Morphisms which are continuous on a
neighborhood of the base of a groupoid
Madalina Roxana Buneci
University Constantin Br^ancusi of T^argu-Jiu
to appear in Studia Sci. Math. Hungar. 42(3) (2005), 265-276.
Abstract
Kirill Mackenzie raised in 3] (p. 31) the following question: given a
morphism F : ! , where and are topological groupoids and F
is continuous on a neighborhood of the base in , is it true that is
continuous everywhere?
This paper gives a negative answer to that question. Moreover, we
shall prove that for a locally compact groupoid with non-singleton orbits and having open target projection, if we assume that the continuity
of every morphism F on the a neighborhood of the base in implies the
continuity of F everywhere, then the groupoid must be locally transitive.
AMS 2000 Subject Classication: 22A22, 28C99
Key Words: topological groupoid, morphism
0
0
1 Introduction
In order to establish notation, we give some denitions that can be found (in
equivalent form) in several places: 3],5],4]. To dene the term groupoid we
begin with two sets, and B , called respectively the groupoid and base, together
with two maps and , called respectively the source and target projections,
a map " : u ! u~ from B to called the object inclusion map, and a partial
multiplication (x y) ! xy in dened on the set
= f(x y) 2 : (x) = (y)g (the set of composable pairs ),
all subject to the following conditions:
1. (xy) = (y) and (xy) = (x) for all (x y) 2 .
2. x (yz ) = (xy) z for all x y z 2 such that (x) = (y) and (y) = (z ).
This work is supported by the MECT-CNCSIS grant (At 127/2004).
1
3. (~u) = (~u) = u for all u 2 B .
4. x g
(x) = x and g
(x)x = x for all x 2 .
5. Each x 2 has a (two-sided) inverse x;1 such that
;
x;1 = (x) and x;1 x = g
(x), xx;1 = g
(x ) .
;
x;1 = (x),
The elements of the base B will usually be denoted by letters as u v w
while arbitrary elements of the groupoid will be denoted by x y z .
The bres of the target and the source projections will be denoted =
;1 (fug) and = ;1 (fvg), respectively. More generally, given the subsets
U , V B , we dene = ;1 (U ), = ;1 (V ) and = ;1 (U )\ ;1 (V ).
The relation u v i 6= is an equivalence relation on B . Its equivalence
classes are called orbits and the orbit of a unit u is denoted u]. A groupoid is
called transitive i it has a single orbit. The set , obviously a group under
the restriction of the partial multiplication in , is called isotropy group at u.
A subgroupoid of is a pair of subsets 0 , B 0 B , such that (0 ) 0
B , (0 ) B 0 , u~ 2 0 for all u 2 B 0 and 0 is closed under the partial
multiplication and inversion in . The inner subgroupoid (the isotropy group
bundle ) of is the subgroupoid G = . The base subgroupoid of is
the subgroupoid B~ = fu~ : u 2 B g. An element u~ of B~ is called the unity
corresponding to u.
A function F : ! 0 , where and 0 are groupoids, is called morphism
if for all composable pairs (x y) 2 , (F (x) F (y)) ;is a composable
pair in
0 and F (xy) = F (x) F (y). It is useful to note that F x;1 = F (x);1 for all
x 2 and that F (~u) belongs to the base subgroupoid of 0 for all u~ in the base
subgroupoid of .
Groupoids are generalizations of groups, but there are many other structures
which t naturally into the study of groupoids. We give some basic examples
that will be needed in the subsequent considerations.
Examples of groupoids:
u
v
U
U
V
V
u
v
u
u
u
u
u
1. Groups: A group G is a groupoid with base B = feg (the unit element)
and G G = G G.
2. Spaces. A space X may be regarded as a groupoid on itself with =
= id and every element is a unity.
