Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions

CHARA CTERIZA TION OF A BANA CH-FINSLER MANIF OLD IN TERMS OF THE ALGEBRAS OF SMOOTH FUNCTIO NS J.A. JARAMILLO, M. JIM ´ ENEZ-SEVILLA AND L. S ´ ANCHEZ-GONZ ´ ALEZ Abstra ct. In this note w e giv e sufficient conditions to ensure that the w eak Finsler stru ct ure of a complete C k Finsler manifold M is determined b y the normed algebra C k b ( M ) of all real-v a lued, b ounded and C k smooth functions with b ound ed deriv ativ e defined on M . As a consequen ce, w e obtain: (i) the Finsler stru cture of a finite- dimensional and complete C k Finsler manifold M is determined by the algebra C k b ( M ); (ii) the weak Finsler structure of a separable and complete C k Finsler manifo ld M mo deled on a Banac h sp ace with a Lipschitz and C k smooth bump function is d etermined by the alg ebra C k b ( M ); (iii) the wea k Finsler structure of a C k uniformly bumpable and complete C k Finsler manifol d M mo deled on a W eakly Compactly Generated (WCG) Banac h space with an (equiv alen t) C k smooth norm is determined by the algebra C k b ( M ); and (iv) th e isometric structure of a WCG Banach space X with an C 1 smooth bump fun ction is determined b y the al gebra C 1 b ( X ). 1. Introduction a n d Pre l iminaries In this note, we are inte rested in charac terizing the Finsler structure of a Finsler manifold M in terms of the sp ace of real-v a lued, b ounded and C k smo oth functions with b ounded d er iv ativ e defi ned on M . The problem of the in terrelation of the top o- logica l, m etric and smo oth stru cture of a space X and the algebraic and top olo gical structure of the sp ace C ( X ) (the set of real-v alued con tin uous functions defi ned on X ) has b een largely stud ied. These results are u sually referred as Banach-Stone typ e the or ems . Recall the celebrated Banac h-Stone theorem, asserting that th e compact spaces K an d L are h omeomorphic if and only if the Banac h spaces C ( K ) and C ( L ) endo w ed with the su p-norm are isometric. F or more information on Banac h -Stone t yp e theorems see the su rv ey [ 10 ] and references therein. The My ers-Nak ai theorem state s that the structure of a complete Riemann ian manifold M is c haracterized in terms of the Banach algebra C 1 b ( M ) of all real-v alued , b ound ed and C 1 smo oth fu nctions with b ounded d er iv ativ e defin ed on M endo w ed with th e sup-norm of the fun ction and its d eriv ativ e. More sp ecifical ly , t w o complete Riemannian manifolds M and N are equiv alen t as Riemannian manifolds, i.e. there is a C 1 diffeomorphism h : M → N su c h th at h dh ( x )( v ) , dh ( x )( w ) i h ( x ) = h v , w i x Date : August, 2011. 2010 Mathematics Subje ct Classific ation. 58B10, 58B20, 46T05, 46T20, 46E25, 46B20, 54C35. Key wor ds and phr ases. Finsler manifolds, algebras of smooth functions, geometry of Banac h spaces. Supp orted in part by DGES (Spain) Pro ject MTM2009-07 848. L. S´ anchez-Gonz´ alez has also b een sup p orted by gran t MEC AP2007-00868. 