Local and 2-local isometries between absolutely continuous function spaces

Ope a o s and Ma ices Volume 15, Numbe 4 (2021), 1461–1468 doi:10.7153/oam-2021-15-91 LOCAL AND 2–LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES MALIHEH HOSSEINI AND JUAN J. FONT (Communica ed by L. Moln´a ) Abs ac . In his pape we gi e a comple e desc ip ion o local and 2-local isome ies de ined be ween spaces o scala - alued absolu ely con inuous unc ions on a bi a y (no necessa ily compac ) subse s o he eal line wi h a leas wo poin s. 1. In oduc ion In he las yea s conside able wo k has been done on local maps, an a ea whose main p oblem o esea ch is whe he he local ac ions o some impo an classes o ans o ma ions (like de i a ions, au omo phisms, isome ies) on a gi en space de e - mine he class unde conside a ion comple ely. We e e he eade o [13] o mo e in o ma ion. In pa icula , gi en wo Banach spaces Aand B, he space o all bounded linea ope a o s om A o B,L(A,B)and S⊆L(A,B), a linea map T:A−→ Bis said o be locally in Si o each a∈A, he e exis s a map Ta∈Ssuch ha Ta =Taa. Simila ly, a map T:A−→ B(which is no assumed o be linea ) is said o be 2-locally in Si o any pai a,a′∈A, he e is a map Ta,a′∈Ssuch ha Ta =Ta,a′(a)and Ta′=Ta,a′(a′). Local isome ies a e an ac i e esea ch a ea cen e ed in he s udy o he alge- b aic e lexi i y o ce ain unc ion spaces. Le us ecall ha a Banach space Ais algeb aically e lexi e i any linea map on Abelonging locally o he g oup o all su jec i e linea isome ies is su jec i e. Fo example, i is known (see [14] and [2]) ha C(X,C)( esp. C0(X,C)) is algeb aically e lexi e p o ided Xis a i s coun able compac space ( esp. Xis locally compac space whose one-poin compac i ica ion is me izable). Besides, C(X,E)is algeb aically e lexi e i Xis a i s coun able com- pac space and Eis a ini e-dimensional complex Banach space o Eis a uni o mly con ex and algeb aically e lexi e Banach space (see [9]). One can also ind ecen esul s ela ed o he algeb aic e lexi i y o Banach algeb as o Lipschi z unc ions in Ma hema ics subjec classi ica ion (2020): P ima y 47B38; Seconda y 46J10, 47B33. Keywo ds and ph ases: Linea su jec i e isome y, absolu ely con inuous unc ions, local isome y, 2-local isome y. This wo k was pa ially suppo ed by a g an om he IMU-CDC. J. J. Fon was suppo ed by Spanish AEI P ojec PID2019-106529GB-I00/AEI/10.13039/501100011033 and by Uni e si a Jaume I (P ojec e UJI-B2019-08). c  D l , Zag eb Pape OaM-15-91 1461 1462 M. HOSSEINI AND J. J. FONT [15]. I is wo h men ioning ha li le is known conce ning spaces o eal- alued unc- ions since he main ool o he abo e esul s, he Gleason-Kahane- ˙ Zelazko heo em, only applies o complex algeb as. On he o he hand, 2-local maps we e in oduced by ˇ Sem l in [16] when ying o d op he linea i y assump ion o ce ain local maps. Subsequen ly, many au ho s ha e wo ked on hese maps. Namely Gy¨o y ([4]) p o ed ha e e y 2-local isome y on C0(X,C)is a su jec i e linea isome y p o ided Xis a i s coun able, σ -compac , locally compac space. La e , in [1], Al-Halees and Fleming gene alized