3. Equivalence relations. Let E X X be an equivalence relation on
the set X . We give E the structure of a groupoid on X in the following
way: is the projection onto the second factor of X X and is the
projection onto the rst factor the object inclusion map is x ! (x x)
and the partial multiplication is (x y) (y0 z ) = (x z ) dened if and only
if y = y0 . The inverse of (x y) is (y x). Two extreme cases deserve to be
single out. If E = X X , then E is called the trivial groupoid on X , while
if E = diag (X ), then E is called the co-trivial groupoid on X (and may
be identied with the groupoid in example 2).
X
2
4. Transformation groups. Let G be a group acting on a set X such that
for x 2 X and g 2 G, gx denotes the transform of x by g. Let e be the unit
element of G. Then G X becomes a groupoid on X in the following way:
is the projection onto the second factor of G X and is the action
G X ! X itself the object inclusion map is x ! (e x) and the partial
multiplication is (g; x) (g0 y) = (gg0 y) dened if and only if x = g0 y. The
inverse of (x g) is g;1 gx .
A topological groupoid consists of a groupoid on B together with topologies on and B such that the maps which dene the groupoid structure are
continuous: the projections , : ! B , the object inclusion map " : B ! ,
" (u) = u~, the inversion map x ! x;1 from to , and the partial multiplication from to , where has the subspace topology from , are
continuous.
As a consequence of the denition of a topological groupoid, the inversion
x ! x;1 is a homeomorphism, the object inclusion map is a homeomorphism
onto the base subgroupoid B~ , the projections are identication maps. If is
Hausdor, then B~ is closed in . If B is Hausdor, then is closed in .
Kirill Mackenzie raised in 3] (p. 31) the following question: given a morphism F : ! 0 , where and 0 are topological groupoids and F is continuous
on a neighborhood of the base in , is it true that F is continuous everywhere?
Obviously, the answer is armative if the groupoid is a group or if is a
co-trivial groupoid. Also if is a "principal "topological groupoid, Mackenzie
has proved that the answer is armative (Proposition 1.21/p. 30 3]) A "principal" groupoid (Denition 1.18/p. 28 3]) is a topological transitive groupoid
on B which satises the following conditions:
1. For each v 2 B , the map : x 2 , is open.
2. For each v 2 B , the map : for all (x y) 2 , is open.
v
v
v
v
v
v
! B, dened by v
(x) = (x) for all
! , dened by v
v
(x y) = xy;1
v
We shall prove that the result remains true for a locally transitive groupoid
. By a locally transitive groupoid we shall mean a topological groupoid on B for which the map : ! B is open, for each v 2 B . Also we
shall prove that in general the answer to the question raised by Mackenzie
is negative. Moreover, we shall show that for a locally compact groupoid with non-singleton orbits and having open target projection, if we assume that
the continuity of every morphism F at all u~ 2 B~ implies the continuity of F
everywhere, then the groupoid must be locally transitive. Furthermore, if
we suppose that B~ is an open subset of and the answer to the question of
Mackenzie is armative, then B must be a discrete space. In particular, if is the transformation group groupoid associated with the action of a discrete
group on a locally compact second countable Hausdor space B , then has the
property that every morphism which is continuous on a neighborhood of the
base must be globally continuous if and only if B is a discrete space.
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2 An example of a groupoid morphism which is
continuous on a neighborhood of the base but
which is not continuous everywhere
We shall construct a morphism (in the algebraic sense) F : ! 0 of topological
groupoids with the property that the continuity of F on a neighborhood of the
base in does not imply the continuity of F everywhere on . Thus we give a
negative answer to a question raised by K. Mackenzie in 3] (p. 31).
Let R be the set of real numbers and Z the set of integer numbers. Let be the groupoid on R associated with the following equivalence relation
(x y) 2 R2 : x ; y 2 Z
R R.
If is endowed with the subspace topology from R R, is a locally
compact Hausdor space. The base subgroupoid of B~ = f(x x) : x 2 Rg
is an open subset in . Let 0 be the group of real numbers under addition.