1 2 J.A. JARAMILLO, M. JIM ´ ENEZ-SEVILLA AND L. S ´ ANCHEZ-GONZ ´ ALEZ for ev ery x ∈ M and v , w ∈ T x M if and only if the Banac h alge bras C 1 b ( M ) and C 1 b ( N ) are isometric. Th is r esult was fir st p r o ved by S. B. My ers [ 22 ] for a compact and Riemann ian manifold and by M. Nak ai [ 23 ] for a fi nite-dimensional Riemannian manifold. V ery recen tly , I. Garrido, J.A. J aramillo and Y.C. Rangel [ 12 ] ga ve an extension of the My ers-Nak ai theorem for ev ery in finite-dimensional, complete Rie- mannian m anifold. A similar result f or the so-called finite-dimensional Riemannian- Finsler manifolds is given in [ 14 ] (see also [ 26 ]). Our aim in this work is to extend the Mye rs-Nak ai theorem to the con text of Finsler manifolds. On the one hand, we obtain the Myers-Nak ai th eorem for (i) finite-dimensional and complete Finsler manifolds, and (ii) WCG Banac h sp aces with a C 1 smo oth bu mp function. On the other hand , w e stud y for k ≥ 1 the algebra C k b ( M ) of all real-v alued, b oun ded and C k smo oth functions w ith b ounded first der iv ativ e defin ed on a complete Finsler manifold M . W e pr o ve that these algebras determine the w eak Finsler structur e of a complete Finsler m anifold when k = 1 and the Finsler structure when k ≥ 2. In particular, w e obtain a wea k er v ersion of the My er s -Nak ai theorem for (i) separable and complete Finsler m anifolds mo deled on a Banac h space with a Lip s c h itz and C k smo oth b ump fun ction, and (ii) C k uniformly bum p able and complete Finsler manifolds mo deled on W CG Banac h spaces with an equiv alen t C k smo oth norm. In the pro of of these resu lts w e will use the ideas of the Riemannian case [ 12 ]. The notatio n w e u s e is standard. The norm in a Banac h space X is denoted b y || · || . The dual sp ace of X is denoted b y X ∗ and its dual norm b y || · || ∗ . The op en b all with cen ter x ∈ X and radiu s r > 0 is d enoted by B ( x, r ). A C k smo oth bump function b : X → R is a C k smo oth f unction on X with b ounded, non- empt y supp ort, where supp( b ) = { x ∈ X : b ( x ) 6 = 0 } . If M is a Banac h m anifold, we denote by T x M the tangen t space of M at x . Recall that the tangen t b undle of M is T M = { ( x, v ) : x ∈ M and v ∈ T x M } . W e refer to [ 6 ], [ 8 ], [ 19 ] and [ 7 ] for additional definitions. W e will sa y that the norms || · || 1 and || · || 2 defined on a Banac h space X are K -equiv alent ( K ≥ 1) whether 1 K || v || 1 ≤ || v || 2 ≤ K || v || 1 , for every v ∈ X . Let us b egin b y recalling the definition of a C k Finsler manifold in the sense of P alais as well as some basic prop erties (for more information ab out these manifolds see [ 25 ], [ 7 ], [ 27 ], [ 24 ], [ 13 ] and [ 18 ]). Definition 1.1. L et M b e a (p ar ac omp act) C k Banach manifold mo dele d on a Ba- nach sp ac e ( X , || · || ) , wher e k ∈ N ∪ {∞} . L et us c onsider the tangent bund le T M of M and a c ontinuous map || · || M : T M → [0 , ∞ ) . We say that ( M , || · || M ) is a C k Finsler manifold in the sense of Palais if || · || M satisfies the fol lowing c onditions: (P1) F or every x ∈ M , the ma p || · || x := || · || M | T x M : T x M → [0 , ∞ ) is