Gy¨o y’s esul s o C0(X,E) o σ -compac me ic spaces Xunde ce ain condi ions on he Banach space E. Simila esul s ha e been achie ed o o he unc ion spaces such as, o ins ance, (poin ed) Lipschi z spaces ([10,11,12]), uni o m algeb as ([5,12]) and spaces o unc ions o bounded a ia ion ([7]). In his pape we gi e a comple e desc ip ion o local and 2-local isome ies de- ined be ween spaces o scala - alued absolu ely con inuous unc ions on a bi a y (no necessa ily compac ) subse s o he eal line wi h a leas wo poin s. In pa icula , we ex end some p e ious esul s in [6,7] o a noncompac amewo k. We would like o ema k ha , in mos o he pape s men ioned abo e, he compaci y o he unde lying spaces (o local compaci y wi h unc ions anishing a in ini y) plays a c ucial ole. 2. P elimina ies Le Xbe a subse o he eal line Rwi h a leas wo poin s. Le us ecall ha a scala - alued unc ion on Xhas bounded a ia ion i he o al a ia ion V( )o is ini e, ha is, V( ):=sup(n ∑ i=1 | (xi)− (xi−1)|:n∈N,x0,x1,...,xn∈X,x0<x1< .. . < xn)<∞. Mo eo e , a scala - alued unc ion on Xis said o be absolu ely con inuous i gi en ε >0, he e is a δ >0 such ha n ∑ i=1 | (bi)− (ai)|< ε , o each ini e amily o non-o e lapping open in e als {(ai,bi):i=1,···,n}whose ex eme poin s belong o Xand ∑n i=1(bi−ai)< δ . We deno e by ACb(X) he space o all scala - alued absolu ely con inuous unc ions o bounded a ia ion on X, endowed wi h he no m k · k =max{k · k∞,V(·)}, whe e k · k∞s ands o he sup emum no m o a unc ion. No e ha when Xis bounded, each absolu ely con inuous unc ion is au oma ically o bounded a ia ion. In he sequel, by Xwe deno e he closu e o Xin R. Also, o any ∈ACb(X), le be he unique absolu ely con inuous ex ension o o he closu e Xo X, which exis s by [8, Lemma 3.1]. Gi en wo Banach spaces Aand B, we shall deno e by Iso(A,B) he se o all su jec i e linea isome ies om Aon o B. When A=B, we shall w i e Iso(A)ins ead o Iso(A,A). LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES 1463 3. Local isome ies o absolu ely con inuous unc ion spaces In he sequel, Xand Ywill be wo a bi a y (no necessa ily closed no bounded) subse s o Rwi h a leas wo poin s. By Iso1(ACb(X),ACb(Y)) we deno e he se o all su jec i e linea isome ies T: ACb(X)−→ ACb(Y)such ha T1 is bounded away om ze o, i.e., he e exis s >0 such ha , o each y∈Y, we ha e |T1(y)|⩾ . Indeed, by [8, Theo em 3.13], we know ha Iso1(ACb(X),ACb(Y)) is he se o all su jec i e linea isome ies T:ACb(X)−→ ACb(Y)o he o m o a weigh ed composi ion ope a o . The ollowing esul , which is used se e al imes in ou p oo s, desc ibes he o m o hese isome ies. THEOREM 3.1. ([8, Theo em 3.13]) I T ∈Iso1(ACb(X),ACb(Y)), hen he e exis a unimodula scala λ and a mono onic homeomo phism ϕ :Y−→ X such ha T = λ ◦ ϕ o all ∈ACb(X). Mo eo e , i is wo h poin ing ou ha , acco ding o Co olla y 3.15 in [8], o he case X(and so Y) is connec ed we ha e Iso1(ACb(X),ACb(Y)) = Iso(ACb(X),ACb(Y)). Le us nex adap he concep s men ioned in he in oduc ion o ou con ex : DEFINITION 3.2. A linea map T:ACb(X)−→ ACb(Y)which is locally in Iso1(ACb(X),ACb(Y)) is called a