )-module, given the inner product
above. Define the left regular representation λ : Cc (G) → L(L2 (G)) by λ(f )ξ = f ∗ ξ for all
f, ξ ∈ Cc (G) (f thought of as an element of the convolution algebra and ξ as an element of
the Hilbert C0 (G(0) )-module). The reduced groupoid C ∗ -algebra Cλ∗ (G) is the norm closure
of λ(Cc (G)) in L(L2 (G)).
Remark 4.45. Elements in the image of λ are only defined on the dense subset Cc (G) of
L2 (G), and so, given an f ∈ Cc (G), to extend λ(f ) to all of L2 (G) it is required to show
that λ(f ) is bounded and apply 4.43. It should also be noted that λ(f ) is adjointable with
adjoint λ(f ∗ ), and the lemma implies that the extension will also be adjointable. Above, it
is assumed that λ is in fact this extension.
Boundedness of λ(f ) depends on (4.1), (4.2). Indeed,
||λ(f )||2 = sup ||f ∗ ξ||2
||ξ||=1
X
= sup sup
ξ
|f ∗ ξ(γ)|2
x∈G(0) γ∈Gx
2
X X
f (α)ξ(β)
= sup sup
x
ξ
γ
αβ=γ


X X
X

≤ sup sup
|f (α)|2
|ξ(β)|2 
ξ
≤ sup
x
x
γ
α∈Gr(γ)
X X
γ
by the Cauchy Schwarz inequality
β∈Gs(γ)
|f (α)|2
since ||ξ|| = 1
α∈Gr(γ)
<∞
by (4.1), (4.2) and since supp(f ) is compact.
Since G is an étale groupoid, Gx is discrete, so let l2 (Gx ) be the usual Hilbert space
of square summable sequences indexed by Gx . Define the representation λx : Cc (G) →
L(l2 (Gx )) by
X
(λx (f )ξ)γ =
f (γβ −1 )ξ(β).
β∈Gx
Lemma 4.46. For any f ∈ Cc (G),
||f ||Cλ∗ (G) = sup ||λx (f )||L(l2 (Gx )) .
x∈G(0)
Proof. Assume η, ξ ∈ Cc (G) such that ||η||L2 (G) = ||ξ|| = 1 and y ∈ G(0) .
16
2
X
X
X
2
≤
|η(γ)|
|f ∗ ξ(γ)|2
η(γ)f
∗
ξ(γ)
γ∈Gy
γ∈Gy
γ∈Gy
X
≤
|(λy (f )ξ)γ|2 ||η||2L2 (G)
γ∈Gy
= ||λy (f )ξ||2l2 (Gy ) ≤ ||λy (f )||2L(l2 (Gy )) ||ξ||2l2 (Gy ) ≤ ||λy (f )||2L(l2 (Gy ))
The computation above shows that
|| hη, f ∗ ξi ||C0 (G(0) ) ≤ sup ||λx (f )||L(l2 (Gx ) .
x∈G(0)
Better still, since ξ and η were chosen arbitrarily,
||f ||Cλ∗ (G) = sup ||f ∗ ξ||L2 (G) = sup sup || hη, f ∗ ξi ||C0 (G(0) ) ≤ sup ||λx (f )||L(l2 (Gx ) .
||ξ||=1
||ξ||=1 ||η||=1
x∈G(0)
The opposite inequality will follow by definition (of the norms) provided that for fixed
y ∈ G(0) and any η ∈ Cc (Gy ) with ||η||l2 (Gy ) = 1 there is an extension η̂ ∈ Cc (G) such
that ||η̂||L2 (G) = 1. The extension exists by 7.68, and if the norm is not equal to 1, then
normalize.
Example 4.47.
(1) Let G = Z2 × Z2 as a pair groupoid. The reduced C ∗ -algebra Cλ∗ (G) ∼
= M2 (C).
As Z2 × Z2 is discrete, the space C(Z2 × Z2 ) is isomorphic to C4 as a vector space
under the map f 7→ (f (1, 1), f (0, 0), f (1, 0), f (0, 1)). However, considering C(Z2 × Z2 )
as a convolution algebra, composition is not preserved under the above map (thinking
of pointwise multiplication in C4 ).
But, upon observation of the convolution, it appears to act as matrix multiplication.
This is in fact the case and the map
f (1, 1) f (1, 0)
f 7→
f (0, 1) f (0, 0)
is a ∗-isomorphism between C(Z2 × Z2 ) and M2 (C). Knowing that M2 (C) has a C ∗ algebra norm, can endow C(Z2 ×Z2 ) with the same norm, thus constructing a C ∗ -algebra
from Z2 × Z2 .
This construction does not follow the general construction precisely, but the result must
be isomorphic as Cλ∗ (Z2 ×Z2 ) (in the general construction) is the completion of the image
of a faithful representation of C(Z2 × Z2 ).
1
(2) Let G = X o Z2 with X = [0, 1] × {0, 1} and the action is given by (x, i) 7→ (x, i + 1)
0
and (x, i) 7→ (x, i). The reduced C ∗ -algebra Cλ∗ (G) ∼
= M2 (C([0, 1])).
17
C(X o Z2 ) is better understood given the groupoid isomorphism X o Z2 ∼
= [0, 1] × (Z2 ×
Z2 ). In the latter, Z2 × Z2 is taken as the pair groupoid, but the other product does
not denote a pair groupoid. Composition is only taken between elements that match in
the first coordinate and can compose in the second, and is given by (x, (i, j))(x, (j, k)) =
(x, (i, k)). Inverse is given by (x, (i, j))−1 = (x, (i, j)−1 ). Range and source of (x, (i, j))
are (x, (i, i)) and (x, (j, j)) respectively.
