NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (2004), 41-45

SEPARATE AND JOINT CONTINUITY OF HOMOMORPHISMS DEFINED ON TOPOLOGICAL GROUPS

J i l i n g C a o a n d W a r r e n B . M o o r s

(Received February 2002)

Abstract. In this paper we prove the following theorem. “Let H be a count- ably Cech-complete topological group, X be a topological space and ( G ,- ,r) be a topological group. I f f : H x X —> G is a separately continuous mapping with the property that for each x 6 X, the mapping h i—> f (h, x ) is a group homomorphism, then for each q-point x q E X the mapping / is jointly con­tinuous at each point of H x {x o } .” We also present some applications of this result.

1. Introduction

In this short note we prove a theorem more general than the following. “Let H and G be topological groups and let X be a topological space. If H is countably Cech-complete, X is a q-space and / : H x X —> G is a separately continuous mapping that possesses the property that for each x G X , the mapping h f ( h , x ) is a group homomorphism, then / is jointly continuous on H x X .”

We begin with some definitions. If X , Y and Z are topological spaces and f : X x Y Z is a function then we say that / is quasi-continuous with respect toY at (x,y) if for each neighbourhood W of f (x , y ) and each product of open sets U x V C X x Y containing (x, y) there exists a non-empty open subset U' C U and a neighbourhood V'oi y such that f {U ' x V') C W and we say that / is separately continuous on X x Y if for each xq € X and yo G Y the functions y i—► f (xo ,y) and x f(x,yo) are both continuous on Y and X respectively. We will call a topological space X , countably Cech-complete [3] if there exists a sequence {A n : n G N} of open coverings of X with the property that every monotonically decreasing sequence {F n : n E N} of non-empty closed subsets of X has flneN Fn 0 , provided each Fn is An-small, i.e., for each n G N there exists an A n € An such that Fn C A n. Finally, we say that a point x E X is a q~~point if there exists a sequence {Un : n G N} of neighbourhoods of x such that every sequence {x n : n G N} in X with x n G Un for all n G N, has a cluster-point in X .

2. Main Result

We shall use the following key results.

Lemma 2.1 ([5, Lemma 1]). Let X be a countably Cech-complete space, Y a topo­logical space and Z a regular space. If f : X x Y —> Z is a separately continuous

2000 A M S Mathematics Subject Classification: Primary 54C05, 22A10; Secondary 54E52, 39B99. K ey words and phrases: Separate continuity; joint continuity; homomorphism; group; i?-module.

42 JILING CAO AND WARREN B. MOORS

mapping then for each q-point y £ Y the mapping f is quasi-continuous with respect to Y at each point of X x {y }.Remark 2.2. We have, for the sake of simplicity, not stated the previous lemma in its most general form. For a more general form of the result see Lemma 1 in [5], where g point is replaced by “g^-point” and countably Cech-complete is replaced by “strongly Baire” .Lemma 2.3 ([6, Lemma 3.2]). Let (G , -,r) be a topological group. Then the topol­ogy on G is determined by the set of all continuous left-invariant pseudo-metrics d on G for which the mapping (G , r) x (G, d) —> (G, d) defined by, (g,h) i—► g • h is continuous.

Recall that a pseudo-metric d defined on a topological group (G, •, r) is call left- invariant if for each g , h and k in G , d(kg, kh) = d(g, h) and it is called continuous if the topology generated by d is coarser than r.

One immediate consequence of this Lemma is the Banach-Steinhaus [8, Theorem2]-Proposition 2.4 (Banach-Steinhaus Theorem). If f : H —>• G is a Baire-l (i.e., the pointwise limit of a sequence of continuous functions) group homomorphism acting from a Baire topological group H into a topological group G then f is con­tinuous.

Proof. Let d be any continuous left-invariant pseudo-metric on G. In light of Lemma 2.3 it is sufficient to show that / : H —► (G, d) is continuous. By Osgood’s Theorem [4, p. 86] (i.e., Baire-l functions defined on Baire spaces and mapping into pseudo-metric spaces are continuous on dense G$ subsets of their domains) there exists a residual set Ho in H on which / is d-continuous. Let h be any element of H and let ho e H0. Then for any net {ha : a € D } in H converging to h we have that f ( h a) = f(hhQ 1 )f(h0h~ 1 ha), with f ( h 0h~ 1 ha) —> f(ho) in (G,d) since hoh~1 ha —> ho in H. Therefore, f(h a) —> f{hhQ1 )f(ho) = f(h) in (G,d); which completes the proof. □