local isome y. In ac , Tis a local isome y i o each ∈ACb(X) he e is a su jec i e linea isome y T ∈Iso1(ACb(X),ACb(Y)) (de- pending on ) such ha T =T . We can now p o ide a comple e desc ip ion o local isome ies de ined be ween spaces o absolu ely con inuous unc ions in a noncompac amewo k, which is a gen- e aliza ion o [6, Theo em 2.1]. THEOREM 3.3. I T :ACb(X)−→ ACb(Y)is a local isome y, hen he e exis a mono onic homeomo phism ϕ :Y−→ X , and a scala λ wi h | λ |=1such ha T (y) = λ ( ϕ (y)) ( ∈ACb(X),y∈Y). P oo . By [8, Lemma 3.1], each absolu ely con inuous unc ion has a unique ab- solu ely con inuous unc ion ex ension o he closu e o he unde lying space wi h he same o al a ia ion. This allows us o conside he map S:ACb(X)−→ ACb(Y)de- ined by S( ) = T o all ∈ACb(X). I is easy o see ha Sis a local isome y. The e o e we can assume, wi hou loss o gene ali y, ha Xand Ya e closed subse s o he eal line. Mo eo e , since Tis a local isome y, we in e om Theo em 3.1 ha T1 is a unimodula cons an unc ion. Hence, by conside ing T T1ins ead o T, we can assume, wi hou loss o gene ali y, ha T1=1. We now con inue he p oo h ough se e al s eps. S ep 1. Fo each ∈ACb(X),kT k∞=k k∞and kT k=k k. Since Tis a local isome y, i is an immedia e consequence o Theo em 3.1. 1464 M. HOSSEINI AND J. J. FONT REMARK. Be o e con inuing wi h he p oo , le us no ice ha i we had assumed compaci y, we could ha e exploi ed he densi y o ACb(X)in C(X)and would ha e ob ained he ep esen a ion o T om he amous Holsz y´nski Theo em (see, e.g., [3, Theo em 2.3.10]). Howe e , in his noncompac amewo k, such densi y is no alid (see [8, Rema k 3.4]) and his o ces us o use ano he app oach in o de o desc ibe T. Be o e s a ing he nex s ep, we need o ix some no a ion. Fo each x∈X, we se Fx:={ ∈ACb(X):k k∞=1= (x)} which is clea ly non-emp y. Mo eo e , we also de ine Ix:= {MT : ∈Fx}, whe e MT :={y∈Y:|T (y)|=1=kT k∞}. S ep 2. Gi en x∈X,Ixis a non-emp y subse o Y. Le us de ine e Ix:= {M T : ∈Fx}, whe e T is he unique ex ension o T o he S one- ˇ Cech compac i ica ion, β Y, o Y. And, as expec ed, M T ={y∈ β Y:| T (y)|=1=k T k∞}, which is ob iously a non-emp y compac subse o β Y. We now p o e, by means o a s anda d echnique in a compac amewo k (see, e.g., [8, Lemma 3.8]), ha e Ixis a nonemp y subse o β Y. To his end, le 1,..., n∈ Fx. De ine =∑n i=1 i n. Clea ly, ∈ACb(X)wi h k T k∞=kT k∞=k k∞=1, by S ep 1. Hence he e is a poin y∈ β Ysuch ha | T (y)|=1, and so 1=| T (y)|= n ∑ i=1 T i(y) n⩽ n ∑ i=1 | T i(y)| n⩽ n ∑ i=1 k T ik∞ n= n ∑ i=1 k ik∞ n=1, which implies ha | T i(y)|=1 o e e y i∈ {1,...,n}. Thus y∈n T i=1 M T i. The e o e, he amily {M T : ∈Fx}has he ini e in e sec ion p ope y and hen e Ix6=/0, as desi ed. Finally, le us check ha Ix=e Ix. Take ∈Fxwi h compac suppo and { ∈ X:| ( )|=1}={x}. F om he ep esen a ion gi en by Theo em 3.1, i ollows ha T has compac suppo in Y. As a consequence, e Ix⊆Ybecause i is clea ha e Ix⊆Supp(T ). The e o e, Ix=e Ixis a non-emp y subse o Y, as equi ed. S ep 3. I x∈Xand ∈ACb(X)wi h (x) = 0, hen T (y) = 0 o all y∈Ix. This s ep is e i ied by a a gumen