I ntnat. J. Mh. Math. Si.
Vol. 3 No. 3 (1980) 491-504
491
ON CERTAIN GROUPS OF FUNCTIONS
J.S. YANG
Department of Mathematics,
Computer Science, and Statistics
University of South Carolina
29208
U.S.A.
Columbia, South Carolina
(Received October 18, 1979)
ABSTRACT.
Let C(X,G) denote the group of continuous functions from a topological
space X into a topological group G with the pointwise multiplication and the
compact-open topology.
We show that there is a natural topology on the collection
e} which is
{f
C(X,G): f(p)
P
analogous to the hull-kernel topology on the commutative Banach algegra C(X) of
of normal subgroups A(X) of
C(X,G) of the M
all continuous real or complex-valued functions on X.
morphisms between groups
We also investigate homo-
C(X,G) and C(Y,G).
KEY WORDS AND PHRASES. Continuous functions, topological group, compact-open
’topology, hl-kernel topology, normal subgroups, S-pair, S-topology, Banach
algebra, struce space.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES;
Secondary 22A05, 20F30.
i.
Primary 54C35, 54HI0,
INTRODUCTION AND NOTATION.
Suppose X is a compact topological space and suppose C(X) is the algebra of
J.S. YANG
492
all continuous real or complex-valued functions on X with the usual pointwise
operations and the supremum norm.
Then C(X) is a regular commutative Banach
algebra with identity and X is homeomorphic to the maximal ideal space A(C(X)) of
the algebra C(X), where A(C(X)) is endowed with the Gel’fand topology which coincides with the hull-kernel topology since C(X) is regular, [3].
If X is a topo-
logical space and G is a topological group, let C(X,G) be the topological group
of all continuous functions from X into G under pointwise multiplication and the
In Section 2 of this paper, we study spaces of normal sub-
compact-open topology.
groups of C(X,G).
There is a natural topology, analogous to the hull-kernel topo-
fogy in Banach algebra, for the collection of normal subgroups of the form
e} where e is the identity element of G;
{f e C(X,G): f(p)
M (X,G)
P
the resulting topological space will be denoted by A(X). We show that, with some
M
P
mild restriction on X and G, X is homeomorphic to
A(X), and that A(X*)
is the one-
point compactification of A(X), where X* is the one-point compactification of the
locally compact space X.
Some theorems on homomorphisms and extension of homomor-
phisms in C(X,G) are considered in Section 3.
a theorem originally stated in
We also prove a correct version of
[7, theorem 8].
All spaces considered in this paper are assumed to be Hausdorff unless speciFor topological spaces X and Y, the function space F
fled.
=
C(X,Y) is under-
stood to be endowed with the compact-open topology whenever it is referred to
topologically.
10(X,G)
or simply I
0
if no confusion should occur, will denote
the identity element of the group C(X,G).
2.
THE STRUCTURE SPACES.
For a topological space X and a topolog.ical group G, let P
X is compact and G is a Lie group, then
manifolds (c.f. [i]).
where a
r
C(X,G).
If
p
X, are in general
P
It is easy to see that M is locally contractible at I 0,
a
and M
2
X, if X is a locally compact group locally contractible at a, and that
493
CERTAIN GROUPS OF FUNCTIONS
Mp,
every
X, is n-simple for every positive integer n if X is a locally com
p
pact contractible space.
It is also easy to see that the topological group Y is
a group with equal left and right uniformities if so is the group G, and that, if
G is the projective limit of the inverse system of topological groups
{ (G,
fss):
{(M_(X,G
u)
,8
A}, then M
p
fo ):
P
8
A}
is the projective limit of the inverse system
where
p
fo (f)
f
M (X,G).
f for every f
Bc
p
Throughout this paper the spaces X and G will be subject to the following
condition.
DEFINITION i [6].
A pair (X,G) of a topological space X and a topological
group G is called an S-pair if for each closed subset A of X and x
exists f e F such that f(x)
# e and Z(f)
{x:
e}
f(x)
n
% A,
there
A.