Let : R ! R be dened by (x) = 0 for all x 12 and (x) = 1 otherwise.
Let F : ! R be dened by
F (x y) = (y) ; (x) .
For all ((x y) (y0 z )) 2 we have
F ((x y) (y z )) = F (x z ) = (z ) ; (x)
= (z ) ; (y) + (y) ; (x)
= F (y z ) + F (x y)
Therefore F is a morphism.
Obviously, F is continuous at every (x x) in , but
;
it is not continuous at 21 23 .
Remark 1 The groupoid is isomorphic and homeomorphic to the transformation group groupoid obtained by letting Z act on R. In this setting the morphism
(cocycle) F is a coboundary determined by the discontinuous function .
3 Sucient conditions for the continuity of a
groupoid morphism
Theorem 2 Let be a topological groupoid on B: Let us assume that for each
v 2 B , the map : ! B, dened by (x) = (x) for all x 2 , is open.
Let 0 be a topological groupoid and F : ! 0 be a morphism (in the algebraic
sense) which is continuous at every u~ 2 B~ . Then F is continuous everywhere
on .
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Proof. Let (x ) be a net converging to x in . Let v = (x) and let
us note that ( (x )) converges to (x) = (x) in B . Since : ! B
is open, we may pass to a subnet and assume that there exists a net
; (y )
in converging to x such that (y ) = (x ) for all i. The net y x;1
converges to xx;1 = g
(x). From the continuity of F at g
(x) it follows that
;
;
1
lim F y x = F g
(x) , or equivalently lim F (y ) = F (x). On the other
;
hand, the net y;1 x converges to x;1 x = g
(x). Now from the continuity of
; ;1 g
F at (x) it follows that lim F y x = F ag
(x) . Thus
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;;
;
;
lim F (x ) = lim F y y;1 x = lim F y y;1 x
;
(x)
= lim F (y ) F y;1 x = F (x) F ag
= F ( x) .
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4 Necessary conditions for the continuity of a
groupoid morphism
Let on B be a locally compact (Hausdor) groupoid. This means that is
a topological groupoid whose topology is locally compact (Hausdor). A subset A of is called -relatively compact if A \ ;1 (K ) is relatively compact
for any compact subset K of B . Jean Renault proved that if is a locally
compact groupoid whose base B is paracompact, then the base subgroupoid B~
admits a fundamental system of -relatively compact neighborhoods (Proposition II.1.9/p. 56 5]).
Throughout this section we shall x a system of positive Radon measures
indexed by B B
f (u v) 2 B Bg
u
v
satisfying the following conditions:
1. supp ( ) = for all u v.
2. = 0 if = .
3. sup (K ) < 1 for all compact K .
u
v
u
v
u v
R
u
v
u
v
u
v
4. f (y) d
( )
x
v
R
(y) = f (xy) d
( )
x
v
(y) for all x 2 and v
2 (x)].
A construction of such system of measures can be found in 6]. In Section
1 of 6] Jean Renault constructs a Borel Haar system f u 2 B g for the inner
subgroupoid G. One way to do this is to choose a function F0 continuous
u
u
5
with -relatively compact support which is nonnegative and equal to 1 at each
~ Then for each u 2 B choose a left Haar measure on so the integral
u~ 2 B:
of F0 with respect to is 1:
R
Renault denes = x if x 2 (where x (f ) = f (xy) d (y), as
usual). If z is another element in , then x;1 z 2 , and since is a left
Haar measure on , it follows that is independent of the choice of x. If K
is a compact subset of , then sup (K ) < 1.
u
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u v
u
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Lemma 3 Let be a locally compact groupoid on B, whose base subgroupoid
B~ has a fundamental system of -relatively compact neighborhoods. Let v 2 B
and f : G ! R be a continuous function with compact support. Then for each
" > 0 there is a symmetric a-relatively compact neighborhood W of B~ , such
"
that:
Z
f (y) d
( )
x
v
Z
(y) ; f (y) d
)
(y < "
( )
x
v
( )
x
v
(UL) for all x 2 W
"
where L is the support of f and U is a symmetric -compact neighborhood of
B~ .