simila o he p oo o Lemma 3.10 in [8]. Con a y o wha we claim, suppose ha x∈Xand ∈ACb(X)wi h (x) = 0, bu T (y)6=0 o some y∈Ix. Le >k k∞and choose h∈Fxsuch ha 0 ⩽h⩽1 LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES 1465 and k| |+ hk∞=k ± hk∞= , by [8, Lemma 3.6 (1)]. Since y∈Ix,|Th(y)|=1, and so we ha e =k ± hk∞=kT( ± h)k∞ ⩾max{|T (y)+ Th(y)|,|T (y)− Th(y)|} > , which is impossible. This comple es he p oo o S ep 3. S ep 4. Fo any wo dis inc poin s xand x′in X,IxTIx′=/0. Assume, on he con a y, ha he e exis s a poin yin IxTIx′. Choose a unc ion ∈Fxwi h (x′) = 0. By S ep 3, T (y) = 0 because y∈Ix′. On he o he hand, |T (y)|=1 since ∈Fxand y∈Ix, which is impossible. This a gumen yields IxTIx′=/0. We can now in oduce he ollowing nonemp y subse o Y: Y0:={y∈Y:y∈Ix o some x∈X}. Besides, we can de ine a mapping ϕ :Y0−→ Xsuch ha ϕ (y) = xi y∈Ix. Acco ding o he p eceding s ep, ϕ is well-de ined. Mean ime, i is clea ha ϕ is su jec i e. S ep 5. Fo each ∈ACb(X)and y∈Y0, we ha e T (y) = ( ϕ (y)). Le ∈ACb(X)and y∈Y0. Since ( − ( ϕ (y)))( ϕ (y)) = 0, we ha e T( − ( ϕ (y)))(y) = 0 by S ep 3. Hence T (y) = T( ( ϕ (y)))(y) = ( ϕ (y)), as desi ed. S ep 6. Y0=Y. Le us i s de ine he unc ion h:X−→ Rby h(x) = (2+1 x−1i x∈(−∞,0)∩X, 1 x+1i x∈[0,+∞)∩X. Clea ly, his an injec i e unc ion which belongs o ACb(X)wi h V(h)⩽2. Since T is a local isome y, he e exis a mono onic homeomo phism ϕ h:Y−→ X, and a scala λ hwi h | λ h|=1 such ha Th =Thh= λ h(h◦ ϕ h). Hence, om S ep 5, we ge h( ϕ (y)) = λ hh( ϕ h(y)) (y∈Y0), which aking in o accoun ha | λ h|=1 and h⩾0, implies ha h( ϕ (y)) = h( ϕ h(y)) (y∈Y0). Hence ϕ (y) = ϕ h(y) o all y∈Y0because his injec i e. Now we can p o e ha Y0=Y. O he wise, he e would exis a poin y0∈Y Y0. Se x0= ϕ h(y0). Since ϕ is su jec i e, he e is a poin y∈Y0such ha x0= ϕ (y). Hence, om he abo e discussion, we conclude ha ϕ h(y0) = ϕ h(y), which con adic s he injec i i y o ϕ h. The e o e, Y0=Y. Then, as obse ed abo e, ϕ = ϕ his a mono onic homeomo phism om Yon o X, and also we ha e T (y) = λ ( ϕ (y)) ( ∈ACb(X),y∈Y), which comple es he p oo o he heo em.  1466 M. HOSSEINI AND J. J. FONT 4. 2-Local isome ies o absolu ely con inuous unc ion spaces In his sec ion we p esen a comple e desc ip ion o 2-local isome ies be ween ab- solu ely con inuous unc ion spaces. Fi s le us ecall he de ini ion o 2-local isome- ies in he ollowing. DEFINITION 4.1. A map T:ACb(X)−→ ACb(Y)(no linea i y no su jec i i y a e assumed) is called a 2-local isome y i i belongs 2-locally o Iso1(ACb(X),ACb(Y)), i.e., o e e y ,g∈ACb(X) he e exis s a T ,g∈Iso1(ACb(X),ACb(Y)) such ha T = T ,g and T g =T ,gg. THEOREM 4.2. Fo each 2-local isome y T :ACb(X)−→ ACb(Y), he e exis a mono onic homeomo phism ϕ :Y−→ X , and a scala λ wi h | λ |=1such ha T (y) = λ ◦ ϕ o all ∈ACb(X). P oo . We will use a modi ica ion o he p oo p o ided in [7, Theo em 2.5]. As in he beginning o he p oo o Theo em 3.3, we can assume, wi hou loss o gene ali y, ha Xand Ya e closed subse s o R. Since T1 is a unimodula cons an unc ion, we can assume, wi hou loss o gene ali y ha Tis uni al, i.e., T1=1. Then, ollowing