Now, similar to the above example, it can be shown under the mapping
f |[0,1]×(0,0) f |[0,1]×(0,1)
f 7→
f |[0,1]×(1,0) f |[0,1]×(1,1)
that C(X o Z2 ) ∼
= M2 (C([0, 1])). Again, obtain a C ∗ -algebra from X o Z2 that must
be isomorphic to Cλ∗ (X o Z2 ) without following the general construction; i.e. through
direct computation.
(3) Consider the subgroupoid H = [0, 1) × (Z2 × Z2 ) ∪ [0, 2] × {(0, 0), (1, 1)} of G in the
previous example (where now X = [0, 2] × {0, 1} for simplicity). H is no longer compact
as the groupoids in the previous examples, however for each f ∈ Cc (G) the map
f |[0,2]×(0,0) f |[0,2]×(0,1)
φ
f 7→
f |[0,2]×(1,0) f |[0,2]×(1,1)
still embeds Cc (G) into M2 (C([0, 2])). Now to determine the image of φ, for each
f ∈ Cc (G),
(
C([0, 2])
if i = j
f |[0,2]×(i,j) ∈
Cc ([0, 1)) if i 6= j.
So,
φ(Cc (G)) =
f11 f12
f21 f22
: f11 , f22 ∈ C([0, 2]) and f12 , f21
∈ Cc ([0, 1)) ,
a ∗-closed subalgebra of M2 (C([0, 2])).
This shows that the image of Cc (G) under the left regular representation, λ, is ∗isomorphic to φ(Cc (G)), the subalgebra of M2 (C([0, 2])). The isomorphism is in fact
isometric by 4.46, but still does not determine the reduced C ∗ -algebra since φ(Cc (G))
is not complete. Determine the completion of λ(Cc (G)) ⊆ L(L2 (G)) by considering the
completion of φ(Cc (G)) ⊆ M2 (C([0, 2])).
As all norms on a finite dimensional space, in particular M2 (C([0, 2])), are equivalent,
use the matrix max norm on φ(Cc (G)) to establish its completion. The computations will
not be carried out here, but notice that the max norm converts the completion question
on the matrix algebra to a completion question for each entry. From here, the answer
is clear as C([0, 2]) is complete in supremum norm and C0 ([0, 1)) is the completion of
Cc ([0, 1)) in supremum norm. Hence, the completion of φ(Cc (G)) can be given as,
f11 f12
: f11 , f22 ∈ C([0, 2]) and f12 , f21 ∈ C0 ([0, 1)) .
f21 f22
18
Consider a general transformation groupoid G := Y o Γ, where Y is a compact Hausdorff
space and Γ is a finite group acting on Y by homeomorphisms. Write X := Y /Γ for the
quotient space associated with the given action, and π : Y → X the associated quotient
map.
Proposition 4.48. The quotient space, X, is compact and Hausdorff.
Proof. Compactness is clear since the topology on the quotient space is taken as the finest
topology in which the quotient map is continuous. And so, the quotient space is the image
of a compact space under a continuous function, hence compact.
The Hausdorff property will follow from the finiteness of Γ. Take arbitrary points, x1 , x2 ∈
X, and consider the sets π −1 (x1 ) and π −1 (x2 ). These sets are finite since Γ is finite, and
so, can use the Hausdorff property of Y to separate the given preimages by open sets (by
taking finite intersections of open sets), say U and V . Thus, π(U ) and π(V ) are open sets
that separate x1 and x2 .
The preimages of each x ∈ X are the orbits of Γ on Y , so the action of Γ on Y restricts
to an action on π −1 (x) for each x ∈ X. Form the transformation groupoids,
Hx = π −1 (x) o Γ
for all x ∈ X. Since Γ is finite, each Hx is finite so the subspace topology it inherits from G
is the discrete topology.
Let A be the collection of functions
G
f :X→
Cλ∗ (Hx )
(4.3)
x∈X
such that f (x) ∈ Cλ∗ (Hx ) for all x ∈ X and supx∈X ||f (x)||Cλ∗ (Hx ) < ∞. With the pointwise
∗-algebra operations coming from each Cλ∗ (Hx ) and norm
||f ||A = sup ||f (x)||Cλ∗ (Hx ) ,
x∈X
A becomes a C ∗ -algebra.
For each f ∈ Cc (G) define
fˆ : X →
G
Cλ∗ (Hx )
x∈X
by fˆ(x) = f |Hx .
Lemma 4.49. For any f ∈ Cc (G) and x ∈ X,
||f |Hx ||Cλ∗ (Hx ) ≤ sup |f (γ)||Γ|2 .
γ∈G
In particular, the upper bound is independent of x, so fˆ is in A. Further,
sup |f (γ)| ≤ sup ||f |Hx ||Cλ∗ (Hx ) .
γ∈G
x∈X
19
Proof. Fix x ∈ X. Define δ(g,y) : G → C by
(
1 if γ = (gy, g, y)
δ(g,y) (γ) =
0 otherwise
(the Dirac masses on G). It is straightforward to show that λ(δ(g,y) )∗ = λ(δ(g−1 ,gy) ), and with
this, to show that λ(δ(g,y) )λ(δ(g,y) )∗ and λ(δ(g,y) )∗ λ(δ(g,y) ) are projections in L(L2 (G)). Hence
λ(δ(g,y) ) is a partial isometry, which implies ||λ(δ(g,y) )||Cλ∗ (Hx ) = 1 for all g ∈ Γ and y ∈ Y .
