Partial dynamical systems and AF C*-algebras

34 Pages·0.31 MB·English

by  Justin R. Peters

PARTIAL DYNAMICAL SYSTEMS AND AF
C∗-ALGEBRAS
3
0
0
JUSTIN R. PETERS AND RYAN J. ZERR
2

n
a Abstract. We obtain a characterization in terms of dynamical
J systems of those r-discrete groupoids for which the groupoid C∗-

8 algebra is approximately finite-dimensional (AF). These ideas are
2 thenusedtocomputetheK-theoryforAFalgebrasbyutilizingthe

actions of these partial homeomorphisms, and these K-theoretic
]
A calculations are applied to some specific examples of AF algebras.
O Finally, we show that, for a certain class of dimension groups,
. a groupoid can be obtained directly from the dimension group’s
h structurewhoseassociatedC∗-algebrahasK0 groupisomorphicto
t
a the original dimension group.
m
[

1
v
7 1. Introduction
3
3 TheuseofdynamicalsystemstostudyC∗-algebrashasreceivedmuch
1 attention. In particular, [19] discusses the C∗-algebraic interpretations
0
of a variety of dynamical systems concepts and [5, 12, 13] have consid-
3
0 ered, at least in part, AF subalgebras of crossed product C∗-algebras
/
h and used the properties of the dynamical system associated with the
t crossed product to gain information about these algebras. In addition
a
m to this, the theory of nonselfadjoint subalgebras of AF C∗-algebras has
: direct connections with dynamical systems.
v
i By a partial homeomorphism φ on a metric space X we mean that
X
Dom(φ) and Ran(φ) are open subsets of X and φ is a homeomorphism
r
a from Dom(φ) to Ran(φ). Suppose X is a 0-dimensional space and
R = {(x,φn(x)) : x ∈ Dom(φn),n ∈ Z}
φ
is the associated groupoid (φ0 = id| ). Then R is AF if and only if
X φ
for any open set U containing Dom(φ), ∞ φn(U) = X [10, Corollary
n=0
4.7].
S
The results of §2 arose from the question of how to generalize this to
the case where the groupoid is the union of the graphs of a countable
1991 Mathematics Subject Classification. Primary 47L40, 46L80; Secondary
37A55.
1
2 JUSTIN R.PETERS AND RYANJ. ZERR
collection of partial homeomorphisms. Note that in the above result,
{σ = φn| : n ∈ Z,V a clopen set in Dom(φn)}
V,n V
is such a countable collection of partial homeomorphisms. Thus, the
setting in §2 generalizes that of [10].
In this paper we will consider partial homeomorphisms for which
the domain (and hence the range) are clopen subsets of the ambient
0-dimensional space. The main result of §2 is a dynamical characteri-
zation of AF inverse semigroups (Definition 2.4). We use this theorem
(Theorem 2.7) in Example 2.11 to show that the inverse semigroup of
the odometer map is not AF. Previously, the proof of this fact had
been via K-theory.
The importance of K-theory in the study of AF algebras makes it
natural to then consider, in §3, how our dynamical system characteri-
zation can be used to compute the K group of an AF algebra directly
0
from a consideration of its groupoid. Motivation for this is provided
in [6, 14, 17]. In [17] and [6], the K-theory of AF subalgebras of certain
crossedproductsisstudied, andthisK-theoryiscomputedthroughuse
of the short exact sequence of Pimsner and Voiculescu [11]. The com-
putations done here are more closely related to those of [15], but valid
for the more general partial dynamical systems considered as part of
the characterization in §2.
The K-theoretic calculations of §3 are applied in §4 to some specific
examples of AF algebras, including the CAR and GICAR algebras
(Examples 4.2 and 4.5). It is here we find that, at least in some cases,
the K group of an AF algebra can be realized as a group of continuous
0
functions acting on a 0-dimensional compact Hausdorff space.
For certain types of simple dimension groups, we discuss in §5 a
“Pontryagin-like” duality for groupoids which uses the structure of the
resulting character group to directly construct an inverse semigroup
of partial homeomorphisms and its associated groupoid. The results
of §3 can then be used to prove that this groupoid is not only AF,
but that the K group of the groupoid C∗-algebra is isomorphic to the
0
original dimension group. This method brings full-circle our ideas in
§3 by reversing the process described there. That is, whereas in §3
we compute the K group directly from the groupoid, for the types of
0
dimensiongroupsconsideredin§5,weareabletocomputethegroupoid
directly from the K group.