It is clear that (X,G) is an S-pair if X is completely regular and G is path
It is also clear that X is completely
connected or if X is zero-dimensional.
regular if (X,G) is an S-pair, and that
(X,Gu)
AGa
X
a
eA
is also an S-pair whenever
Magill called a space X a V-space, [4], if
is an S-pair for each u e A.
for points p, q, x, and y of X, where p # q, there exists a continuous function
x and f(q)
f of X into itself such that f(p)
=
y, and has shown that every
completely regular path connected space and every zero-dimensional space is a VX
G) is an S-pair if each (Xa,G), a A, is
a
uA
is an S-pair and if G is a V-space. If G is a topological group such that (G,G)
space.
It is easy to see that
is an S-pair, G may not be a V-space.
For example, let G
of real numbers with the usual topology and let G
with the discrete topology, then (G
(G2,G2)
are S-pairs.
I
G
I
GI,
G
I
G
2
be the additive group
be any non-trivial finite group
2)
Since the topological group G
the identity component isomorphic to
from [4, Theorem 3.5].
G
G2,
2
I
is an S-pair since
I
G
2
(GI,G I)
and
is not connected with
is not a V-space as it follows
It is pointed out in [7] that X is hemicompact and G is
metrlzable if (X,G) is an S-pair, G is a V-space, and F is first countable.
J.S. YANG
494
It is well-known (c.f. [2]) that, for every topological space X, there
C(Y) is (algebraically) isomorphic
exists a completely regular space Y such that
to
is the ring of continuous real-valued function on the space Z.
C(X), where C(Z)
Using the similar argument mutatis mutandis as used in the construction of the
space Y, it is a straightforward to see that, for every topological space X and a
YG
topological group G, there is a completely regular space
such that
C(YG,G)
is
continuously isomorphic to C(X,G), and that, in the case G is path connected,
(YG,G)
is independent of the group G
YG
is an S-pair and the associated space
The latter means that
within the category of path connected topological groups.
YGI YG2
whenever G
YG
struction of the space
.
S-pair.
For a collection
If U c
THEONEM i.
Just
and M
=
M1 n
either M
M2,
PROOF:
U* u V*
c
2
M2.
=
sV, then (U n V)*
If
M
=
nU}, let
"*"
I:
*
.
U, (U*)*
M
=
M1
0
*
U*,
Hence
M2}.
[
as
I
"*"
,
I,
then
and that
is a closure operator if
7..
sU, and
Now if
Hence we have the theorem.
is a closure operator on
ing topology, not necessarily Hausdorff, on
"*"
if and only if whenever M
(U u V)* for subsets U and V of
{M
.
C(X,G), we define
are intersections of some subsets of
(U u V)* for subsets U and V of
DEFINITION 2.
:
{M
let U*
It is clear that U*
and only if U* u V*
M2
,
and M
1
=
is an S-pair.
made above, we shall now assume that (X,G) is an
is a closure operator on
where M
M1 of M
(X,G)
if
of normal subgroups of F
and U #
"*"
YG
that X
Because of the remarks
follows:
It follows from the con-
I and G 2 are path connected groups.
I,
we shall refer the result-
as the S-topology, and the resulting
.
space will be referred to as a G-structure space, or simply structure space, of
the space X.
COROLLKRY
If
[
admits the S-topology, so is every subset of
495
CERTAIN GROUPS OF FUNCTIONS
REMARK 2.
If G is path connected, we may speak of structure spaces for the
space X without referring to the group since C(X,G) and C(X,R) are isomorphic in
this case.
LEMMA 3.
If a collection of normal subgroup
then a subset A of
M
r
of
0
:
such that
= M0}.
M
In fact, M
hA.