Proof. Since is a locally compact groupoid whose base subgroupoid B~ has
a fundamental system of -relatively compact neighborhoods, by Proposition
3.4/p. 40 1], it results that f is left "uniformly continuous". Hence for each
" > 0 there is a symmetric -relatively compact neighborhood W of B~ , such
that
"
x2W
) jf (xy) ; f (y)j < " for all y 2 "
( )
x
Let U be a symmetric -compact neighborhood of the base subgroupoid B~
and assume, without loss of generality, that W U . Then
jf (xy) ; f (y)j < "1
"
(y ) ,
UL
for all x 2 W y 2 follows that
"
Z
f (xy) d
( )
x
v
( )
x
(y ) ;
, where 1
Z
UL
f (y) d
( )
x
v
Z
< "1 (y) d
UL
denotes the indicator function of UL. It
(y) ( )
x
v
Z
jf (xy) ; f (y)j d
(y) = "
( )
x
v
( )
x
v
(y )
(UL)
for all x 2 W .
"
Proposition 4 Let be a locally compact groupoid on B, whose base subgroupoid B~ has a fundamental system of -relatively compact neighborhoods.
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For each v 2 B and each continuous function with compact support, f : ! R,
let us denote by F : ! R the map dened by
v
f
Z
F (x) = f (y) d
v
f
( )
x
v
Z
(y) ;
f (y) d
( )
x
v
(y) for all x 2 .
Then
1. F is a morphism of groupoids.
2. For each " > 0 there is a symmetric neighborhood W of the base subgroupoid B~ such that:
v
f
"
F (x) ; F (~u) = F (x) < " for all x 2 W :
v
f
v
f
v
f
"
Consequently, F is a morphism of groupoids which is continuous at every
unity u~ 2 B~ .
v
f
Proof. For all (x y) 2 we have
Z
F (xy) =
v
f
Z
=
f (z ) d
( )
f (z ) d
( )
y
v
y
v
Z
Z
(z ) ; f (z ) d
Z
(z ) ; f (z ) d
(z )
( )
(z ) +
y
v
+ f (z ) d
= F ( y ) + F (x )
( )
x
v
v
f
( )
x
v
(z ) ;
Z
f (z ) d
( )
x
v
(z )
v
f
Therefore F is a morphism. Let L be the support of f .
According to the preceding lemma, it follows that for each " > 0 there is a
symmetric neighborhood W of the base subgroupoid B~ such that:
v
f
"
F (x) ; F (~u) < "
v
f
v
f
( )
x
v
(UL) for all x 2 W
"
F (x) ; F (~u) < " sup (UL) for all x 2 W
v
f
v
f
2
u
B
u
v
"
Hence F is continuous at every u~ 2 B~ .
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f
Proposition 5 Let be a locally compact groupoid on B, whose base sub-
groupoid B~ has a fundamental system of -relatively compact neighborhoods.
Let v 2 B . Then the following conditions are equivalent:
1. The map : ! B (x) = (x) is open (or equivalently, the map
: ! B a (x) = (x) is open).
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v
v
v
v
v
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2. For each continuous function with compact support, f : ! R, the map
v Z
u !f f (y) d (y) : B ! R]
u
v
is continuous at every u 2 v].
Proof. 1 => 2. Let f : ! R be a continuous function with compact
support. Let us denote by F : ! R the map dened by
v
f
Z
F (x) = f (y) d
v
f
( )
x
v
(y ) ;
Z
f (y) d
( )
x
v
(y) for all x 2 .