0
2. AF Groupoids and Partial Dynamical Systems
In this section we will develop a characterization of those groupoids
that come from AF C∗-algebras. The conditions which describe this
PARTIAL DYNAMICAL SYSTEMS AND AF C∗-ALGEBRAS 3
characterization are dynamical in nature, and therefore give a dynam-
ical systems perspective on the nature of these AF groupoids.
Given an AF algebra A = A , where {A } is an increasing
n≥1 n n n≥1
sequence of finite-dimensional subalgebras of A, consider D = D
S n≥1 n
where D is the subalgebra of A spanned by the diagonal matrix units
n n
in A . As described in [16], we can identify the matrix unitsSof each
n
subalgebra A with partial homeomorphisms on the Gelfand space X
n
of D. The set X is a 0-dimensional compact Hausdorff space, and the
equivalence relation RA on X that results from taking the union of the
graphs of these partial homeomorphisms is an r-discrete groupoid and
will be called the groupoid of the AF algebra A.
In the theory of groupoid C∗-algebras, [8, 18], one forms the C∗-
algebra of a groupoid R, C∗(R), by considering the continuous func-
tions of compact support on R. In the case of the groupoid RA, the
C∗-algebra of the groupoid is A itself. So, through a consideration of
thisgroupoid, we can establish necessary conditions onthosegroupoids
that give rise to AF algebras. The main result of this section (Theo-
rem2.7)willconsidertheconverse. Thatis, wewillestablishconditions
that ensure a given groupoid gives rise to a C∗-algebra which is AF.
This leads us to make the following definition.
Definition 2.1. A groupoid R will be called an AF groupoid provided
C∗(R) is AF.
To begin, we let X be a 0-dimensional compact Hausdorff space.
That is, X is compact Hausdorff and has a basis consisting of clopen
(n)
sets. Suppose that {σ : n ≥ 0,1 ≤ r ≤ m ,1 ≤ s ≤ κ(r,n)} is a
r,s n
countable collection of partial homeomorphisms on X where {m }∞
n n=0
is a sequence of positive integers (with m = 1), and, for each n ≥ 1
0
and 1 ≤ r ≤ m , κ(r,n) ∈ Z+. Further suppose that for any partial
n
(n) (n)
homeomorphism in this collection, say σ , Dom(σ ), the domain of
r,s r,s
(n)
σ , is the clopen set B(r,n) ⊂ X. That is to say, for n and r fixed,
r,s
(n) (n)
σ ,...,σ share a common domain of B(r,n).
r,1 r,κ(r,n)
We now form the clopen subsets U , for all n ≥ 1, of X by
n
mn
U = B(r,n),
n
r=1
[
and suppose that for each n ≥ 1, the collection {B(r,n)}mn partitions
r=1
U . In particular, for each fixed n ≥ 1, the sets {B(r,n)}mn are
n r=1
pairwise disjoint. For notational convenience, let U = B(1,0) = X.
0
4 JUSTIN R.PETERS AND RYANJ. ZERR
With the notation thus established, we assume that for each fixed
(n)
n ≥ 1 the collection {σ : 1 ≤ r ≤ m ,1 ≤ s ≤ κ(r,n)} satisfies the
r,s n
following conditions.
Conditions 2.2.
(n)
(i) σ (x) = x, for all 1 ≤ r ≤ m and x ∈ B(r,n);
r,1 n
(ii) for each 1 ≤ r ≤ m and 1 ≤ s ≤ κ(r,n), there exists an i,
n
(n)
1 ≤ i ≤ m , such that σ (B(r,n)) ⊂ B(i,n−1); and
n−1 r,s
(iii) mn κ(r,n)σ(n)(B(r,n)) is a partition of U .
r=1 s=1 r,s n−1
Wewillconsidertheinversesemigroupgeneratedbythesepartialhome-
W W
omorphisms (Definition 2.3).
Remark 1. From conditions (i) and (ii) it follows that the sequence
{U }∞ is nested decreasing.
n n=1
Now, given a collection T of partial homeomorphisms on a topologi-
calspace X, let A(T) bethe set of partialhomeomorphisms onX given
by
A(T) = {ρ : ∃σ ∈ T and clopen subset B ⊂ Dom(σ) with ρ = σ| }.
B
Note that T ⊂ A(T). We then make the following definition.
