\input amstex \documentstyle{amsppt} \magnification=\magstep 1 \TagsOnRight \NoBlackBoxes \leftheadtext{M. Megrelishvili (Levy), P. Nickolas and V. Pestov} \rightheadtext{Locally connected groups} \pageheight{23.8truecm} \pagewidth{15.8truecm} \voffset=-7mm % Macros \def\N{{\Bbb N}} \def\R{{\Bbb R}} \def\T{{\Bbb T}} \def\Z{{\Bbb Z}} % Endmacros % Counters for references \newcount\refHT \newcount\refHS \newcount\refHR \newcount\refItz \newcount\refItzk \newcount\refM \newcount\refP \newcount\refRD \newcount\refno \refno=0 \advance\refno by 1\refHT=\refno \advance\refno by 1\refHS=\refno \advance\refno by 1\refHR=\refno \advance\refno by 1\refItz=\refno \advance\refno by 1\refItzk=\refno \advance\refno by 1\refM=\refno \advance\refno by 1\refP=\refno \advance\refno by 1\refRD=\refno \topmatter \title Uniformities and uniformly continuous functions on locally connected groups \endtitle \author Michael Megrelishvili (Levy) $\P$, Peter Nickolas $\dag$ \\ and Vladimir Pestov $\ddag$ \endauthor \affil $\P$ Bar-Ilan University, {\smc Israel}\\ $\dag$ University of Wollongong, {\smc Australia}\\ $\ddag$ Victoria University of Wellington, {\smc New Zealand} \endaffil \address Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel \endaddress \email {\tt megereli$\@$bimacs.cs.biu.ac.il} \endemail \address Department of Mathematics, University of Wollongong, Wollongong, NSW 2522, Australia \endaddress \email {\tt peter$\_$nickolas$\@$uow.edu.au} \endemail \address Department of Mathematics, Victoria University of Wellington, P.O. Box 600, Well\-ing\-ton, New Zealand \endaddress \email {\tt vladimir.pestov$\@$vuw.ac.nz} \endemail \abstract{We show that the left and the right uniformities on a locally connected topological group $G$ coincide if and only if every left uniformly continuous real-valued function on $G$ is right uniformly continuous.} \endabstract \thanks Research supported by the New Zealand Ministry of Research, Science and Technology through the project ``Dynamics in Function Spaces'' of the International Science Linkages Fund, by the Israel Ministry of Science, and by the Mathematical Analysis Research Group of the University of Wollongong. \endthanks \subjclass 22A05 \endsubjclass \keywords {SIN-group, locally connected, neutral set, uniformly discrete} \endkeywords \endtopmatter \document A topological group $G$ is said to be a {\it SIN-group} if the left and the right uniform structures on $G$ coincide. Such is every compact and every abelian topological group; the linear Lie group $SL_2(\R)$ provides the best known specimen of a topological group that is not SIN. (Cf. \cite{\the\refHR, \the\refRD}.) An obvious corollary of the SIN property is that every left uniformly continuous real-valued function on $G$ is right uniformly continuous (and {\it vice versa}). Rather surprisingly, it is still unknown if the converse holds true. \proclaim{Open Question} \rom{(Itzkowitz, \cite{\the\refItzk})} Is a topological group $G$ SIN whenever every left uniformly continuous real-valued function on $G$ is right uniformly continuous? \endproclaim Firstly Itzkowitz \cite{\the\refItz} has shown that if a locally compact group is either unimodular or metrizable then the answer is ``yes''. For arbitrary locally compact groups the affirmative answer was obtained independently by Milnes \cite{\the\refM}, Itzkowitz \cite{\the\refItzk} and Protasov \cite{\the\refP}; moreover, Protasov proved that the answer to the problem is in the affirmative for a wider class of almost metrizable groups in the sense of Pasynkov (Cf. \cite{\the\refRD}.) Recently, Hansell and Troallic \cite{\the\refHT} have established an analogous result for a somewhat larger class of {\it q-groups}. In the present note we answer Itzkowitz's question in the affirmative for a class of topological groups going in a completely different direction. Recall that a topological group $G$ is {\it locally connected} if the connected open subsets form a