0
Suppose A=
is closed, then A
PROOF:
admits the S-topology,
is closed if and only if there exists a normal subgroup
which is the intersection of some subset of
{M
A
r
of
,
:
{M
nA
M
Conversely, suppose that there exists a normal subgroup M
M0
0U for some U c
:
{M
=
M
hA}
THEOREM 4.
then
M
and I
PROOF:
12
I
are
dlsoint open
I
U 2, A
I
3, we have A
=
I I, M
2
U
12
.
and
2
,,
:
A.},I
M
I
M
M
12
I
=
2
such that M
I
2
1,2.
and I
I
M
c
UI,
If we let I
n
12
I
M
M
and M
M2.
I
1,2, Ml
{M
M
i
:
B
and V
1
2
= II,
M2
= 12,
ll},
i
1,2.
I.
If we let V
and M
V
Then there are subsets U
2
then either M
2
.
B
M
I
12,
Then B
To see that V
B
I
or M
B
2.
1
i
M2
Now M
and U
Ii,
2
2
Am
i
Ai,
i
2
BI, V
,
I
12,
I
Thus there
eemma
Using
1,2, then
.
MI,
M
such
2
such that if I
I
12
I
I
12,
I.
of
and I
and
2
AI,
are closed by Lmma 3, M
2
n V
l
1
I # M2.
M
Conversely, assume that the stated property holds, and let
that M
M
12,
2
,
2
are closed and
i
admits the S-topology,
Ii,
I
MI,
I
2
r
of
M2, there are I
such that M
in
2
and A
12
where
Then
I,
M2
is Hausdorff, and let
and U
I
U
Ul,
U l, then A
{M
i
=
sets U
of
Hence A is closed.
Suppose that
2
M0}.
M
If a collection of normal subgroups
nUl,
I
Ii,
2
A.
is Hausdorff if and only if for
where I
:
such that A ={M
0
, M0}.
Let B
BI,
B 2, then M
I
M
B2,
2
VI,
M2
V2,
it suffices to show that if M
implies M
=
I
I
12
This means
J.S. YANG
496
that either M
M
B
2.
=
I
I
or M
=
12
since
[
B
Hence M
admits the S-topology.
or
I
This completes the proof.
If we denote by A(X) the collection of all normal subgroups of F of the form
{f
M
P
C(X,G):
X, then the following theorem states that A(X)
e}, p
f(p)
admits the S-topology and that the S-topology is Hausdorff if
THEOREM 5.
X:
{q
02
Mq
V}.
M
q
(POlnM) (k02)’
n
i and q 2"
Then f(q
/
{P
01
n
M
M and g
q
PO 1 P
i’
For if q e
u 0
01
pOlMp and Mq
Mq
then either
.
U} and
M
X:
e since f(q
kO2
M
kO 2
then there is a net
f(q), and hence f(q)
Hence q
and let
p
It is, by Theorem i, sufficient to show that, if
wise, then there exist f
q
is an S-pair.
A(X) admits the Hausdorff S-topology.
Let U and V be subsets of A(X)
PROOF:
(X,G)
This implies that
q
{q}
in O
Z(h) but h(q)
e.
contradiction.
Hence either M
This would show that h
q
01Mp)
P
I
such that
e for each s.
But (X,G) is an S=pair, let h
2
Suppose other-
or
Similarly,
F such that
Mp)
(POln
M
n
(kO 2)
(k 02)’
q
q/q.
but
and A(X)
admits the S-topology.
Next to show that the S-topology is Hausdorff.
q.
and
01 02
I
1
let
T2,
If C
Mq 12.
01
,
X
2
n
kO 2
To see that
and
kC I
F(C 2)
.
Since X is
p
12
Mp 12
e but f(p)
Similarly, we have M
q
and
O
I
02
1
X
I
I
2
note that p
02
I
I.
then p
C
A(X), where
C2,
O1,
I and q
A(X) since C 1 u C
2
X,
hence there exists f
C
2.
q
02
If
Mp Ii,
and
F such that
This shows that M
M
but f
P
P
kC 2
This completes the proof that A(X) is T2, by
Thus f
e.