Definition 2.3. Given a collection T of partial homeomorphisms on a
topological space X, the inverse semigroup generated by T will be the
smallest set of partial homeomorphisms on X which contains A(T) and
is closed for inverses and compositions (wherever those compositions
make sense).
(n)
Forthecollection{σ },wewillnowgiveadescriptionofthisinverse
r,s
semigroup. To begin, let S be the inverse semigroup generated by
(n) (1) (1)
{σ } and define τ = σ for all 1 ≤ r ≤ m and 1 ≤ s ≤ k(r,1) =
r,s r,s r,s 1
(1) (2)
κ(r,1). By then taking the compositions τ ◦ σ of the partial
r1,s1 r2,s2
(2) (1)
homeomorphisms {σ } with the partial homeomorphisms {τ }, we
r,s r,s
obtain partial homeomorphisms {τ(2) }, for 1 ≤ r′ ≤ m and 1 ≤
r′,s′ 2
s′ ≤ k(r′,s) for some k(r′,2) ∈ Z+, which are in S and for which
Dom(τ(2) ) = B(r′,2) for all r′ and s′. We proceed inductively to
r′,s′
(n)
obtain,foralln ≥ 1,thepartialhomeomorphisms {τ }where1 ≤ r ≤
r,s
m and 1 ≤ s ≤ k(r,n) for some k(r,n) ∈ Z+, such that Dom(τ(n)) =
n r,s
(n)
B(r,n), for all r and s, and where each τ is an element of S.
r,s
Remark 2. The collection {τ(n) : n ≥ 1,1 ≤ r ≤ m ,1 ≤ s ≤ k(r,n)}
r,s n
of partial homeomorphisms so constructed is also a generating set for
(n)
S since it contains σ for every n, r, and s.
r,s
PARTIAL DYNAMICAL SYSTEMS AND AF C∗-ALGEBRAS 5
Remark 3. For each n ≥ 1,
mn k(r,n)
τ(n)(B(r,n))
r,s
r=1 s=1
_ _
forms a partition of X, and the sequence of partitions thus obtained is
a sequence of successively finer partitions of X.
Now, let R be an r-discrete groupoid which is an equivalence rela-
tion on X. We will say that an inverse semigroup of partial home-
omorphisms Z generates R if the union of the graphs of the partial
homeomorphisms in Z equals R. This leads us to make the following
definition.
Definition 2.4. We will call an inverse semigroup Z of partial homeo-
morphisms an AF inverse semigroup provided the groupoid generated
by Z is AF.
When a groupoid is generated by an inverse semigroup Z, we will
assume the groupoid has a topology with basis consisting of the graphs
of elements of Z restricted to clopen subsets in X. For the inverse
semigroup S constructed above and the groupoid R generated by it,
make note of the fact that the collection
∞ mn k(r,n)
{Γ(ρ) : ∃B ⊂ Ran(τ(n)),clopen, with ρ = τ(n) ◦τ(n)−1| }
r,s′ r,s r,s′ B
n=1r=1s,s′=1
_ _ _
forms a basis for this topology on R, where Γ(ρ) represents the graph
of ρ as a subset of X × X. We will show that any such groupoid is
AF. Before establishing this fact, however, we pause briefly to give a
heuristic account of why the converse is true.
(n) (n)−1
The partial homeomorphisms τ ◦ τ should be thought of as
r,s r,s′
playing a role corresponding to the matrix unit partial homeomor-
(n) (n)−1
phisms in the AF groupoid RA. In particular, τr,s ◦τr,s′ is the partial
homeomorphism corresponding to the matrix unit es′,s ∈ Mk(r,n) where
the AF algebra A = A is such that
n≥1 n
ASn = Mk(1,n) ⊕···⊕Mk(mn,n).
Also, the sets B(r,n) act in the same way as the supports of the upper
left diagonal matrix units in the subalgebras A when considering the
n
groupoid of the AF algebra A. With these motivating ideas in mind,
it is fairly straightforward to show that for each AF algebra A, there
exists a collection of partial homeomorphisms satisfying Conditions 2.2
6 JUSTIN R.PETERS AND RYANJ. ZERR
which generate RA in the sense just described. Therefore, Conditions
2.2 describe necessary conditions on AF groupoids.