Since Hx is finite, f |Hx has finite support, so it can be represented as a linear combination
of the Dirac masses; that is,
X X
f |Hx =
f (gy, g, y)δ(g,y) .
y∈π −1 (x) g∈Γ
Therefore,
||fˆ(x)||Cλ∗ (Hx )
X X
= ||f |Hx || = f (gy, g, y)δ(g,y) y∈π−1 (x) g∈Γ
X X
≤
sup |f (γ)|
y∈π −1 (x) g∈Γ
γ∈G
≤ sup |f (γ)||Γ|2 .
γ∈G
For the first inequality, fix γ ∈ G. There is some x ∈ X such that γ ∈ Hx . Let δγ , δs(γ)
be Dirac masses in L2 (Hx ). Note that both have norm 1.
|| δγ , f ∗ δs(γ) ||C0 (Hx(0) )
X
X
δγ (γ0 )
f (α)δs(γ) (β) = |f (γ)|
= sup (0)
y∈Hx γ0 ∈Gy
αβ=γ0
This shows that, for the chosen γ and corresponding x, |f (γ)| ≤ ||f |Hx ||Cλ∗ (Hx ) . Therefore,
sup |f (γ)| ≤ sup ||f |Hx ||Cλ∗ (Hx ) .
γ∈G
x∈X
Proposition 4.50. The function
Ψ : Cc (G) → A
defined by Ψ(f ) = fˆ is an injective ∗-homomorphism. Also, the Cλ∗ (G) norm on Cc (G) is
equal to
sup ||f |Hx ||Cλ∗ (Hx ) ,
x∈X
and so, Ψ is isometric. Moreover, Cc (G) is complete in the Cλ∗ (G) norm.
20
Proof. Showing that Ψ is a ∗-homomorphism will not be shown here, but note when evaluating any f ∈ Cc (G), (or a the product of two such elements or even the involution
of such an element), at some (gy, g, y) ∈ G the result depends only on some collection
{(y1 , h, y2 ) : h ∈ Γ, y1 , y2 ∈ orbit of y} ⊆ Hx where x = π(y).S
Showing that Ψ is injective depends on the fact that G = x∈X Hx . If f ∈ ker(Ψ), then
f |Hx is zero for all x ∈ X. Hence f ≡ 0 on all of G by the first statement.
From 4.46,
||f ||Cλ∗ (G) = sup ||λy (f )||L(l2 (Gy )) .
y∈G(0)
The right hand side is the same as
!
sup ||λy (f )||L(l2 (Gy ))
sup
x∈X
.
(4.4)
y∈π −1 (x)
(0)
Note for any x, π −1 (x) = Hx and given y ∈ π −1 (x), Gy = (Hx )y . Hence, the action of any
λy (f ) on l2 (Gy ) only realizes values of the restriction f |Hx . With this, the supremum in the
parenthesis of (4.4) becomes
sup ||λy (f |Hx )||L(l2 ((Hx )y )) = ||f |Hx ||Cλ∗ (Hx )
by 4.46.
(0)
y∈Hx
Putting everything together,
||f ||Cλ∗ (G) = sup ||f |Hx ||Cλ∗ (Hx ) .
x∈X
Now, the above equality and 4.49, show that the Cλ∗ (G) norm on Cc (G) is equivalent to
the supremum norm which is complete since G is compact.
Corollary 4.51. Let X be a compact topological space and Γ a finite group. Let G =
[X × Γ] o Γ be the transformation groupoid with action given by
g : (x, h) → (x, gh).
Then Cc (G) is ∗-isomorphic to the set of continuous functions in A (as defined above, Equation 4.3), denote this set by B.
In this case, for any x ∈ X, the groupoid Hx is isomorphic to Γ o Γ by the following map
Φx : Γ o Γ → Hx ,
(γα, γ, α) 7→ ((x, γα), γ, (x, α)).
And so, to clarify what is meant by continuous functions in A, use the above identification
for each x. So that B is the set of all continuous functions from X to Cλ∗ (Γ × Γ).
For the desired isomorphism, adjust Ψ, as in 4.50, so that any f ∈ Cc (G) maps to f˜
where f˜(x) = f |Hx ◦ Φx .
21
Multiplier Algebras and Morita Equivalence of C ∗ -Algebras
This section discusses multiplier algebras for the purpose of defining Morita equivalence
of C ∗ -algebras. The most important piece for the purpose of this paper is that the multiplier
algebra of an ètale groupoid C ∗ -algebra can be taken as follows,
M (Cλ∗ (G)) = T ∈ L(L2 (G)) : T a, aT ∈ Cλ∗ (G) for all a ∈ Cλ∗ (G) .
The material in this section is based on notes from Rufus Willett, in which is cited Blackadar’s Operator Algebras: Theory of C ∗ -Algebras and Von Neumann Algebras [1, II.7.3].
Definition 5.52. The multiplier algebra of A, M (A) is the C ∗ -algebra L(EA ).
Definition 5.53. A projection p in the multiplier algebra M (A) is said to be full if ApA :=
span {apb : a, b ∈ A} is dense in A.
Definition 5.54. Two C ∗ -algebras A and B are said to be elementary Morita equivalent,
eM
denoted A ∼ B, if there is a full projection p in M (A) such that pAp := {pap : a ∈ A} is
∗-isomorphic to B.
M
Two C ∗ -algebras A and B are Morita equivalent, denoted A ∼ B, if there is a finite
sequence of C ∗ -algebras A1 , A2 , . . . , An such that
eM
eM
eM
A∼
= A1 ∼ A2 ∼ · · · ∼ An ∼
= B.
For the purpose of this paper, it is not tractable to use the multiplier algebra as defined
in 5.52. However, the following will give a host of C ∗ -algebras which are ∗-isomorphic to
M (A).