Mp, Mq
be open sets in X such that P
and C
then I
Let
12
Theorem 4.
Note that the S-topology defined above for A(X) is analogous to the hullkernel topology, which coincides with the Gel’fand topology, on the maximal ideal
space of the commutative Banach algebra C(X).
CERTAIN GROUPS OF FUNCTIONS
I, let A be a closed set of a structure
For each
Lemma 3, there exists a normal subgroup M
{M
such that A
subset of
E
;
uIM,
subgroup of I" generated by
.
Then, by
of F which is the intersection of some
M }.
M
sace
497
If we denoted by [ U M ] the normal
then we have the following 1 whose proof is
straightforward and hence omitted.
LEMMA 6.
{M
n A
tion of normal subgroups
Z,
{N
el
PROOF:
Suppose
u N ]
A
of X is compact if and only if every collec-
r,
of
each of which is the intersection of some
[
Z
M for each M
x
]
has a finite subcollection.
[.
M for each M
is compact, and let {N
}
,
compactness of
Z,
there exist
n
0
:
M
A
i’ =2
n
[:
such that {M
M m
M
N )
o
such that
n
u
:
{M
I, let A
If, for each e
Lemma 3, and
e
Sl e2’
[.
,
be a collection of normal sub-
each of which is the intersection of some subset of
M for each M
for ch M e
I
such that [ u N
i=l
n
is closed in
ist
[ u M]}
M
n
N
r,
{N}e_l
such that [ u N ]
N2
groups of
[:
A structure space
THEOREM 7.
subset of
E
$.
.
n A
i=l
N
]}
i=l
such that
M
m
N }
then
Hence, by h
i.e., there
@;
1
n
u N
Hence
]
M
[.
Conversely
suppose that
[
has the stated property
and let {A
a sI
be a
collection of closed sets with the finite intersection property, where A
{M
[:
.
M
N
and N
is the intersection of some subset of
Then {M
A
[: M [ u N ]
aI
ael
Thus, by the hypothesis, there ist el’
each M e
[.
n
This would imply tt
A
$, hence [ u N ]
[.
Suppose that
W
M for each M
u
such that [ u N ] W M for
2’
n
i=l al
$, a contradiction. Hence [ is
aeI
c ompac t.
COROLLARY.
A structure space
[
of F not contained in any element of
of X is compact if every normal subgroup N
[
contains a finitely generated normal sub-
498
J.S. YANG
group of N not contained in any element of
Assume that the stated property holds in
PROOF:
r,
collection of normal subgroups of
subset of
[
such that [ u N ]
M for every M in
N
such that B
[,
and let {N }
be a
each of which is the intersection of some
7..
M for every M in
[.
n
Then [ u N
n.
[,
[ u N ]. Then
Let N
. .
thus N contains a finitely generated normal subgroup B
M for each M
i, 2,
i
.
s.
i= I 1
[asl
Let B
]
a
a
s
Sn
2
], where a
Hence
M for every M in
N
si
si
is compact by
Theorem 7.
r
We shall call a normal subgroup N of
f(p)
free if there is no p
X such that
N.
e for each f
A(X) is compact if every free normal subgroup N of F contains
COROLLARY.
a finitely generated free normal subgroup.
THEOREM 8.
:
The mapping
X
A(X) defined by (x)
/
Mx,
x
X, is a
homeomorphlsm.
PROOF: Clearly,
is one-to-one and onto.
For the continuity of $, let A
r
subgroup M of
0
Mx
M
0
for each s.
imply that f(x)
# e,
If
Mx
0
n
Mx.
a contradiction since f(x
We claim that $(C)
(C) is closed.
Mx {Mx
f
A(X):
Thus x
{Mx
It is clear that (C)
Mx = M0}.
C(X,G) such that f(C)
diction.
-1
M 0, there exists f
Next to show that $ is a closed map.