In order to show that these conditions are sufficient we begin by
defining, for any n ≥ 1, the set
mn k(r,n)
R = Γ τ(n) ◦τ(n)−1 ⊂ R,
n r,s r,s′
r[=1s,[s′=1 (cid:16) (cid:17)
and give R the topology inherited from R.
n
Lemma 2.5. The sequence {R }∞ is a nested increasing sequence of
n n=1
groupoids such that R = ∞ R .
n=1 n
Proof. First, for any n ≥S1 let (x,y) ∈ Rn. That is to say, there exists
r,s,s′ such that (x,y) ∈ Γ τ(n) ◦τ(n)−1 . So, y = τ(n) ◦τ(n)−1(x). It
r,s r,s′ r,s r,s′
follows that (cid:16) (cid:17)
τ(n)−1(y) = τ(n)−1(x) ∈ B(r,n).
r,s r,s′
(n+1)
By construction, there exists B(r,n+1) and τ such that
r,s
τ(n)−1(y) = τ(n)−1(x) ∈ τ(n+1)(B(r,n+1)).
r,s r,s′ r,s
So, for some z ∈ B(r,n+1),
τ(n) ◦τ(n+1)(z) = y
r,s r,s
and
(n) (n+1)
τ ◦τ (z) = x.
r,s′ r,s
Again, by construction, there exist s and s such that
1 2
τ(n+1) = τ(n) ◦τ(n+1)
r,s1 r,s r,s
and
(n+1) (n) (n+1)
τ = τ ◦τ .
r,s2 r,s′ r,s
(n+1) (n+1)−1
Hence, we see that y = τ (z) and z = τ (x), which leaves
r,s1 r,s2
(n+1) (n+1)−1
y = τ ◦τ (x). Therefore,
r,s1 r,s2
mn+1k(r,n+1)
(x,y) ∈ Γ τ(n+1) ◦τ(n+1)−1 = R .
r,s r,s′ n+1
r[=1 s,[s′=1 (cid:16) (cid:17)
So, {R }∞ is nested increasing. Since R = ∞ R is immediate, it
n n=1 n=1 n
remains to show that each R is a groupoid.
n
S
To show R is a groupoid, the only nontrivial part is to show that
n
if (x,y),(y,z) ∈ R then (x,z) ∈ R , which we leave to the reader.
n n
(cid:3)
PARTIAL DYNAMICAL SYSTEMS AND AF C∗-ALGEBRAS 7
Remark 4. By using arguments similar to those in Lemma 2.5, one can
show that if B = Γ τ(n1) ◦τ(n1)−1 and B = Γ τ(n2) ◦τ(n2)−1 are
1 r1,s1 r1,s′1 2 r2,s2 r2,s′2
two basis elements o(cid:16)f R then eithe(cid:17)r B1 and B2 a(cid:16)re disjoint or o(cid:17)ne is
a subset of the other. Hence, the topology of R has a basis consisting
of clopen sets. It also follows easily that each basis element is in fact
compact. Thus, since each R is a finite union of compact opensubsets
n
of R, it follows that R is compact for all n ≥ 1.
n
For n ≥ 1 fixed, consider C (R ) = C(R ), the continuous functions
c n n
(of compact support) on R . We endow this set with the groupoid
n
convolution multiplication [8] in order to obtain a ∗-algebra structure.
Let 1 ≤ r ≤ m , g ∈ C(B(r,n)), and 1 ≤ i,j ≤ k(r,n) be given, and
n
(r)
define f ⊗g ∈ C(R ) by
i,j n
(n)−1 (n) (n)−1
g ◦τ (x), if (x,y) ∈ Γ τ ◦τ
f(r) ⊗g(x,y) = r,i r,j r,i
i,j
( 0, otherwise. (cid:16) (cid:17)
Inthefollowinglemma,whichissomewhatsimilartoLemma2.2of[15],
we show that C(R ) is spanned by functions of this form.
n
Lemma 2.6. The algebra C(R ) is isomorphic to
n
mn
M (C(B(r,n))).
k(r,n)
r=1
M
Proof. First, we take g ∈ C(R ). Let Γ(r) = Γ τ(n) ◦τ(n)−1 . If we
n i,j r,j r,i
let (cid:16) (cid:17)
(r)
g(x,y), if (x,y) ∈ Γ
g| (x,y) = i,j
Γ(r) 0, otherwise,
i,j
(cid:26)
then it is clear that
mn k(r,n)
g = g| .