Let u 2 v] and let (u ) be a net in B converging to u. Let x 2 be such
that (x) = u. Since : ! B is open, we may pass to a subnet and
assume that there exists a net (x ;) in converging to x such that (x ) = u
for all i. Thus, we obtain a net x x;1 converging to u~ in . According to
Proposition 4, F is continuous at u~. Hence
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v
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v
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v
f
;
lim F x x;1 = 0
i
that is
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f
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lim F (x ) ; F (x) = 0.
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f
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Since F (x ) = ( (x )) ; ( (x )) = (v) ; (u ) and F (x) = (v) ;
(u), it follows that lim (u ) = (u).
2 => 1. Let U be an open set in and u0 2 (U \ ). Let x0 2 U
be such that u0 = (x0 ). Let us choose a compactly supported nonnegative
continuous function, f : ! 0 1], such that f (x0 ) > 0 and f vanishes outside
U . Since the map
v
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f
v
Z
u 7! f (y) d (y) : B ! R]
u
v
is continuous at u0 , it follows that
Z
Z
W = u : f (y) d (y) > 21 f (y) d 0 (y)
is a neighborhood of u0 . It is easy to see that W is contained in the set
u
v
u
v
Z
u : f (y) d (y) 6= 0
u
v
which is contained in (U \ ). Therefore (U \ ) is a neighborhood of
u0 .
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Theorem 6 Let be a locally compact groupoid on B, whose base subgroupoid
B~ has a fundamental system of -relatively compact neighborhoods. Suppose
that , and hence , are open maps. For each v 2 B and each continuous
function with compact support, f : ! R, let us denote by F : ! R the
v
f
map dened by
Z
F (x) = f (y) d
v
f
( )
x
v
(y ) ;
Z
f (y) d
( )
x
v
! R the map dened by
Z
(u) = f (y) d (y) u 2 B .
(y ) x 2 and let us denote by : B
v
f
v
f
u
v
For each continuous function with compact support, f : ! R, assume that
F is continuous on . If v] is not singleton, then is continuous at every
u 2 v].
Proof. Let u 2 B \ v]. Because v] contains at least two elements, there
is v0 2 v] such that v0 6= u. Since u 6= v0 , there are two open subsets of B ,
u 2 U0 and v0 2 V0 such that U0 \ V0 = . Let g : B ! 0 1] be a continuous
v
f
v
f
function with the following properties:
1. g (w) = 1 for all w in a compact neighborhood U of u contained in U0
2. g (w) = 0 for all w 2= U0 .
For each continuous function with compact support, f : ! R, let us set
f = f g . For each w 2 B , we have
v
U
U (w ) =
Z
v
f
Z
f (y) d (y) = f (y) g ( (y)) d (y) =
w
v
U
Z
w
v
= g (w) f (y) d (y) = g (w) (w)
w
v
v
f
Therefore (w) = U (w) for all w 2 U and (w) = 0 for all w 2 V0 .
Let (u ) be a net converging to u. Without loss of generality we may assume
that u 2 U for all i, and hence, (u ) = U (u ) for all i. So to prove that is continuous in u it suces to show that lim U (u ) = U (u). Let us choose
x such that (x) = u and (x) = v0 . Because : ! B is open, we may
pass to a subnet and assume that there exists a net (x ) in converging to x
such that (x ) = u for all i. Since F U is continuous on ,
lim F U (x ) = F U (x) = U (v) ; U (u) = ; U (u).
(1)
On the other hand, lim (x ) = (x) = v0 and for large i we may assume
that (x ) 2 V0 . Hence, U ( (x )) = 0 and from F U (x ) = U ( (x )) ;
U ( (x )), it follows that
F U (x ) = ; U ( (x )) .
(2)
By 1 and 2, it results that lim U (u ) = U (u), as required.