Definition 5.55. A representation of a C ∗ -algebra A is a ∗-homomorphism
π : A → L(E)
from A to the adjointable operators on a Hilbert module E.
Proposition 5.56. Let π : A → L(E) be a representation. Then
Mπ (A) := {T ∈ L(E) : T π(a), π(a)T ∈ π(A) for all a ∈ A}
is a C ∗ -subalgebra of L(E).
The proof will not be given, but the most interesting part of the proof is the fact that
Mπ (A) is norm closed. This follows from 1.22.
This C ∗ -subalgebra will be called the multiplier algebra of A associated to π and its
elements will be called multipliers. The following will show that L(EA ) is the multiplier
algebra of A associated to the left multiplication representation.
Proposition 5.57. If π : A → L(EA ) is the ∗-homomorphism defined by π(a) : ξ → aξ
for all a ∈ A, called the left multiplication representation, then π(A) = K(EA ), the set of
compact operators.
22
Proof. Let θξ,η be a rank one operator. Then, for all ζ ∈ EA ,
θξ,η (ζ) = ξ hη, ζi = ξη ∗ ζ = π(ξη ∗ )ζ.
This implies that θξ,η ≡ π(ξη ∗ ), and so, K(EA ) ⊆ π(A).
For the other inclusion, let {ui } be an approximate identity in A, 1.19. Note that {u∗i }
is also an approximate identity since ||a|| = ||a∗ || for any a ∈ A. Now, for any a ∈ A,
||θa,ui − π(a)||L(EA ) = sup ||θa,ui ξ − π(a)ξ||EA
ξ∈EA
||ξ||≤1
= sup ||au∗i ξ − aξ||A ≤ sup ||au∗i − a||A ||ξ||A
ξ∈A
||ξ||≤1
ξ∈A
||ξ||≤1
≤ ||au∗i − a||A
Since {ui } is an approximate identity, the final expression tends to zero, and so, π(a) is the
limit of rank one operators. Thus, π(A) ⊆ K(EA ) as this subspace is norm closed.
Definition 5.58. A representation
π : A → L(E)
is nondegenerate if the subspace
( n
)
X
π(A) · E :=
π(aj )ξj : n ∈ N, a1 , a2 , . . . , an ∈ A, ξ1 , ξ2 , . . . , ξn ∈ E
j=1
of E is dense in E.
Example 5.59.
(1) The left multiplication representation is nondegenerate. This is easily seen considering an
approximate identity, {ui } ⊂ A. For arbitrary ξ ∈ EA , the sequence {π(ui )ξ} ⊂ π(A)·EA
converges to ξ. Hence, π(A) · EA is dense in EA .
(2) Take X to be locally compact Hausdorff, µ any radon measure on X, and E = L2 (X, µ)
the corresponding Hilbert space (Hilbert C-module). The natural representation of
C0 (X) on E by multiplication operators is non-degenerate.
Proof. Let π denote the natural representation of C0 (X) on E by multiplication operators. Since µ is a Radon measure, Cc (X) is dense in L2 (X), so it is sufficient to prove
that π(C0 (X)) · L2 (X, µ) ⊇ Cc (X).
If µ is a finite measure, then χX ∈ L2 (X) (the characteristic function on the whole
space). Let h ∈ Cc (X) ⊆ C0 (X). Then π(h) · χX ≡ h and we are done.
If µ is not finite and h ∈ Cc (X) with support k, then let fk ∈ C0 (X) such that fk ≡ 1
on k and vanishes outside some compact subset of X (existence of such a function is by
7.68). Then π(fk ) · h ≡ h.
23
(3) If G is a locally compact, Hausdorff, étale groupoid, then the natural representation of
its convolution algebra Cλ∗ (G) on the Hilbert module L2 (G) is nondegenerate.
Proof. Let π denote the natural representation of Cr∗ (G) on L2 (G) by left multiplication
(convolution). Since Cc (G) is dense in L2 (G), it is sufficient to show that π(Cr∗ (G))·L2 (G)
contains Cc (G). In fact, it will be shown that Cc (G) ∗ Cc (G) := {ξ ∗ ζ : ξ, ζ ∈ Cc (G)} ⊇
Cc (G), which would be obvious if Cc (G) was unital.
Fix ζ ∈ Cc (G) and let k = s(supp(ζ)) ⊆ G(0) , where s is the source map. Since s
is continuous and supp(ζ) is compact, k is also compact. Let f be the characteristic
function on k, clearly continuous. By the locally compact version of Tietze’s Extension
Theorem 7.68, extend to some F ∈ Cc (G). Since G(0) is clopen, in particular open, can
take such an F which vanishes outside of a compact subset of G(0) (1). Claim: ζ ∗ F = ζ.
Note that,
X
ζ ∗ F (γ) =
ζ(α)F (β) =
X
ζ(γβ −1 )F (β),
β∈Gs(γ)
αβ=γ
where Gs(γ) = {β ∈ G : s(β) = s(γ)} . By (1), there is only one such β where F (β) is
non-zero, β = s(γ), in which case F (s(γ)) = 1. And so,
ζ ∗ F (γ) =
X
ζ(γβ −1 )F (β) = ζ(γ).
β∈Gs(γ)
Proposition 5.60. If A is unital, a representation π : A → L(E) is nondegenerate if and
only if it is unital.
Proof. If π is a unital representation of A, then it is clearly nondegenerate as π(1)(E) = E.