Γ(r)
i,j
r=1 i,j=1
X X
(r)
It is also straightforward to see that g| = f ⊗g˜, with
Γ(r) i,j
i,j
g˜ = g ◦r−1 ◦τ(n) : B(r,n) → C,
r,i
(r) (n)
where here r : Γ → τ (B(r,n)) is projection onto the first coordi-
i,j r,i
(r)
nate and restricted to the subset Γ of R. The map r is essentially
i,j
(n)
the range map of Renault [18], and since both r and τ are homeo-
r,i
morphisms in this case, we see that g˜ ∈ C(B(r,n)). Hence, it follows
8 JUSTIN R.PETERS AND RYANJ. ZERR
that C(R ) is equal to the span of the collection
n
(r)
f ⊗g : 1 ≤ r ≤ m ,1 ≤ i,j ≤ k(r,n),g ∈ C(B(r,n)) .
i,j n
n o
Now, for 1 ≤ r < r ≤ m , g ∈ C(B(r ,n)), g ∈ C(B(r ,n)),
1 2 n 1 1 2 2
1 ≤ i ,j ≤ k(r ,n), and 1 ≤ i ,j ≤ k(r ,n) we have
1 1 1 2 2 2
f(r1) ⊗g ∗ f(r2) ⊗g = 0
i1,j1 1 i2,j2 2
(cid:16) (cid:17) (cid:16) (cid:17)
(n) (n)
since τ (B(r ,n)) ∩ τ (B(r ,n)) = ∅, where ∗ is the convolution
r1,j1 1 r2,i2 2
product on C(R ).
n
Next, let 1 ≤ r ≤ m be fixed and take 1 ≤ i,j,l,m ≤ k(r,n). If j 6=
n
(n) (n)
l then τ (B(r,n))∩τ (B(r,n)) = ∅, and so, for g ,g ∈ C(B(r,n)),
r,j r,l 1 2
we again have
(r) (r)
f ⊗g ∗ f ⊗g = 0.
i,j 1 l,m 2
(cid:16) (cid:17) (r)(cid:16) (cid:17) (r)
However, if j = l take (x,z) ∈ Γ and (z,y) ∈ Γ . Then,
i,j l,m
(r) (r) (r) (r)
f ⊗g ∗ f ⊗g (x,y) = f ⊗g (x,z)·f ⊗g (z,y)
i,j 1 l,m 2 i,j 1 l,m 2
(cid:16) (cid:17) (cid:16) (cid:17) (n)−1 (n)−1
= g ◦τ (x)·g ◦τ (z).
1 r,i 2 r,l
(r) (n)−1
Also, we see that f ⊗(g ·g )(x,y) = (g ·g )◦τ (x) since
i,m 1 2 1 2 r,i
y = τ(n) ◦τ(n)−1(z) = τ(n) ◦τ(n)−1 ◦τ(n) ◦τ(n)−1(x)
r,m r,l r,m r,l r,j r,i
= τ(n) ◦τ(n)−1(x).
r,m r,i
So,
(r) (n)−1 (n)−1
f ⊗(g ·g )(x,y) = g ◦τ (x)·g ◦τ (x).
i,m 1 2 1 r,i 2 r,i
(n) (n)−1 (n)−1 (n)−1
Since z = τ ◦τ (x) we see that τ (x) = τ (z) due to the
r,j r,i r,i r,l
fact that j = l. Hence, for all (x,y) ∈ R ,
n
(r) (r) (r)
f ⊗(g ·g )(x,y) = f ⊗g ∗ f ⊗g (x,y).
i,m 1 2 i,j 1 l,m 2
(cid:16) (cid:17) (cid:16) (cid:17)
In particular,
(r) (r) (r)
f ⊗χ ∗ f ⊗χ = δ f ⊗χ ,
i,j B(r,n) l,m B(r,n) j,l i,m B(r,n)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
where δ = 1 if j = l and 0 if j 6= l.
j,l
Next, for r,g,i,j as above, we see that
(n)−1 (r)
(r) g ◦τ (x), if (x,y) ∈ Γ
f ⊗g(x,y) = r,j j,i
j,i
( 0, otherwise
PARTIAL DYNAMICAL SYSTEMS AND AF C∗-ALGEBRAS 9
and
(n)−1 (r)
(r) g ◦τ (y), if (y,x) ∈ Γ
f ⊗g(y,x) = r,i i,j
i,j
( 0, otherwise.
