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ANOTE ON THE ARENS−ROYDEN THEOREM
FOR REAL BANACH ALGEBRAS
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N．L． Alling and I．． A． Campbe皿11ave血st fbmulated and pro鴨d the Arens−Royden theore皿f（）r real Banach algebras丘1［2］． Theiエmethod also has given another proof fbr the complex case． in the present note we 9ive a simple proofof this theorem for the rea1 Banach algebra， and the assumption血［2］that the algebra is semi−s血1Ple， is removed． ln our proof， we use the Arens−Royden theorem for complex algebras and a theorem on the eXistence of a solution fbr a㏄rtain equation血a complex Banach algebra． §1． Prelimillaries． Let B be a com皿1utative complex Banach algebra with identity 1≠0，and Xits maxima1 ideal space． Let B＊ denote the group of Umits of B． The Are血s− Royden theorem fbr the complex case’states that B＊／ exp（B）is cononically isomorphic to H1（X；Z）， the first 6ech cohomology group of X with integer coefUcients． Let A be a commutative real Ba皿㏄h algebra with identity， and．A＊the group of units of A． Let B＝A⑧CI）e the compleXification of A；Y and X max血al idea1 spaoes of A and B respectively， both compact；reY＝｛ハr∈r；A／N≡R｝，a closed subset of Y． The ．map p；X→Y is defined as p（M）＝M∩Af（）r M∈X（s㏄【1】）． ln the folloWing we re− gard A as being isometrically imbedded in B． Let∠4幸＝｛〃∈A＊；諺 is positive on p−1（reη｝，where ti∈C（X）is the Gelfand transform of u∈A⊂B and C（iX） denotes the血g of a皿complex・valued continuous f迦ctions on X． For b＝u十ン＝了y（〃， v∈A）σ（b）＝u−∨ノ＝iγ， then 6 is an R−automOrphism of B of order two haVing A as the set of a皿its fixed elements． Thisσinduces an祖volutionτ on X． For， f∈C（X），1etδσ）（M）＝∫（τ（M））（M∈X）， thel1∂is an R−automorPhism of C（X）of order two for whiCh A“ consists of its 6xed elements， and the fbllowmg diagram is commutative： B△c（x） 1・ la BムC（幻． 1 in the foilowing we denote a again 1）Y a for the sake of simplicity， and also regard that G＝ ｛1，τ｝acts on」Xas the topological transfbnnation group． In this case， poτ＝pand】r is the quotient space of ．M by G． For〃∈A， put ＊ Received June 15，1975 5f
6
K．FURUTAN【
1 ex・ω一￡。（2πμn！）n∈A・． The Arens−Royden theorem for rea1 algebras is given as follows（［2］）．T田・剛・乃・佗・酷白ψ・力伽i・g・鋤・i・al・i・・m・励紘
．A＃／expω…≡」m（γ；Z（う，in whi・h za d・n・tes th・sheaf・ηr功励な㎞ば妙磁㎎功・ω〃蹴’吻㎡Z。n Xas．加
§2 and is〃o’〃ecessarily coπ8如η’W4 1’ §2・ Lemmas・ Let r8 and｛B’＊ be the sheaves of continuous complex−valued functions on X and th・t・f・v・rywh・・e n・n−zer・fun・ti・n・re・p㏄ti・ely， th・n・th・f・ll・wi・g、eq。，n㏄。f sheaves on X is exact： o−z＿＿▽里▽・＿＿o タ whe「・Ker（剛i・㎞・面・tdy・㏄n n・t t・b・th・・heaf・Z・fr・m th・d・血iti・n。f th。 maP exp in§1・but it may be ident近ed with it． This fact plays a．fUndamental roll fbr the com． Plex case． Next， we de丘ne the sheaf（7a on y as fb皿ows： 1et gG（の＝｛f∈r（P−1（の，9）；・（∫）＝∫・np−・（の｝ wher・7i・an arbit町・Pe・・et i・y・nd・r（P−・の，▽）d斑・t・・th・・㏄ti・n・・。 p−・（V）， °「the funct’°ns°n’t・ The「i・f°「・∈Yth・・眺・n…9；一’・g；：皿…’σ（n・・Th・・h・av・・ Zσand（9＊）G・re d・血・d血th・・am・w・y． L・t影・b・a・ub・he・f。f（9・）・，u。h th。t 「（v，se’＃）＝｛∫∈「（P−1（の・9＊）；・ω一∫・・p−・（V）and∫i・p・・iti…np−・（・er）｝．Then，