If π is a nondegenerate representation of A, then π(A) · E P
is dense in E. Since π is a
homomorphism, every element in π(A)·E can be written as π(1) j π(aj )ξj . Hence, π(1)(E)
is dense in E. Now, use the fact that π(1) is idempotent and bounded. Any ξ ∈ E is the
limit of a sequence {π(1)ξi } in the image of π(1). So,
ξ = lim π(1)(ξi ) =⇒ π(1)(ξ) = π(1) lim π(1)(ξi ) = lim π(1)(ξi ) = ξ.
Therefore π(1) is the unit in L(E).
Definition 5.61. A closed, two sided ideal A of a C ∗ -algebra D is called essential if the set
A⊥ := {d ∈ D : da = 0 for all a ∈ A}
consists of only the zero element.
Lemma 5.62. Let π : A → L(E) be a nondegenerate representation. Then π(A) is an
essential ideal of Mπ (A).
24
Proof. Assume T ∈ Mπ (A) is such that T π(a) = 0 for all a ∈ A. Since T is adjointable,
it is continuous, so it suffices to show that T is equivalent to the zero operator on a dense
subset of E. As π is nondegenerate, π(A) · E is dense in E. The fact that T π(a) = 0 for all
a ∈ A implies that T is zero on π(A) · E, and thus, equivalent to the zero operator on all of
E. Therefore, π(A) is an essential ideal of Mπ (A).
Theorem 5.63. Let A be an essential ideal in a C ∗ -algebra D. If π : A → L(E) is any
faithful nondegenerate ∗-homomorphism, there is a unique faithful extension
π̃ : D → L(E)
of π with image contained in Mπ (A).
Proof. Let {ui } be an approximate identity in A, 1.19. Since A is an ideal of D, dui ∈ A
for every d ∈ D and for every i. Claim that for d ∈ D and ξ ∈ E
lim π(dui )ξ
i
(5.5)
exists.
First, show that {π(dui )} are equicontinuous. This follows from the contractive nature
of π, 1.20, and the Banach algebra property of the norm. Indeed, for any ξ ∈ E,
||π(dui )ξ||E ≤ ||π(dui )||L(E) ||ξ||E ≤ ||dui ||A ||ξ|| ≤ ||d||A ||ui ||A ||ξ|| ≤ ||d||||ξ||.
This ensures that the existence of (5.3) can be checked on a dense subset of E, in particular
π(A) · EPas π is nondegenerate.
Let nj=1 π(aj )ξj ∈ π(A) · E. Then
X
X
X
lim π(dui )
π(aj )ξj =
π(d lim ui aj )ξj =
π(daj )ξj .
i
i
So, (5.3) exists for all ξ ∈ E.
Define the extension π̃ : D → L(E) by
π̃(d) : ξ → lim π(dui )ξ.
i
It is straightforward to check, for any d ∈ D, that π̃(d) is linear, bounded by ||d||, and
adjointable, with adjoint π̃(d∗ ). It is also easy to check that π̃ is a ∗-homomorphsim.
For uniqueness, assume that σ is another ∗-homomorphic extension of π. Fix d ∈ D, as
σ(d) is also continuous consider its action on the dense subset π(A) · E.
X
X
σ(d)
π(aj )ξ =
σ(d)(π(aj )ξj )
X
=
σ(daj )ξj
as σ ≡ π on A.
X
=
π(daj )ξj
since A is an ideal of D.
This shows that σ(d) ≡ π̃(d) on π(A) · E, and so σ ≡ π̃ on D.
To show π̃ is injective, again fix 0 6= d ∈ D. Since A is an essential ideal in D, there is
an element a ∈ A such that da 6= 0. Since π is faithful, there is an element ξ ∈ E such that
π(da)ξ 6= 0. Hence, π̃(d)(π(a)ξ) 6= 0 which shows that π̃(d) 6≡ 0.
25
Corollary 5.64. Let A be a C ∗ -algebra, and π : A → L(E), σ : A → L(F ) be faithful nondegenerate representations. Then the multiplier algebras Mπ (A) and Mσ (A) are canonically
∗-isomorphic.
Proof. Note that π is a ∗-isomorphism of A and π(A), denote the inverse of this isomorphism
by π −1 . Lemma 5.62 shows that π(A) is an essential ideal of Mπ (A). So, the composition
σ ◦ π −1 : π(A) → L(F ) is a faithful, nondegenerate ∗-homomorphism; further, the image
of the composition is σ(A) ⊆ Mσ (A). Hence, by Theorem 5.63, the composition σ ◦ π −1 :
π(A) → Mσ (A) extends uniquely to a faithful representation
σ̃ : Mπ (A) → Mσ (A).
Similarly, the composition π ◦ σ −1 : σ(A) → Mπ (A) extends uniquely to a faithful representation
π̃ : Mσ (A) → Mπ (A).
It remains to show that σ̃ and π̃ are mutual inverses, and by Lemma 5.62, it suffices to
show that σ̃ ◦ π̃ is a faithful ∗-homomorphism that extends the identity on the essential ideal
σ(A) ⊆ Mσ (A) as the opposite composition will follow in similar fashion. The fact that σ̃ ◦ π̃
is a faithful ∗-homomorphism is clear since it is the composition of such maps.
Now, take σ(a) ∈ σ(A),
σ̃ ◦ π̃(σ(a)) = σ̃ ◦ π ◦ σ −1 (σ(a)) = σ ◦ π −1 (π(a)) = σ(a).
This suffices since the identity map is another faithful ∗-homomorphic extension from σ(A)
to Mσ (A), so the result follows from the uniqueness statement in Lemma 5.62.
This corollary proves what was claimed in the introduction since it has been shown that
the multiplier algebra of an étale groupoid C ∗ -algebra, M (Cλ∗ (G)), is associated to the left
multiplication representation, 5.57, which is faithful and nondegenerate, 5.59. And so, the
corollary states that M (Cλ∗ (G)) is ∗-isomorphic to the multiplier algebra associated to the
identity representation (another faithful, nondegenerate representation), which is exactly the
set stated in the introduction.