a norm on the tangent sp ac e T x M such that for eve ry chart ϕ : U → X with x ∈ U , the norm v ∈ X 7→ || dϕ − 1 ( ϕ ( x ))( v ) || x is e quivalent to || · || on X . (P2) F or e v ery x 0 ∈ M , every ε > 0 and every chart ϕ : U → X with x 0 ∈ U , ther e is an op e n neighb orho o d W of x 0 such that if x ∈ W and v ∈ X , then (1.1) 1 1 + ε || dϕ − 1 ( ϕ ( x 0 ))( v ) || x 0 ≤ || dϕ − 1 ( ϕ ( x ))( v ) || x ≤ (1 + ε ) || dϕ − 1 ( ϕ ( x 0 ))( v ) || x 0 . ALGEBRAS OF SMOOTH FUNCTIONS ON BANACH-FINSLER MANIFOLDS 3 In terms of e quivalenc e of norms, the ab ove ine qualities yield the fact that the norms || dϕ − 1 ( ϕ ( x ))( · ) || x and || dϕ − 1 ( ϕ ( x 0 ))( · ) || x 0 ar e (1 + ε ) -e quivalent. Let u s r ecall that Banac h spaces and Riemannian manifolds are C ∞ Finsler manifolds in the sense of Pal ais [ 25 ]. Let M b e a Finsler man if old, we denote by T x M ∗ the dual space of the tangen t space T x M . Let f : M → R b e a differentia ble fun ction at p ∈ M . The norm of d f ( p ) ∈ T p M ∗ is given b y || d f ( p ) || p = su p {| d f ( p )( v ) | : v ∈ T p M , || v || p ≤ 1 } . Let us consider a different iable fun ction f : M → N b et w een Finsler manifolds M and N . The n orm of the deriv ativ e at the p oint p ∈ M is defin ed as || d f ( p ) || p = su p {|| d f ( p )( v ) || f ( p ) : v ∈ T p M , || v || p ≤ 1 } = = su p { ξ ( d f ( p )( v )) : ξ ∈ T f ( p ) N ∗ , v ∈ T p M and || v || p = 1 = || ξ || ∗ f ( p ) } , where || · || ∗ f ( p ) is the du al norm of || · || f ( p ) . Recall that if ( M , || · || M ) is a Finsler manifold, the length of a piecewise C 1 smo oth path c : [ a, b ] → M is defin ed as ℓ ( c ) := R b a || c ′ ( t ) || c ( t ) dt . Besides, if M is connected, th en it is connected by piecewise C 1 smo oth paths, and the asso ciated Finsler metric d M on M is defined as d M ( p, q ) = inf { ℓ ( c ) : c is a piecewise C 1 smo oth path connecting p and q } . It was sho w n in [ 25 ] that the Finsler metric is consistent with the top ology giv en in M . Th e op en ball of cen ter p ∈ M and radius r > 0 is denoted by B M ( p, r ) := { q ∈ M : d M ( p, q ) < r } . The Lipschitz constan t Lip( f ) of a Lipschitz function f : M → N , wher e M and N are Finsler manifolds, is defi ned as Lip( f ) = sup { d N ( f ( x ) ,f ( y )) d M ( x,y ) : x, y ∈ M , x 6 = y } . W e shall only consider conn ected manifolds. Let us recall the follo wing “mean v alue inequ alit y” for Finsler m anifolds [ 1 , 18 ]. Lemma 1.2. L et M and N b e C 1 Finsler manifolds (in the sense of Pala is) and f : M → N a C 1 smo oth function. Then, f i s Lipschitz if and only if || d f || ∞ := sup {|| d f ( x ) || x : x ∈ M } < ∞ . F urthermor e , Lip( f ) = || d f || ∞ . W e will also need the follo wing result related to the (1 + ε )-bi-Lipsc hitz lo cal b ehavio r of the c harts of a C 1 Finsler manifold in the sense of P alais [ 18 , Lemma 2.4]. Lemma 1.3. L et us c onsider a C 1 Finsler manifold M (in the sense of Palais). Then, for every x 0 ∈ M and every chart ( U, ϕ ) with x 0 ∈ U satisfying ine quality ( 1.1 ) , ther e exists an op e n neighb orho o d V ⊂ U of x 0 satisfying (1.2) 1 1 + ε d M ( p, q ) ≤ ||| ϕ ( p ) − ϕ ( q ) ||| ≤ (1 + ε ) d M ( p, q ) , for every p, q ∈ V , wher e ||| · ||| is the (e quivalent) norm || dϕ − 1 ( ϕ ( x 0 ))( · ) || x 0 define d on X . No w, let us recall the concept of uniformly bump able manifold , in tro duced b y D. Azagra, J. F errera and F. L´ op ez-Mesas [ 1 ] for Riemannian manifolds. A natural extension to Finsler manifolds is defined in the same w a y [ 18 ]. Definition 1.4. A C k Finsler manifold in the sense of Palais M is C k uniformly bump able whenever ther e ar e R > 1 and r > 0 such that for every p ∈ M and δ ∈ (0 , r ) ther e exists a C k smo oth f unction b : M → [0 , 1] su ch that: 4 J.A. JARAMILLO, M. JIM ´ ENEZ-SEVILLA AND L. S ´ ANCHEZ-GONZ ´ ALEZ (1) b ( p ) = 1 , (2) b ( q ) = 0 whenever d M ( p, q ) ≥ δ , (3) sup q ∈ M || db ( q ) || q ≤ R /δ . Note that this is not a restrictiv e definition: D. Aza gra, J. F errera, F. L´ op ez- Mesas and Y. Rangel [ 3 ] prov ed that ev er y separable Riemannian manifold is C ∞ uniformly b umpable. This r esult was generalized in [ 18 ], where it was pr o ved that ev ery C 1 Finsler man if old (in th e sense of Pal ais) mo d eled on a certain class of Banac h spaces (suc h as Hilb ert spaces, Banac h spaces with separable dual, clo sed subspaces of c 0 (Γ) for every set Γ 6 = ∅ ) is C 1 uniformly bump able. In particu- lar, ev ery Riemannian manifold (either sep arab le or non-separable) is C 1 uniformly bumpable. It is straigh tforw ard to v erify that if a C k Finsler manifold M is mo deled on a Banac h space X and M is C k uniformly bum p able, then X admits a Lipschitz C k smo oth bu m p fu nction. Besides, a sep ar able C k Finsler manifold M is mo d eled on a Banac h space with a Lipsc hitz, C k smo oth bump function if and only if M is C k uniformly bumpable [ 18 ]. Nev ertheless, we d o not kno w w hether this equiv alence holds in the non-separable case. F rom now on, w e shall refer to C k Finsler man if olds in the sense of P alais as C k Finsler manifolds, and k ∈ N ∪ {∞} . W e shall use the stand ard notatio n of C k ( U, Y ) f or the set of all k -times con tinuously differentiable functions defined on an op en subs et U of a Banac h sp ace (Finsler manifold) taking v alues into a Banac h space (Finsler manifold) Y . W e sh all wr ite C k ( U ) whenever Y = R . No w, let us recall the concept of we akly C k smo oth fu nction. Definition 1.5. L et X and Y b e Banach sp ac e s and c onsider a function f : U → Y , wher e U is an op en subset of X . The f u nction f is said to b e we akly C k smo oth at the p oint x 0 whenever ther e is an op en neig hb orho o d U x 0 of x 0 such that y ∗ ◦ f is C k smo oth at U x 0 , for every y ∗ ∈ Y ∗ . The function f i s said to b e we akly C k smo oth on U whenever f is we akly C k smo oth at every p oint x ∈ U . On the one hand , J. M. Guti ´ errez an d J .L. G. Lla vona [ 15 ] pr ov ed that if f : U → Y is wea kly C k smo oth on U , then g ◦ f ∈ C k ( U ) for all g ∈ C k ( Y ). Th ey also pro v ed that if f : U → Y is wea kly C k smo oth on U , then f ∈ C k − 1 ( U ). F or k = 1, the ab o v e yields