M
Mx MO}.
(X):
For this purpose, let {x } be a net in
is closed.
Then
{Mx
such that A
A(X) be closed.
c
Then
C, and we have M
X
We shall see that
(A) converging to x
M
0
X.
Mx which would
f(x) and f(x)
e for each s.
Mx = M0}, which would imply
{Mx A(X): Mx = MO}. Now let
A(X):
c
q-l(A)
Let C be closed in X, and let
Mx = M0.
e but f(x)
/
Then there exists a normal
#.e.
Suppose x
Hence f
that
C, then there exists
M
0
but f
Mx,
a contra-
(C).
If A is a commutative Banach algebra without identity, and if A(e) is the
algebra obtained by adjoining an identity to A, then the maximal ideal space
499
CERTAIN GROUPS OF FUNCTIONS
A(A(e)),
with the Gel’ land topology, is the one-point compactification of A(A)
Using the previous results, we can also state the following theorem whose proof
is now trivial.
THEOREM 9.
If X is a locally compact space and X* its one-point compactifi-
cation, then A(X*) is the one-point compactification of A(X), and A(X*)
{M},
A(X)
D
where
M=
3.
HOMOMORPHISMS
OF
{f
f(=)
C(X*,G):
e}.
C(X,G).
In this section, we shall study homomorphisms of the group C(X,G) into the
group C(Y,G) which leads us to have another version of a theorem originally
announced in [7].
We shall also, at theend of the section, consider extensions
of homomorphisms of the group
C(X,G).
All pairs (Z,G) are again assumed to be
S-pairs.
DEFINITION 3.
C(X,G)
(i) A homomorphism
of the group C(Y,G) into the group
is said to be a constant-preserving if
# maps every constant function on
Y into the corresponding constant function on X.
A homomorphism
(2)
the property that
-i
of the group
(A(X))
C(Y,G)
into the group
C(X,G) which has
A(Y) is called an F-homomorphism.
c
C(Y,G)
It is easy to construct an example of a homomorphism #:
which is an F-homomorphism but is not constant-preservlng.
/
C(X,G)
The following example
[5], shows that the converse does not hold either.
EXAMPLE.
([-i,i]
f
E
Then
x
Let Y
[0,I] be the closed
{0}) u ({0}
C(Y,R), define (f)
unit interval, and let X
(0,i]) considered as a subspace of R
E
2.
For each
C(X,R) by
1
#(f)(t,0)
f(
(t + I)),
(f)(0,s)
f(
(s + i)) + f(), s
t
is a constant-preserving isomorphism of
[-I,i]
[0,i]
C(Y,R) onto C(X,R).
If g e C(X,R),
J.S. YANG
500
then
g(4y-l,0)
-i
(g) (Y)
g(0,2y-l) + g(l,0)
then g e M
.
-I
(0,1/2)’ but Z( (g))
THEOREM I0.
:
Suppose that
(0,1/2)}
Hence
C(Y,G)
A(Y), and
induces a continuous map j of X into Y such that j (x)
by h(x)
g(y) for each g
,
f(x), f
X.
x
C(X,G)
C(X,G), and let MX
-I
Then ker
h(x)
A(Y) by (Mx)
+
Clearly
M
Y
m
MI},
closed set in A(Y).
(Mx),
If
m
My
e A, thus
A.
z
If
Mx
(Mz)
hy.
I
-i
My
(My)
for some
,
let A
is the intersection of some subset U of A(y), be
We claim that
(M__)x
My
-i
e
M
Y
-i
{Mx
(A)
(U) of A(X).
ker hy
(A).
M e A for some y
y
Mx (MI)} and that
In fact, let
Mx (MI)"
A(X):
m
m
ker (hx
)
Conversely, let
Y, then Hy
Mz
hz
-l(Mx
-i
D
(A).
hence
M
1
Then
(MI)
It is easy to see that (MI) is the intersection of the subset (U) of A(X).