L鴫1．πeW〃θ〃cθげsheaves on Y
・−z・＿9・蟹許＿。
岳徽・L疏・θ舵屹㎡⑭・）xpa・i・・ua’urqlly definedfr・励・卿exp・9」・9・．凧．
吻’exp。σ＝σ。exp．）・取・・F・臨tus＆・t pr・ve・that・expa（gny⊂99＃． lf∫∈r（K⑳，∫i，噺．曲。d。n
〆（γ∩・eη・孤d…）xpσ（∫）＝Σ（2・tf）・／n！i・p・・i輪・n it． H・nce、exp・（殉⊂eW・．C・n…sely・㎞an・ighb・血・・dγ・f・nyfix・dp・intア・∈Y，1・tg∈r（K影・）b，9ive。．
Th・n・w・・regard・9…。・ntinu…fun・ti・n皿ρ一・（v）・u・h th・t8（M）＝9（T（M））fb，」lf∈ P−1（の・nd p・・itive・n p−1（V∩ren．恥・γ・， the・f・皿・wing・tw・ca・e・tak。 place、 （1）アo∈re】フ（2）アo申eヱ 麺hC血・tば・・as・9（P−1ω）＞Oby t曲gγ・副・n・・9h，・・e、m・y、as・㎜・th。tg（P−・ （の）i・c・nt・in・d in th・right half pl・砥S垣・e・w・・cap・tak・the・1・9・・ithm・in・th。⑰t h。lf P1・ne・th鵬exi・ts∫∈r（V， 9e such that exp・（∫）＝9： Ass㎜・n・w th・t②h・ld・・Sinc・・rel’i・，d・・ed， we・皿ass㎜・th・t V∩・ey＝Oby’
ANOTE ON THE ARENS−ROYDEN THEOREM FOR REAL
7
taking Vsmall enough． Let p−1（yo）＝｛xo， x1｝．As p is a two−to−one local homeomorphism on p−1（V），1）y taking 7 fUrther small enough if n㏄essary， we May assume that p−1（V）is ・devided into two open sets Uo and Ui such that Uo∩ひt＝の，τ（Uo）＝＝ひt and xi∈Ui， Ui is homeomorphic to V under the map p for i ＝1，0， and g（Uo）is contained in a simply connected open set in C−｛0｝．Then let us de丘nef∈r（V，｛9ea）in such a way that 炸｛篇器， where／b（x）＝logl8・（X）］for x∈Uo．’lhis f satisfies that ext）G（の＝・g． So we have proved ’that expa（（7a）：＝99＃． The exactnesses of other places can be proved without di丘icU lty． REMARK． If re］r＝の， then彫＃＝（9＊）a． From Lemma 1． we get the following lernma． LEMMA 2． 7heノわ〃bwing sequence is exact： ∂σ0→∬oα；za）→五〇（Y；｛i7a）→HO（γ；．ev＃）一一m（τ；ZG）→m（Y；▽（り＝0．
咀ere∂G is the co皿㏄ting holnomorphism． Note that H1（γ；▽G）＝O is deduced from the fact that Hl（x；▽）＝Oand Hl（r；9a）・→1担（X；（7）is i叫jective． We also n㏄d the fbUowing proposition which is fUndamental to our proof of the itheorem． Its proof can be fbund f（）r instance in［3】．PR（疋osmON． Le’βbεa eomplex Ba耽h algebra・∠∬u〃2θ吻’Σ墾一〇舗＝0カr
s・me・Z∈C（X）， whe・θα毒∈3・ぷ卿・se・m・励yε励α’Σ纂、・蹴一i is zer・η・吻rθZ〃X・カε励θr・醐ぴω∈βぷuch吻’φ＝z and ￡ ？一。 ait・i＝0仇B．
§3．Proof・of・the theorem．0→丑・α；Z）→五・α；9）聖H・（X；卵一・H・（X；Z）→O
ll ↑ ↑ ll ベ ベ0→、HO（X；Z）→8−一→β＊、 →孕α；Z）→0
。−H。（し、Zり一｝＿＿−2。ユ＿」、（皇Zり一。
ll ↓ ↓ ∂∋l
OtHO（r；ZG）→HO（γ；▽G） 〈 【∂g is the restriction of the connecting homoniorphism∂ p＊is canonically deiined 1）y the map p：X→η 〈We consider the following commutative diagram：
∂ →HO（γ；Eee＃）→Hl（y；za）→0． ato A＃， and the homomorphism In the diagra叫AA⊂Ho（Y；r㌘G）〈＝Ho（2r；（7）＝C（X）and（∠4＃）⊂Ho（γ；彩ク＃）⊂1望o（X；《♂＊）， The exactness of the Second row is the very theorem for the complex algebra． The血st ｛md the fbrth rows are also exact． P・tδ＝∂9・〈・A・→傘→U（y；zり．th。。 t。 P，。v。 th。 i，。m。rphi、m， A・1。xP （A）≡1昇（γ；Zσ）in our theorem， it is sufficient to show that Ker（δ）＝exp（∠）and that δiS SUIjective．8