(r) (r)
Thus, since (x,y) ∈ Γ implies not only that (y,x) ∈ Γ but also
j,i i,j
(n)−1 (n)−1
that τ (x) = τ (y), we see that in fact,
r,j r,i
(r) (r)
f ⊗g(x,y) = f ⊗g(y,x),
j,i i,j
for all (x,y) ∈ R . Therefore,
n
∗
(r) (r) (r)
f ⊗g (x,y) = f ⊗g(y,x) = f ⊗g(y,x)
i,j i,j i,j
(cid:16) (cid:17) (r)
= f ⊗g(x,y).
j,i
∗
(r) (r)
So, f ⊗χ = f ⊗χ .
i,j B(r,n) j,i B(r,n)
F(cid:16)inally, becaus(cid:17)e the images of the partial homeomorphisms {τ(n)}
r,i
partition X, we have that
mn k(r,n)
(r)
f ⊗χ = χ
i,i B(r,n) ∆(XA)
r=1 i=1
X X
is the identity on C(R ).
n
This shows that for each 1 ≤ r ≤ m , the collection of functions
n
(r) k(r,n)
{f ⊗χ } ⊂ C(R ) forms a system of matrix units for a full
i,j B(r,n) i,j=1 n
k(r,n)×k(r,n) matrix algebra. This completes the proof.
(cid:3)
Now, considering the sequence of algebras {C(R )}∞ , define, for
n n=1
each n ≥ 1, the map φ : C(R ) → C(R ) by
n,n+1 n n+1
f(x,y), if (x,y) ∈ R
φ (f)(x,y) = n
n,n+1 0, otherwise.
(cid:26)
Since R is a subset of R , it follows that φ (f) is in fact an
n n+1 n,n+1
elementofC(R ). Itisalsofairlyclearthatφ isaunitalinjective
n+1 n,n+1
∗-homomorphism. We can therefore view C(R ) as a subalgebra of
n
C(R ), and in fact, via the map Φ : C(R ) → C (R), where
n+1 n n c
f(x,y), if (x,y) ∈ R
Φ (f)(x,y) = n
n 0, otherwise,
(cid:26)
we embed C(R ) as a ∗-subalgebra of C (R). As such, we have
n c
∞
C(R ) ⊂ C (R).
n c
n=1
[
10 JUSTIN R.PETERS AND RYANJ. ZERR
If, however, we let f ∈ C (R), then by the compactness of supp(f),
c
the support of f, there exists an N ≥ 1 such that
mN k(r,N)
supp(f) ⊂ Γ τ(n) ◦τ(n)−1 .
r,s r,s′
r[=1s,[s′=1 (cid:16) (cid:17)
So, f| ∈ C(R ) and we see that Φ (f| ) = f. Therefore,
RN N N RN
∞
C(R ) = C (R).
n c
n=1
[
Now, if W is a 0-dimensional compact Hausdorff space, there exists a
sequence {E }∞ of successively finer partitions of W which generate
n n=1
thetopology. Thus, C(W) isthe closed unionof anincreasing sequence
of subalgebras, {C(E )}∞ , where C(E ) consists of those functions
n n=1 n
that are constant on elements of the partition E . Hence, C(W) is AF.
n
It follows that for k ≥ 1, M (C(W)) is AF. Therefore, for all n ≥ 1,
k
C(R ) is an AF algebra, and so, by [2],
n
C∗(R) = C (R)
c
is an AF algebra. We summarize in the following theorem.
Theorem 2.7. Suppose there exists a nested decreasing sequence of
clopen subsets {U }∞ of X (0-dimensional), clopen sets {B(r,n)}mn
n n=0 r=1
for each n ≥ 0 which partition U (with U = B(1,0) = X), and a
n 0
countable collection
{σ(n) : n ≥ 0,1 ≤ r ≤ m ,1 ≤ s ≤ κ(r,n)}
r,s n
of partial homeomorphisms on X such that Dom(σ(n)) = B(r,n) for
r,s
all n, r, and s, and which satisfy Conditions 2.2. Then, the r-discrete
groupoid corresponding to the inverse semigroup generated by this col-
lection of partial homeomorphisms is AF.
This theorem, in conjunction with our earlier remarks leads to the
following corollaries, the first of which is a generalization of [10, Corol-
lary 4.7].
Corollary 2.8. The r-discrete groupoid R is AF if and only if it is
an equivalence relation on the 0-dimensional space X and there exists
a countable collection of partial homeomorphisms (as in Theorem 2.7)
satisfying Conditions 2.2 which generate R.
Corollary 2.9. An inverse semigroup of partial homeomorphisms on
X (0-dimensional) is AF if and only if it is generated by a subset
satisfying Conditions 2.2.