Morita Equivalent Étale Groupoids Give Rise to Morita Equivalent
Reduced C ∗ -Algebras
Proposition 6.65. Let X be a locally compact, Hausdorff space with open cover U = {Ui }i∈I
and subcover (of U) V. If XU and XV are the localization groupoids associated with the covers
U and V respectively, then Cλ∗ (XU ) is Morita equivalent to Cλ∗ (XV ).
Proof. Let p : XU → C be the characteristic function of the subset
{(i, x, i) : Ui ∈ V}
(0)
of XU .
Note that for any f ∈ Cc (XU ),
26
p ∗ f (n, x, m) =
X
k
x∈Uk
and
f ∗ p(n, x, m) =
X
k
x∈Uk
(
f (n, x, m)
p(n, x, k)f (k, x, m) =
0
if Un ∈ V
otherwise
(6.6)
(
f (n, x, m)
f (n, x, k)p(k, x, m) =
0
if Um ∈ V
otherwise
(6.7)
Define λ(p) : Cc (XU ) → L2 (XU ) by λ(p)(f ) = p ∗ f . Linearity of λ(p) is clear, so it will
(0)
be shown that λ(p) is adjointable, in fact self adjoint. Let (i, x, i) ∈ XU and f, g ∈ Cc (XU ).
(The notation r(γ) ∈ V means r(γ) ∈ Ui for some Ui ∈ V.)
hp ∗ f, giCc (XU ) (i, x, i) =
X
p ∗ f (γ)g(γ)
γ∈XU(i,x,i)
=
X
f (γ)g(γ)
by (6.4)
γ∈XU(i,x,i)
r(γ)∈V
and similarly,
X
hf, p ∗ giCc (XU ) (i, x, i) =
f (γ)p ∗ g(γ)
γ∈XU(i,x,i)
X
=
f (γ)g(γ)
γ∈XU(i,x,i)
r(γ)∈V
This proves that λ(p) is self adjoint. Another computation, again refer to (6.4), shows that
λ(p) is idempotent. To satisfy 4.43, in order to extend λ(p) to all of L2 (XU ), it is required to
show λ(p) is bounded. Once this is shown the lemma implies the extension, p̃, is a bounded
projection on L2 (XU ), and by the definition of the extension and the continuity of the inner
product, p̃ ∈ L(L2 (XU )).
Regarding boundedness of λ(p), consider f ∈ Cc (G).
||p ∗ f ||2Cc (XU ) = sup | hp ∗ f, p ∗ f iCc (XU ) (i, x, i)|
(i,x,i)
= sup | hf, p ∗ f iCc (XU ) (i, x, i)|
p is self-adjoint and idempotent
(i,x,i)
≤ sup | hf, f iCc (XU ) (i, x, i)|
by the work above with the inner product
(i,x,i)
= ||f ||2Cc (XU ) .
Now, to show that p̃ is in the multiplier algebra of Cr∗ (XU ),
M (Cλ∗ (XU )) = T ∈ L(L2 (XU )) : T a, aT ∈ Cλ∗ (XU ) for all a ∈ Cλ∗ (XU ) ,
27
let f ∈ Cc (XU ). Recall that p̃ on Cc (XU ) is equivalent to λ(p). Hence, it is sufficient to prove
that p ∗ f and f ∗ p are contained in Cc (XU ), a dense subset of Cλ∗ (XU ), since composition
in Cλ∗ (XU ) is continuous.
S
Assume that supp(f ) = n [in × Kn × jn ] where Kn is compact in X for all n and
{in }, {jn } are finite subsetsSof I.
By (6.1), supp(p ∗ f ) = nk [ink × Knk × jnk ] where {nk } ⊆ {n} such that ink ∈ V for all
k. It is clear that this is a compact set, so it remains to check that p ∗ f is continuous.
−1
S Let W ∈ C be open. Then f (W ), which is open by the continuity of f , has the form
most finitely many
i,j∈I [i × Wi,j × j] where Wi,j are open in X for all pairs i, j. Note that at S
−1
Wi,j are nonempty strict subsets of X. Now, (p ∗ f ) (W ) has the form i,j∈I [i × wi,j × j]
where wi,j is either equal to Wi,j , equal to all of X, or equal to ∅. Hence p ∗ f is also
continuous.
To show that p̃ is a full projection, again, work with the dense subset Cc (XU ) and the
restriction to this subset, p. It suffices to prove that Cc (XU )pCc (XU ) ⊇ Cc (XU ) as the other
inclusion is clear from the work done above.
Fix g ∈ Cc (XU ) and let
Ki,j = [supp(g) ∩ (i × X × j)] .
First, it will be shown that there is a function in Cc (XU )pCc (XU ) that mimics g on i×X×j
and is zero elsewhere. Then, since g is nonzero for finitely many pairs i, j, summing such
functions will show that g ∈ Cc (XU )pCc (XU ).
By 7.68, there exists vi,j ∈ Cc (XU ) such that


if (n, x, m) ∈ Ki,j
= 1
vi,j (n, x, m) ∈ [0, 1] if (n, x, m) ∈ K \ Ki,j


=0
otherwise,
where K ⊂ i × X × j is compact.
Another reduction on g will be made at this point. Claim that it suffices to assume
Ki,j ⊆ i × Uk × j for some Uk ∈ V. If this statement holds, and it was not the case that
Ki,j ⊆ i × Uk × j for only one Uk ∈ V, then consider a partition of unity on Ki,j as follows.