Therefore
(2)
-i
(A) is closed in A(X), and
Let the mapping j:
X
G
/
Now we define a mapping
For the continuity of
where M
C(Y,G)
Define h(x):
hence ker h(x)
Y
is the intersection of the subset
-l(M--)x MI"
G be the evaluation map de-
M
is one-to-one.
A(Y):
/
kerhx.
Such an y is unique and we have h(x)
(X)
y if and only
C(Y,G).
(i) for each x e X, let hx:
hx
y e Y.
(M
Then
(2)
fined by hx(f)
hence
C(X,G) is a continuous constant-pre-
/
induces a one-to-one continuous map of A(X) into
PROOF.
Then
g(0,0) > 0,
and that g(l,0)
(i)
(g)(x)
(MI)
[,I]
is not an F-homomorphism.
C(X,G).
serving F-homomorphism of C(Y,G) into
{M
Y
1
g(0,0), y
Now choose g e C(X,R) such that Z(g)
if
i
[0,]
y e
/
is continuous.
Y be defined by j
y-i o
x,
where
M
z
501
CERTAIN GROUPS OF FUNCTIONS
z:
Z
j(x)
/
A(Z) is the mapping of Theorem 8.
(Mx)
y if and only if
(g)(x)
g(y) for every g
My.
4)
ker(hx
g(y).
#(g)(x)
E
If g
To see that j(x)
M
y if and only if
Y
Thus
y. Then (M
C(Y,G), let j(x)
My,
(c_k)
g(y) for each g
#(g)(x)
g
hx(c__ #(k))
C(Y,G), then, for g
(Mx)
c(k)(x)
(g)
ck for some k
c, while g(y)
C(Y,G).
E
hx
e, and we have
C_My,
G such that g
there exists c
(g)
ck(y)
c__
is the
#(g)
Hence
c.
g(y) for each
hy(g).
g(y)
(g)(x)
where
Now hx
My.
Conversely, if (g)(x)
C(Y,G), hx
My, and we have that j(x)
REMARK:
My,
If g
constant mapping of X into c, hence g
hx
My.
x)
C(Y,G).
Let g
Then clearly j is continuous and
Thus
y.
It is easy to see that, if the mapping
in Theorem i0 is an onto
map, then # is an embedding.
THEOREM ii.
of C(Y,G) into
A continuous homomorphism
C(X,G)
stant-preserving F-homomorphism if and only if there exists f
(k)
k
f for every k
PROOF:
is a con-
C(X,Y) such that
C(Y,G).
It is clear that a homomorphism
of the form
k e C(Y,G) is a constant-preserving F-homomorphism.
(k)
k
Conversely, if
f for every
is a con-
stant-preserving F-homomorphism, and if j is the continuous map of X into Y as
defined in Theorem i0, then, for each k
where j(x)
y,
COROLLARY.
Hence (k)
k
A homomorphism
C(Y,G), (k)(x)
k
j(x),
j for each k e C(Y,G).
of
C(Y,G)
F-homomorphism if and only if there exists f
every k
k(y)
ifo
X(X,G)
is a constant-preserving
C(X,Y) such that #(k)
k
f for
C(Y,G).
PROOF:
Note that the group topologies for C(Y,G) and C(X,G) are not rele-
vant in the proof of Theorem i0.
Hence take discrete topologies for the groups
C(Y,G) and C(X,G), then apply the proof
of Theorem ii.
As a consequence of the discussions made above, we can now state a correct
J.S. YANG
502
version of the theorem originally stated in
THEOREM 12.
C(X,G) which
[7, Theorem 8] in the following.
C(Y,G) and
between groups
If there exists an isomorphism
and #
is constant-preserving such that both
-I
are F-homomorphisms,
then X and Y are homeomorphic.