he p oo o [7, Theo em 2.5] (see also [11, Theo em 2.1]), one can deduce ha o each x∈X, he se Ix:=T ∈ACb(X)Ex, is a single on, whe e Ex, ={z∈Y:T (z) = (x)}. By ψ (x)we deno e he unique poin in Ix. This allows us o de ine an injec i e map ψ :X−→ Ysuch ha T ( ψ (x)) = (x) o all x∈Xand ∈ACb(X). Now, aking Y0:= ψ (X)and ϕ := ψ −1we will ha e he bijec i e map ϕ :Y0−→ Xsuch ha T (y) = ( ϕ (y)) ( ∈ACb(X),y∈Y0). We now claim ha Y0=Y. Le hbe de ined as in he p oo o Theo em 3.2 (S ep 6). Since Tis a 2-local isome y, he e exis s Th,1∈Iso1(ACb(X),ACb(Y)) such ha T1,h(1) = 1 and T h =T1,h(h). Acco ding o Theo em 3.1, he e is a mono onic homeo- mo phism ϕ 1,h:Y−→ Xsuch ha T1,h(h) = h◦ ϕ 1,h. Combining he la e equa ions, we ge h( ϕ (y)) = h( ϕ 1,h(y)) (y∈Y0), which easily implies ha ϕ = ϕ 1,hon Y0because o he injec i i y o h. In o de o p o e ha Y0=Y, le ybe an a bi a y poin in Y. Since ϕ is su jec- i e, he e exis s y0∈Y0wi h ϕ (y0) = ϕ 1,h(y). On he o he hand, om he p e ious pa ag aph we ha e ϕ (y0) = ϕ 1,h(y0), which implies ha ϕ 1,h(y0) = ϕ 1,h(y). Conse- quen ly, y=y0because ϕ 1,his injec i e, which yields y∈Y0. The e o e, we can conclude ha Y0=Y. Ga he ing all he in o ma ion, we in e ha ϕ :Y−→ Xis a mono onic homeo- mo phism ( ϕ = ϕ 1,h) and also T (y) = ( ϕ (y)) ( ∈ACb(X),y∈Y), as equi ed.  LOCAL ISOMETRIES BETWEEN ABSOLUTELY CONTINUOUS FUNCTION SPACES 1467 REMARK 4.3. I should be no ed ha , o he complex case, one can exploi he sphe ical e sion o he Kowalski-Słodkowski heo em p o ided in [12] o ob ain he desc ip ion o 2-local isome ies (see he ema k a he end o [7]). Mo e p ecisely, le T:ACb(X)−→ ACb(Y)be a 2-local isome y. Fo each y0∈Y, de ine he map Ty0:(ACb(X),k · kΣ)−→ Cby Ty0 =T (y0)( ∈ACb(X)), whe e k · kΣ=k · k∞+ V(·). Clea ly, since Tis a 2-local isome y, Ty0is 1-homogeneous, and acco ding o Theo em 3.1, o each pai ,g∈ACb(X), he e exis a scala λ ,g∈Tand a mono onic homeomo phism ϕ ,g:Y−→ Xsuch ha Ty0 = λ ,g ( ϕ ,g(y0)) and Ty0g= λ ,gg( ϕ ,g(y0)). Then Ty0 −Ty0g= λ ,g( ( ϕ ,g(y0))−g( ϕ ,g(y0))) ∈T σ ( −g), whe e σ ( −g)is he spec um o −g. By P oposi ion 3.2 in [12] i ollows ha Ty0is linea . Consequen ly, Tis a linea map since y0was a bi a y, which especially implies ha T:ACb(X)−→ ACb(Y)is a local isome y. Thus he esul ollows om Theo em 3.2. Acknowledgemen s. We hank he e iewe o his/he aluable commen s and sugges ions on he manusc ip . R E F E R E N C E S [1] H. AL-HALEES, R. J. FLEMING,On 2-local isome ies on con inuous ec o - alued unc ion spaces, J. Ma h. Anal. Appl. 354 (2009), 70–77. [2] F. CABELLO S´ ANCHEZ,Local isome ies on spaces o con inuous unc ions, Ma h. Z. 251 (2005), 735–749. [3] R. J. FLEMING, J. E. JAMISON,Isome ies on Banach Spaces: Func ion Spaces, Chapman Hall/CRC Monog . Su . 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Toosi Uni e si y o Technology Teh an, 16315-1618, I an e-mail: [email p o ec ed] Juan J. Fon Depa amen o de Ma em´a icas Uni e si a Jaume I Campus Riu Sec 8029 AP, Cas ell´on, Spain e-mail: [email p o ec ed] Ope a o s and Ma ices www.ele-ma h.com [email p o ec ed]