Given Ki,j , i×V ×j is an open cover. Further, there is some finite subcover i×{Uk }k∈J ×j
(J finite) and a partition of unity, {hk }k∈J , on Ki,j subordinate to i × {Uk } × j consisting of
compactly supported functions 7.70. Define gk ∈ Cc (XU ) by gk (i, x, j) = hk (i, x, j)·g(i, x, j).
Each gk satisfies the assumptions
in the claim, and so assuming the claim is true, gk ∈
P
Cc (XU )pCc (XU ). Thus g ≡ k gk ∈ Cc (XU )pCc (XU ).
From now on assume Ki,j ⊆ i × Uk × j for fixed Uk ∈ V. Now, claim that
fi,j := vi,k ∗ p ∗ vk,i ∗ g ∈ Cc (XU )pCc (XU )
is the desired function; that P
is, fi,j mimics g on i × X × j and is zero otherwise. And, if this
is indeed the case, then g = i,j fi,j ∈ Cc (XU )pCc (XU ) where the sum is taken over all pairs
such that g is nonzero on i × X × j.
28
Take arbitrary (n, x, m) ∈ XU .
vi,k ∗ p∗vk,i ∗ g(n, x, m)
X
=
vi,k ∗ p ∗ vk,i (n, x, l)g(l, x, m)
l
x∈Ul
= vi,k ∗ p ∗ vk,i (n, x, i)g(i, x, j)
X
= g(i, x, j)
vi,k (n, x, l)p ∗ vk,i (l, x, i)
if j = m (and 0 otherwise)
l
= g(i, x, j)p ∗ vk,i (k, x, i)
X
= g(i, x, j)
p(k, x, l)vk,i (l, x, i)
if n = i (and 0 otherwise)
l
= g(i, x, j).
Therefore p̃ is a full projection in M (Cλ∗ (XU )).
It remains to show that p̃Cλ∗ (XU )p̃ is ∗-isomophic to Cλ∗ (XV ). Recall that p̃Cλ∗ (XU )p̃ =
{p̃ap̃ : a ∈ Cλ∗ (XU )}. Define Φ : pCc (XU )p → Cc (XV ) by Φ(pf p) = f |{γ : r(γ),s(γ)∈V} . Note
that Φ is a canonical ∗-isomorphism of dense subsets of the respective C ∗ -algebras. To extend
this isomorphism it is left to show that Φ is isometric, in particular
||f ||L(L2 (XU )) = ||f ||L(L2 (XV ))
where the following notation is assumed for the rest of the proof f = f |{γ : r(γ),s(γ)∈V} .
First, assume ξ ∈ Cc (XV ) with ||ξ|| = 1. Since XV is an open subset of XU and ξ has
compact support, it can be extended to all of XU by giving it value 0 off of XV , denote the
˜ Now compute,
extension by ξ.
||f ∗ ξ||2L2 (XV )
2
X X
= sup
f (α)ξ(β)
(0)
x∈XV γ∈XUx αβ=γ
2
X X
˜
= sup
f (α)ξ(β)
(0)
x∈XV γ∈XUx αβ=γ
2
X X
˜ ≤ sup
f (α)ξ(β)
(0)
x∈X
U
(0)
(0)
since XV ⊆ XU
γ∈XUx αβ=γ
˜ 22
= ||f ∗ ξ||
L (XU ) .
Therefore,
||f ||L(L2 (XU )) ≥ ||f ||L(L2 (XV )) .
(0)
(0)
Second, assume ξ ∈ Cc (XU ) with ||ξ|| = 1. Assume that (i, x, i) ∈ XU \ XV . Since V
(0)
is also a cover of X there exists a j such that (j, x, j) ∈ XV . Let β = (i, x, j) and define
η : XU → C by η(γ) = ξ(γβ −1 ). As η is a translation of ξ, η ∈ Cc (XU ) and ||η|| = 1.
29
Now, consider η|XV , still denoted by η. Clearly ||η||L2 (XV ) = 1, and since XV is an open
subset of XU , η is continuous. Also, due to the discrete nature of the topology on XU and
the fact that V is a subcover of U, η still has compact support. And so, the following
computation will prove ||f ||L(L2 (XU )) ≤ ||f ||L(L2 (XV )) .
X
2
X
f (α)ξ(β) =
γ∈XU(i,x,i) αβ=γ
X
2
X
f (α)η(β)
γ∈XU(j,x,j) αβ=γ
2
X X
f (α)η(β) ≤ ||f ||2L(L2 (XV ))
≤ sup
(0)
y∈X
V
γ∈XUy αβ=γ
2
X X
f (α)ξ(β) ≤ ||f ||2L(L2 (XV ))
=⇒ sup
(0)
y∈X
U
γ∈XUy αβ=γ
P
Note that the first equality relies on the fact that αβ=γ f (α)ξ(β) = 0 unless r(γ) ∈ Vi for
some Vi ∈ V, and so, it is not a problem that η is taken to be the restriction on XV .
Thus, Φ is indeed an isometry, and so, it can be extended as a ∗-isomorphism between
∗
p̃Cλ (XU )p̃ and Cλ∗ (XV ). This completes the proof showing that Cλ∗ (XU ) is Morita equivalent
to Cλ∗ (XU ).
Lemma 6.66. Let G be a locally compact, Hausdorff, étale groupoid. Let U = {Ui }i∈I be an
open cover of G(0) with subcover V. If GU and GV are the localization groupoids associated
with the covers U and V respectively, then Cλ∗ (GU ) is Morita equivalent to Cλ∗ (GU ).