PROOF:
the above corollary to
such that
-I
It is clear that
(k)
k
and
-i
is also constant-preserving if
-i
C(Y,G) and # (k)
j(x)
Consequently, we have that
x and j
To see this suppose that there exists x
C(X,G) such that f(
f
(
-i
(f)
j)(x) # f(x), .and thus (#
Similarly, j
(y)
y.
-i
y for x
X and y
Y.
j(x) # x, then we have
X such that %
j(x) # f(x).
j(x)) # f(x) or f
C(X,G).
for each k
k
(y)
C(Y,X)
C(X,Y) and
there exist functions j
j for each k
Applying
is.
(f))(x) # f(x) which leads
Hence
to
f(x) # f(x).
Hence j is a homeomorphism of X onto Y.
For topological spaces X and Y, it is clear that the space C(X,Y) may be
embedded into the space C(X x
Z,Y) as
homomorphism of the topological group
a retract for any space Z, and that every
C(X,G)
into a topological group L may be
extended to a homomorphism of the topological group C(X x Y,G) into L for any
topological group L.
We shall conclude this paper with the following result
concerning an extension of F-homomorphisms.
Suppose A is a closed subset of X.
THEOREM 13.
serving F-homomorphism h of the topological group
Then every constant-pre-
C(G,G)
into the topological
group C(A,G) may be extended to a homomorphism H of the same kind from the topological group C(G,G) into the topological group C(X,G) such that I
every continuous function f:
where I:
C(X,G)
/
A
/
H
h if
G may be continuously extended to all of X,
C(A,G) be the map
defined by l(f)
f
i for f
C(X,G),
i being the inclusion map of A into X.
PROOF:
f*:
C(G,G)
For necessity, let f:
A
G be any continuous function, and let
C(A,G) be the natural homomorphism induced by f, namely f*(k)
503
CERTAIN GROUPS OF FUNCTIONS
f for each k
k
Then f* is a constant-preservlng F-homomorphlsm, by
C(G,G).
Hence there exists a constant-preservlng F-homomorphlsm H of the
Theorem ii.
topological group C(G,G) into
that H(k)
k
A, 0(a)
I(H(id))(a)
i(a)
C(G,G).
@ for every k
itself, then, for a
(I
H
C(X,G) such that I
If i
denotes the identity map of G into
d
0)(a)
(id
S)(i)(a)
C(X,G) such
Let 8
f*.
H(i d)
H(id)(i(a))
H(id)(a)
f*(id)(a) (id
f(a).
f)(a)
Hence
8 is an extension of f to all of X.
For sufficiency, assume that every continuous function f:
C(G,G)
extended continuously to all of X, and let h:
serving homomorphism of the topological group
C(A,G).
Then there exists f
/
C(X,G) by H(k)
serving F-homomorphism and I
H
G may be
C(G,G) into the topological group
k
f for every k
C(G,G).
to all of X, define a function H:
for each k
k
/
C(A,G) be a constant-pre-
/
C(A,G) such that h(k)
If we denote by f the extension of
C(G,G)
A
h.
C(G,G).
Then H is a constant-pre-
This completes the proof.
In particular, if X is a normal space, and A a closed subset of X, then
every constant-preserving F-homomorphism h of the topological group C(R,R) into
the topological group C(Z,R) may be extended
from the topological group
I
C(R,R)
to
a homomorphism H of the same kind
into the topological group
C(X,G) such that
H=h.
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i.
Geoghegan, R., On spaces of homeomorphisms, embeddings, and functions I,
Topology, Vol. ii (1972), 159-177.
2.
Gillman, L. and Jerrison, M., Rings of continuous functions, Van Nostrand,
N. Y. (1960).
3.
Larsen, R., Banach algebra, Dekker, N. Y. (1973).
4.
Magill, K. D. Jr., Some Homomorphism theorems for a class of semigroups,
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Nickolas, P., Free topological groups and function spaces, Preprint.
504
J.S. YANG
6.
Yang, J. S., Transformation groups of automorphisms of C(X,G), Proc. Amer.
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