base for $G$. \proclaim{Main Theorem} A locally connected topological group $G$ is a SIN-group if and only if every left uniformly continuous real-valued function on $G$ is right uniformly continuous. \endproclaim \vfill\eject Before proceeding to the proofs, we find it convenient to introduce the following {\it ad hoc} concept. \definition{Definition 1} Say that a topological group $G$ is a {\it functionally SIN-group} ({\it FSIN}, for short) if every left uniformly continuous real-valued function on $G$ is right uniformly continuous (or {\it vice versa}, which is the same). \enddefinition Now Itzkowitz's question can be reformulated as follows: are the properties SIN and FSIN equivalent for every topological group? \definition{Definition 2} We say that a subset $A$ of a topological group $G$ is {\it left neutral} in $G$ if for every neighbourhood $V$ of the identity in $G$ there is a neighbourhood $U$ of the identity such that $UA\subseteq AV$. In a similar way we define {\it right neutral} subsets. A subset that is both left and right neutral is said to be {\it neutral}. \enddefinition For example, every left-neutral symmetric subset or a precompact subset of a topological group are always neutral. The property of being a neutral subset appears in the paper \cite{\the\refHS} as the `(*)-property,' but we feel that our present terminology is justified being a straightforward extension of the previously known concept of a {\it neutral subgroup} \cite{\the\refRD}: a subgroup $H$ of a topological group $G$ is neutral if and only if it forms a neutral subset of $G$ in our sense. Every normal subgroup of a topological group is neutral. \definition{Definition 3} We say that a subset $A$ of a topological group $G$ is {\it left uniformly discrete} in $G$ if it is uniformly discrete with respect to the left uniform structure, that is, for a suitable neighbourhood $V$ of the identity the left translates $aV$ and $bV$ are disjoint whenever $a,b\in A$ and $a\neq b$. In a similar fashion we define the {\it right uniformly discrete} subsets. \enddefinition \proclaim{Theorem 1} Every left uniformly discrete subset of a FSIN group is left neutral. \endproclaim \demo{Proof} Let $A$ be a left uniformly discrete subset of a FSIN-group $G$. Let $V$ be an arbitrary neighbourhood of identity in $G$. We will prove that $UA\subseteq AV$ for a suitable neighbourhood $U$ of identity in $G$. Using the left uniform discreteness of $A$ and passing to a smaller neighbourhood if necessary, we can assume without loss of generality that $V$ is a symmetric neighbourhood of identity such that the left translates $aV^2$ and $bV^2$ are disjoint whenever $a,b\in A$ and $a\neq b$. One can choose a left uniformly continuous function $f\colon G\to\R$ with the properties $f(e_G)=1$, $f\vert_{G\setminus V}\equiv 0$, and $0\leq f(x)\leq 1$ for all $x\in G$. (One easy way to construct such a function is to choose a left invariant pseudometric $\rho$ on $G$ whose unit ball centred at zero is contained in $V$, cf. \cite{\the\refHR}, Th. 2.8.2, and set $f(x)=\min\{1,\rho(e_G,x)\}$.) For every $a\in A$, the function $f_a\colon G\to \R$, defined by letting $f_a(x)=f(a^{-1}x)$, is also left uniformly continuous (because $(a^{-1}x)^{-1}(a^{-1}y)=x^{-1}y$) and has the properties $f_a(a)=1$, $f_a\vert_{G\setminus aV}\equiv 0$, and $0\leq f_a(x)\leq 1$ for all $x\in G$. The family of open subsets $aV,a\in A$ is left uniformly discrete in $G$ by virtue of the choice of $V$. Indeed, if $x\in G$ is arbitrary, then the neighbourhood $xV$ of $x$ can meet not more than one set of the form $aV$: assuming $av=xw$ and $bu=xz$ with $v,u,w,z\in V$, one obtains $avw^{-1}=x=buz^{-1}$, where $vw^{-1},uz^{-1}\in V^2$, that is, $aV^2\cap bV^2\neq\emptyset$. As a corollary, the function $$\varphi(x)=\sum_{a\in A}f_a(x)\colon G\to\R$$ is well-defined and is left uniformly continuous. It has the properties $\varphi(a)=1$ for every $a\in A$, $\varphi\vert_{G\setminus