Proof. Let p : GU → C be the characteristic function of the subset
(i, x, i) : Ui ∈ V, x ∈ G(0)
(0)
of GU , analogous to the p defined in the proposition above.
Note that for any f ∈ Cc (GU ) and (n, γ, m) ∈ GU ,
p ∗ f (n, γ, m) =
X
αβ=γ
r(β),s(α)∈Uk
(
f (n, γ, m)
p(n, α, k)f (k, β, m) =
0
if Un ∈ V
otherwise
(6.8)
(
f (n, γ, m)
f (n, α, k)p(k, β, m) =
0
if Um ∈ V
otherwise
(6.9)
and
f ∗ p(n, γ, m) =
X
αβ=γ
r(β),s(α)∈Uk
30
The proof of this lemma follows the same arguments as proposition 6.65 with (6.6) and
(6.7) in place of (6.4) and (6.5). The only difference is present in the argument showing that
p̃ is a full projection, in particular, the following piece.
Cc (GU ) ⊆ Cc (GU )pCc (GU ): Let g ∈ Cc (GU ). Assume as in proposition 6.65 that there
are i, j such that supp(g) ⊂ i × G × j. For any n, let K(r) = {r(γ) : (i, γ, i) ∈ r(supp(g))}.
Similar to proposition 6.65, assume there exists k such that Uk ∈ V and (i × Uk × i) covers
i × K(r) × i. Otherwise, consider a partition of unity {hn } on i P
× K(r) × i subordinate to
{(i × Un × i)} where {Un } ⊂ V is a finite cover of K(r). So g ≡ hn ∗ g where each hn ∗ g
satisfies the assumption above.
Use the Tietze extension theorem 7.68 to obtain a function in Cc (GU ) such that


= 1 if (n, γ, m) ∈ i × K(r) × j
vi,j (n, γ, m) ∈ [0, 1] if (n, γ, m) ∈ K \ (i × K(r) × j)


= 0 otherwise,
where K is a compact set in i × G(0) × j.
Claim that g ≡ vi,k ∗ p ∗ vk,i ∗ g. To show this let (n, γ, m) ∈ GU . Then
vi,k ∗ p ∗ vk,i ∗ g(n, γ, m)
X
=
vi,k ∗ p ∗ vk,i (n, α, l)g(l, α−1 γ, m)
α∈Gr(γ)
l
s(α)∈Ul
=
X
vi,k ∗ p ∗ vk,i (n, α, i)g(i, α−1 γ, j)
if m = j
α∈Gr(γ)
s(α)∈Ui
=
X
α
=
X
g(i, α−1 γ, j)
X
vi,k (n, α0 , l)p ∗ vk,i (l, β 0 , i)
α0 β 0 =α
l
g(i, α−1 γ, j)p ∗ vk,i (k, α, i)
if n = i
α
=
X
g(i, α−1 γ, j)vk,i (k, α, i)
(α
g(i, γ, j) if r(γ) ∈ K(r)
=
0
otherwise.
Theorem 6.67. If G and G0 are locally compact, Hausdorff, étale groupoids which are Morita
equivalent, then Cλ∗ (G) and Cλ∗ (G0 ) are Morita equivalent as C ∗ -algebras.
Proof. Let GU ∼
= G0V be the localization groupoids of G and G0 respectively given by the
definition of Morita equivalence for groupoids. Let U0 and V0 be trivial covers on G and G0
M
M
respectively. By 6.66, Cλ∗ (GU ) ∼ Cλ∗ (GU ∪U0 ) ∼ Cλ∗ (GU0 ) = Cλ∗ (G), and similarly for Cλ∗ (G0V ).
31
Since GU ∼
= G0V , Cλ∗ (GU ) ∼
= Cλ∗ (G0V ), and so
M
Cλ∗ (G) ∼ Cλ∗ (G0 ).
Appendix
Theorem 7.68. (LCH version of Tietze’s extension theorem): [4, 4.34] Suppose that X is
an LCH space and K ⊂ X is compact. If f ∈ C(K), there exists F ∈ C(X) such that
F |K = f . Moreover, F may be taken to vanish outside a compact set.
Definition 7.69. If X is a topological space and E ⊂ X, a partition of unity on E is a
collection {hα }α∈A of functions in C(X, [0, 1]) such that
(1) each x ∈ X has a neighborhood on which finitely many hα ’s are nonzero, and
P
(2)
α∈A hα (x) = 1 for each x ∈ E.
A partition of unity is subordinate to an open cover U of E if for each α there exists
U ∈ U with supp(hα ) ⊂ U .
Proposition 7.70. [4, 4.41] Let X be an LCH Space, K a compact subset of X, and {Uj }n1
an open cover of K. There is a partition of unity on K subordinate to {Uj } consisting of
compactly supported functions.
Theorem 7.71. (Gelfand-Mazur): [11, 18.7] If A is a complex Banach algebra with unit in
which each non zero element is invertible, then A is isometrically isomorphic to C.
Theorem 7.72. (Stone-Weierstrass): [4, 4.51] Let X be a compact Hausdorff space. If
A is a complex closed subalgebra of C(X) that separates points, is closed under complex
conjugation, and contains 1 ∈ C(X), then A = C(X).
Theorem 7.73. (Open Map): [4, 5.10] Let X and Y be Banach spaces. If T : X → Y is a
bounded linear map, then T is open.
Theorem 7.74. (Closed Graph): [4, 5.12] If X and Y are Banach spaces and T : X → Y
is a closed linear map, then T is bounded.
32
Acknowledgment
Thank you to my advisor Rufus Willett for introducing me to this interesting topic and for
the guidance that lead to this work. Thank you also to John Clark Robertson, Efren Ruiz,
and Erik Guentner.
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