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Corollary 7 Let be a locally compact groupoid on B, whose base subgroupoid
B~ has a fundamental system of -relatively compact neighborhoods. Suppose
that , and hence , are open maps. For each v 2 B and each continuous
function with compact support, f : ! R, let us denote by F : ! R the
v
f
map dened by
Z
F (x) = f (y) d
v
f
( )
x
v
(y) ;
Z
f (y) d
( )
x
v
(y) x 2 .
Then each F is a morphism of groupoids which is continuous at every unity
u~ 2 B~ . If all morphisms F are continuous on , then for each v 2 B with
non-singleton orbit, the map : ! B (x) = (x) is open.
v
f
v
f
v
v
v
Proof. According to the preceding theorem, for each v with v] 6= fvg and
for each continuous function with compact support, f : ! R, the map
: B ! R dened by
v
f
Z
(u) = f (y) d (y) u 2 B
v
f
u
v
is continuous at every u 2 v]. Applying Proposition 5, it follows that the map
! B (x) = (x) is open for all v 2 B with v] 6= fvg.
:
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Corollary 8 Let be a locally compact groupoid on B, for which the target
projection is an open map and whose base subgroupoid B~ is paracompact.
Let us assume that for any morphism F : ! R the continuity of F at all
u~ 2 B~ implies the continuity of F everywhere on . Then for each v 2 B with
non-singleton orbit, the map : ! B (x) = (x) is open.
Proof. If the base subgroupoid B~ is paracompact, then B~ has a fundamental system of -relatively compact neighborhoods (see the proof of Proposition
II.1.9/p. 56 5]). Therefore the hypotheses of the preceding corollary are satised.
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Denition 9 A locally compact groupoid is -discrete if its base subgroupoid is
an open subset (Denition I.2.6/p. 18 5]).
The -discrete groupoids are generalizations of transformation groups G Any locally compact
groupoid for which (and hence ) is a local homeomorphism is -discrete.
Such groupoids are called etale groupoids (4] p. 46) or sheaf groupoids ( 2] p.
209).
B
! B, where the group G is assumed to be discrete.
Corollary 10 Let be a locally compact -discrete groupoid, for which the
target projection is an open map and whose base subgroupoid B~ is paracompact.
Let us assume that for any morphism F : ! R the continuity of F on a
neighborhood of B~ implies the continuity of F everywhere on . If for each
v 2 B the orbit v] contains at least two elements, then B is a discrete space.
10
Proof. Applying Lemma I.2.7/p. 18 5], it follows that for each v 2 B, is a discrete space. According to the preceding corollary, for each v 2 B , the
map : ! B (x) = (x) is open. Since (~v ) = v, it follows that fvg
is open in B .
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Remark 11 We have proved that for general topological groupoids the continuity of a morphism on a neighborhood of the base does not necessarily imply the
continuity everywhere. Perhaps, the question of Mackenzie should be reformulate: For what kind of groupoid morphism does the continuity on a neighborhood
of the base imply the continuity everywhere?
References
1] M. Buneci, Haar systems and homomorphisms on groupoids, Operator Algebras and Mathematical Physics: Conference Proceedings Constanta (Romania), July 2-7, 2001, Theta Foundation (2003), 35-50.
2] A. Kumjian, On equivariant sheaf cohomology and elementary C -bundles,
J. Operator Theory, 20 (1988), 207-240.
3] K. Mackenzie, Lie groupoids and Lie algebroids in dierential geometry, LMS
Lecture Note Ser., vol. 124, Cambridge Univ. Press, 1987.
4] P. Muhly, Coordinates in operator algebra, (Book in preparation).
5] J. Renault, A groupoid approach to C - algebras, Lecture Notes in Math.
Springer-Verlag, 793(1980).
6] J. Renault, The ideal structure of groupoid crossed product C - algebras, J.
Operator Theory, 25 (1991), 3-36.
University Constantin Br^ancusi of T^argu-Jiu
Bulevardul Republicii, Nr. 1, 210152 T^argu-Jiu
Rom^ania
e-mail: ada@utgjiu.ro
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