AV}\equiv 0$, and $0\leq \varphi(x)\leq 1$ for all $x\in G$. By force of the assumed FSIN-property, $\varphi$ is right uniformly continuous. Therefore, there is a neighbourhood of identity $U$ in $G$ such that whenever $x,y\in G$ and $xy^{-1}\in U$, one has $\vert \varphi(x)-\varphi(y)\vert<1$. We claim that $UA\subseteq AV$. Indeed, let $u\in U$ and $a\in A$ be arbitrary. Since $(ua)a^{-1}=u\in U$, one has $\vert \varphi(ua)-\varphi(a)\vert<1$, that is,$\vert \varphi(ua)-1\vert<1$, and therefore $ua\in AV$. \qed\enddemo \proclaim{Lemma} Let $V$ be a neighbourhood of the identity in a topological group~$G$. Then there exists a set $A \subseteq G$ such that the left translates $aV$ and $bV$ are disjoint whenever $a, b \in A$ and $a \neq b$, and such that $AVV^{-1} = G$. \endproclaim \demo{Proof} Let $U = VV^{-1}$ and note that $U = U^{-1}$. Let ${\Cal A}$ be the set of subsets $A$ of~$G$ such that $aU \cap A = \{a\}$ for all $a \in A$. Clearly ${\Cal A}$ is non-empty and is partially ordered under set inclusion. If $\{A_\iota\}$ is a chain in ${\Cal A}$, put $A = \bigcup_{\iota} A_\iota$. We claim that $A \in {\Cal A}$. For if we fix $a \in A$, then $aU \cap A = aU \cap \bigcup_{\iota} A_\iota = \bigcup_{\iota} (aU \cap A_\iota)$. But for each $\iota$, $aU \cap A_\iota$ is empty if $a \notin A_\iota$ and is $\{a\}$ if $a \in A_\iota$, and it is clear that $a \in A_\iota$ for at least one value of $\iota$. Hence $aU \cap A = \{a\}$, and $A \in {\Cal A}$ as claimed. Therefore, by Zorn's lemma, there is a maximal set $A_0 \in {\Cal A}$. We claim that $A_0U = G$. If not, pick $a_0 \in G \setminus A_0U$. Then $a_0 \notin A_0$, and we have not only $a_0 \notin aU$ for all $a \in A_0$, but also $a \notin a_0U$ for all $a \in A_0$, by the symmetry of $U$. Therefore $A_0 \cup \{a_0\} \in {\Cal A}$, contradicting the maximality of $A_0$. Therefore $A_0U = G$, as claimed. It is now easy to see that $A = A_0$ is the set we require. For the last assertion proved above shows that $AVV^{-1} = G$, and if $a, b \in A$ and $a \neq b$, then $aV \cap bV = \emptyset$, since otherwise $b \in aVV^{-1} = aU$, contradicting the fact that $aU \cap A = \{a\}$. \qed \enddemo \proclaim{Theorem 2} Let $G$ be a locally connected topological group in which every left uniformly discrete subset is left neutral. Then $G$ is SIN. \endproclaim \demo{Proof} Let $\Cal O$ be an arbitrary neighbourhood of identity in $G$. We wish to find a neighbourhood of identity, $U$, such that $g^{-1}Ug\subseteq \Cal O$ for all $g\in G$. Let $V$ be a connected symmetric neighbourhood of identity in $G$ such that $V^5\subseteq\Cal O$. Choose a subset $A\subseteq G$ as in Lemma. Since $A$ is a left uniformly discrete subset, it is left neutral by the hypothesis. Therefore, one can find a connected neighbourhood of identity $U$ with $UA\subseteq AV$. Let $a\in A$ be arbitrary. We will show that $a^{-1}Ua\subseteq V$. The connected component of the set $AV$, containing $a$, is exactly $aV$, in view of connectedness of $V$. The set $Ua$ is contained in $AV$, is connected, and has a non-empty intersection with $aV$ ($\{a\}\subseteq Ua\cap aV$); therefore, it must be $Ua\subseteq aV$, that is, $a^{-1}Ua\subseteq V$. Now let $g\in G$ be arbitrary. By the choice of $V$, there exists a representation $g=avw$, where $a\in A$ and $v,w\in V$. One has: $$g^{-1}Ug=(avw)^{-1}Uavw= w^{-1}v^{-1}(a^{-1}Ua)vw \subseteq w^{-1}v^{-1}Vvw\subseteq V^5\subseteq \Cal O,$$ as required. \qed\enddemo Now the Main Theorem follows from Theorems 1 and 2 as an immediate corollary. \subheading{Acknowledgements} The third author (V.P.) is grateful to the Faculty of Science of the University of Wollongong and to the Department of Mathematics and Computer Science of the Bar-Ilan University for the hospitality extended in August 1994 and June 1995, respectively. \Refs \widestnumber\key{99} \vskip0.3truecm \ref\key \the\refHT \by G. 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