Theorem 2.5. Let H be a countably Cech-complete topological group, X be a topological space and (G, - , t ) be a topological group. If f : H x X —> G is a separately continuous mapping with the property that for each x E X , the mapping h i—> f(h,x) is a group homomorphism, then for each q-point xq € X the mapping f is jointly continuous at each point of H x {a;o}-Proof. Suppose xo € X is a g-point. As with the Banach-Steinhaus Theorem we will appeal to Lemma 2.3 to deduce that it will be sufficient to prove that for any continuous left-invariant pseudo-metric d on G for which that mapping (G,t) x (G,d) —>■ (G,d), defined by, (g,h) i—> g ■ h is continuous, the mapping f - H x I - » (G, d) is continuous at each point of H x {a?o } - Fix £ > 0 and consider the open set:

Os := (^Jjopen sets U C H :

there is a neighbourhood V of xo with d — diam[f(U x V)] < e}.

We shall show that Oe is dense in H. To this end, let Uq be a non-empty open subset of H and let ho £ Uq. Since, by Lemma 2.1, / is quasi-continuous with respect to

SEPARATE AND JOINT CONTINUITY 43

X there exists a non-empty open subset U of Uq and a neightbourhood V of xq such that f ( U x V) C Bd(f(ho,xo);e /3 ). Therefore, d — diam[f(U x V)] < £ and so 0 ^ U C 0 £C\Uq] which shows that Oe is dense in H. Hence, / is ci-continuous at each point of (PlneN ^ i/n ) x {^o}; which is non-empty. We now show that / is in fact ^-continuous at each point of H x {a^o}. To this end, let ho be any element of flneN Oi /n and let h be any element of H and suppose that {(ha, x a) : a € D } is a net in H x X converging to (h,xo). Then by using the fact that,

f ( h a, x a) = f i h h ^ ^ x ^ f i h o h ^ K . X a )

and hoh~l ha —► h0 we obtain that f ( h a, x a) —>• f(hhQ1 , x 0)f(ho,xo) = f ( h , x 0). [Note: we also used the fact that fihh^ 1 , x a) —> f(hhQl ,x o) in (G, r).] This proves that / is d-continuous at each point of H x {xo}; which in turn implies that / is continuous at each point of H x {a?o} - □

Remark 2.6. Again, as with Lemma 2.1, it is possible to phrase this theorem more generally in terms of q p -points and strongly Baire spaces. Specifically, Theorem 2.5 remains valid (with the same proof) if we replace countably Cech-complete by strongly Baire (or even a-(3 unfavourable) and g-point by g^-point. Let us also mention another way in which this theorem may be refined. If H is Cech-complete then we may relax the hypothesis that “for each f {h ,x) is continuous”to “for each x 6 X , h f ( h , x ) is Souslin measurable” [7].

3. Applications

In this section we present a few sample applications of Theorem 2.5.Let X and Y be arbitrary sets and let A C X and B C Y. We shall write

F(A; B) for the set of all functions from X into Y that map A into B , that is,

F(A-,B) := { / € Y x : f (A ) C B}.If X and Y are topological spaces then the compact-open (pointwise) topology on

is the topology generated by the sets{.F { A - B ) : A e A and B 6 G}

where A. denotes the class of compact subsets (singleton subsets) of X and G denotes the class of open subsets of Y.

For a topological group G we shall denote by Endp(G) the space of all continuous endomorphisms on G endowed with the topology of pointwise converegence on G.

Corollary 3.1. Suppose that G is a countably Cech-complete topological group. If E is a q-subspace of Endp(G) (i.e., each point in E is a q-point) then the mapping 7r : Endp(G) x E —> Endp(G), defined by 7r(m,m') := m' o m is continuous. In particular, 7r is continuous on E.

Proof. Consider the mapping / : G x E —> G, defined by f (g , m) := m(g). Then / satisfies the hypotheses of Theorem 2.5 and so is jointly continuous on G x E. Now, if { (ma,m'a) : a G D } is a net in Endp(G) x E converging to (to, m') and g G G then,

7t(toq,to'q)(^) = m'a (ma(g)) -> rri(m(g)) = n(m,m'){g).

Hence 7r is continuous on Endp(G) x E. □

44 JILING CAO AND WARREN B. MOORS

Let G and H be topological groups. We shall denote by Homp(iJ; G) the space of all continuous homomorphisms from H into G endowed with the topology of pointwise convergence on H.

Corollary 3.2. Let G and H be topological groups and let E be a subset of HornP(H;G) . If H is countably Cech-complete and E is a q-subspace of Homp(^ir;G) then on E the pointwise and compact-open topologies coincide.

Proof. Since the compact-open topology is always finer than the pointwise topol­ogy it will be sufficient to show that for each compact set K C H and open set W C G, F ( K ; W ) is open in the pointwise topology. To this end, let K be a non-empty compact subset of H, W be a non-empty open subset of G and mo G F ( K - W ) (ie ., m o (K ) C W ). From Theorem 2.5 it follows that the mapping / : H x E —► G defined by, /( /i , m) := m(h) is jointly continuous on H x E and so from a simple compactness argument it follows that there exists an open set U in H and an open neighbourhood V of mo in E such that K C U and f ( U x V) C W . In particular this means that tuq G ^ C F ( K ; W ) and so F ( K ; W ) is open in the pointwise topology. □

If (M, +) is an Abelian group endowed with a topology t\ and (R, +, •) is a ring endowed with a topology 72 then we say M is a semitopological R module (over R) if it is an i?-module (over R) and both the mappings (x , y ) x + y and (r, x) i—► r-x are separately continuous on M x M and R x M respectively.

Corollary 3.3. Let M be a semitopological R-module over a ring R. If R is a q-space and M is a countably Cech-complete space then M is a topological R-module, i.e., (M, + ) is a topological group and (r,x) r-x is jointly continuous.

Proof. By Theorem 2 in [5], which states that every group endowed with a count­ably Cech-complete topology for which mulitplication is separately continuous is in fact a topological group, we may deduce that (M, + ) is a topological group. The result then follows from Theorem 2.5 since for each fixed r G R the mapping x i—► r • x is an endomorphism of M . □

Remark 3.4. As with our previous results this result remains valid when count­ably Cech-complete is replaced by strongly Baire and g-space is replaced by qo~ space. Also, if countably Cech-complete is replaced by Cech-complete (in M ) and g-space is replaced by Cech-completeness (in R) then we may relax the hypothesis that (r,x) > r -x is separately continuous to (r, x) r ■ x being separately Souslin measurable [7].

We say that a mapping / : X —> Y acting between Banach spaces X andY is almost C1 on X if for each y G X the mapping x i—> f'(x^y) defined by, f { x \ y ) := weak-lim ^o[/(a; + ty) — f(x)] / t is norm-to-norm continuous on X .

Corollary 3.5. Let f : X —> Y be a continuous mapping acting between Banach spaces X and Y . If f is almost C1 on X then the mapping (x ,y ) i—>• f'(x- ,y) is jointly norm continuous on X x X .

Proof. Fix xq G X . We shall show first that the mapping y (-*• f ' (xo;y) is linear. To do this it is sufficient to show that for each y* G Y* the mapping

SEPARATE AND JOINT CONTINUITY 45

V l—> V* ( f ' {x o', y)) is linear on X . Fix y* G Y* and let g : X —► M be de­fined by, g(x) := (y * o f)(x) , then g is continuous and almost C1 with g'{x\y) = y*(f'(x-,y)) for each y G X . It now follows, as in the finite dimensional case, (see [1], p. 261) that the mapping y »—> g'{xQ-,y) is linear on X [Note: g'(xo,y) is linear on X if it is linear on every 2 dimensional subspace of X], Next, let {tn : n G N} be any sequence of positive numbers converging to 0 and define fn : X +■ Y by, f n(y) := [f(x0 + tny) - f { x 0)\/tn. Then each f n is continuous and weak-lim^^oo f n(y) = f ' ( x o ;y ) for each y G X . Therefore by the Banach- Steinhaus Theorem y i—> f ' ( x o; y) is norm-to-weak continuous. Since y i—> f { x o; y) is linear it follows from the uniformly boundedness theorem that y i—> f ' (x o ;y ) is norm-to-norm continuous. The result then follows from Theorem 2.5. □

Rem ark 3.6. We say that a mapping / : X —> Y acting between Banach spaces X and Y is weakly C1 if for each y G X and y* € Y* the mapping x >■ (y* o f)'(x- ,y) is continuous on X . Now if Y is reflexive and for each fixed xo E X and y € X the mapping T*° : Y* —> M defined by, Ty°(y*) := (y* o f ) ' ( xo ; y ) is a bounded linear functional on Y* (note: this mapping is necessarily linear) then weak-lim£_>o[/(£o + ty) — f(xo)\/t exists. Moreover, by examining the proof of Corollary 3.5 we see that if f is norm-to-weak continuous and weakly C1 then for each xo € X such that each Ty° is bounded for all y G X , the mapping y f ' ( x o; y) is a bounded linear operator between X and Y.

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Jiling CaoDepartment of Mathematics The University of Auckland Private Bag 92019 Auckland N E W ZEALAND cao@math. auckland. ac. nz

Warren B. Moors Department of Mathematics The University of Auckland Private Bag 92019 Auckland N E W ZEALAND moors@mat h. auckland. ac. nz