that eve ry w eakly C 1 smo oth fu nction on U is con tin uous on U . Also, for k = ∞ , every w eakly C ∞ smo oth function on U is C ∞ smo oth on U . M. Bac hir and G. Lancien [ 4 ] p ro v ed that, if the Banac h space Y has the Sch ur prop ert y , then the concept of w eakly C k smo othness coincides with the concept of C k smo othness. On the other hand, there are examples of weakly C 1 smo oth fu nctions that are n ot C 1 smo oth (see [ 15 ] and [ 4 ]). Definition 1.6. L et M and N b e C k Finsler manifolds and U ⊂ M , O ⊂ N op en subsets of M and N , r esp e ctive ly. A function f : U → N i s said to b e we akly C k smo oth at the p oint x 0 of U if ther e exist charts ( W, ϕ ) of M at x 0 and ( V , ψ ) of N at f ( x 0 ) such that ψ ◦ f ◦ ϕ − 1 is we akly C k smo oth at ϕ ( W ) . We say that f : U → N is we akly C k smo oth in U if f is we akly C k smo oth at every p oint x ∈ U . We say that a bije ction f : U → O is a we akly C k diffe omorphism if f and f − 1 ar e we akly C k smo oth on U and O , r e sp e ctively. Notic e that these definitions do not dep e nd on the chosen charts. ALGEBRAS OF SMOOTH FUNCTIONS ON BANACH-FINSLER MANIFOLDS 5 Let us note that there are homeomorphisms whic h are weakly C 1 smo oth but not different iable. Indeed, we f ollo w [ 15 , Examp le 3.9] and defin e g : R → c 0 ( N ) and h : c 0 ( N ) → c 0 ( N ) by g ( t ) = (0 , 1 2 sin(2 t ) , . . . , 1 n sin( nt ) , . . . ) and h ( x ) = x + g ( x 1 ) for ev ery t ∈ R and x = ( x 1 , . . . , x n , . . . ) ∈ c 0 . The fu nction h is an homeomorphism , h − 1 ( y ) = y − g ( y 1 ) for ev ery y ∈ c 0 , and h is w eakly C 1 smo oth on c 0 ( N ). Notice that if h were differen tiable at a p oin t x ∈ c 0 with x 1 = 0, then h ′ ( x )(1 , 0 , 0 , . . . ) = (1 , 1 , 1 , . . . ) ∈ ℓ ∞ \ c 0 , whic h is a cont radiction. No w, let us consider different d efinitions of isometries b etw een C k Finsler mani- folds. Definition 1.7. L et ( M , || · || M ) and ( N , || · || N ) b e C k Finsler manifolds and a bije ction h : M → N . ( MI ) We say that h is a metric isometry for the Finsler metrics, if d N ( h ( x ) , h ( y )) = d M ( x, y ) , for every x, y ∈ M . (FI) We say that h is a C k Finsler isometry if it is a C k diffe omorphism satisfying || dh ( x )( v ) || h ( x ) = || ( h ( x ) , dh ( x )( v )) || N = || ( x, v ) || M = || v || x , for e very x ∈ M and v ∈ T x M . We say that the Finsler manifolds M and N ar e C k e quivalent as Finsler manifolds if ther e is a C k Finsler i sometry b etwe en M and N . ( ω -FI ) We say that h is a we ak C k Finsler isometry if it is a we akly C k dif- fe omorphism and a metric isometry for the Finsler metrics. We say that the Finsler manifolds M and N ar e w e akly C k e quivalent as Finsler manifolds if ther e is a we ak C k Finsler isometry b etwe en M and N . Prop osition 1.8. L et M and N b e C k Finsler manifolds. L et us assume that ther e is a C k diffe omorphism and metric isometry (for the Finsler metrics) h : M → N . Then h is a C k Finsler isometry. Pr o of. Let us fix x ∈ M and y = h ( x ) ∈ N . F or eve ry ε > 0, there are r > 0 and c h arts ϕ : B M ( x, r ) ⊂ M → X and ψ : B N ( y , r ) ⊂ N → Y satisfying inequalities ( 1.1 ) and ( 1.2 ). Since h : M → N is a metric isometry , h is a bijection from B M ( x, r ) on to B N ( y , r ). Let us consider the equ iv alen t n orm s on X and Y defin ed as |||·||| x := || dϕ − 1 ( ϕ ( x ))( · ) || x and ||| · ||| y = || dψ − 1 ( ψ ( y ))( · ) || y , r esp ectiv ely . Since h is a metric isometry , w e obtain f rom Lemma 1.3 , for p , q in an op en neigh b orho o d of ϕ ( x ), ||| ψ ◦ h ◦ ϕ − 1 ( p ) − ψ ◦ h ◦ ϕ − 1 ( q ) ||| y ≤ (1 + ε ) d N ( h ◦ ϕ − 1 ( p ) , h ◦ ϕ − 1 ( q )) = = (1 + ε ) d M ( ϕ − 1 ( p ) , ϕ − 1 ( q )) ≤ (1 + ε ) 2 ||| p − q ||| x . 6 J.A. JARAMILLO, M. JIM ´ ENEZ-SEVILLA AND L. S ´ ANCHEZ-GONZ ´ ALEZ Th us, sup {||| d ( ψ ◦ h ◦ ϕ − 1 )( ϕ ( x ))( w ) ||| y : ||| w ||| x ≤ 1 } ≤ (1 + ε ) 2 . No w, for every v ∈ T x M with v 6 = 0, let u s write w = dϕ ( x )( v ) ∈ X . W e ha v e || dh ( x )( v ) || y = || dψ − 1 ( ψ ( y )) dψ ( y ) dh ( x )( v ) || y = ||| d ( ψ ◦ h )( x )( v ) ||| y = = ||| d ( ψ ◦ h )( x ) dϕ − 1 ( ϕ ( x ))( w ) ||| y = ||| d ( ψ ◦ h ◦ ϕ − 1 )( ϕ ( x ))( w ) ||| y ≤ ≤ (1 + ε ) 2 ||| w ||| x = (1 + ε ) 2 || v || x . Since this inequalit y holds for every ε > 0 and the same argument works for h − 1 , w e conclude that || dh ( x )( v ) || y = || v || x for all v ∈ T x M . Thus, h is a C k Finsler isometry .  Let u s now turn our atten tion to the Banach algebr a C 1 b ( M ), the algebra of all real-v alued, C 1 smo oth and b ounded functions with b ounded deriv ativ e defined on a C 1 Finsler m anifold M , i.e. C 1 b ( M ) = { f : M → R : f ∈ C 1 ( M ) , || f || ∞ < ∞ and || d f || ∞ < ∞} , where || f || ∞ := sup {| f ( x ) | : x ∈ M } and || d f || ∞ := sup {|| d f ( x ) || x : x ∈ M } . The usual norm considered on C 1 b ( M ) is || f || C 1 b = max {|| f || ∞ , || d f || ∞ } for ev ery f ∈ C 1 b ( M ) and ( C 1 b ( M ) , || · || C 1 b ( M ) ) is a Banac h space. Let us notice that, by Lemma 1.2 , w e h a ve || d f || ∞ = Lip ( f ). Recall that ( C 1 b ( M ) , 2 || · || C 1 b ( M ) ) is a Banac h algebra. F or 2 ≤ k ≤ ∞ and a C k Finsler manifold M , let us consider the algebra C k b ( M ) of all r eal-v alued, C k smo oth and b oun ded functions that h a ve b ounded first deriv ativ e, i.e. C k b ( M ) = { f : M → R : f ∈ C k ( M ) , || f || ∞ < ∞ and || d f || ∞ < ∞} = C k ( M ) ∩ C 1 b ( M ) . with the norm || · || C 1 b . Thus, C k b ( M ) is a su balgebra of C 1 b ( M ). Nev ertheless, it is not a Banac h algebra. A function ϕ : C k b ( M ) → R (1 ≤ k ≤ ∞ ) is said to b e an algebr a homomorphism whether for all f , g ∈ C k b ( M ) and λ, η ∈ R , (i) ϕ ( λf + η g ) = λϕ ( f ) + η ϕ ( g ), and (ii) ϕ ( f · g ) = ϕ ( f ) ϕ ( g ). Let us denote by H ( C k b ( M )) the set of all nonzero algebra homomorphism s, i.e. H ( C k b ( M )) = { ϕ : C k b ( M ) → R : ϕ is an algebra homomorphism and ϕ (1) = 1 } . Let us list some of the basic prop erties of the algebra C k b ( M ) and the al gebra homomorphisms H ( C k b ( M )). They can b e c hec k ed as in the Riemannian case (see [ 11 ], [ 12 ] an d [ 17 ]). (a) If ϕ ∈ H ( C k b ( M )), then ϕ 6 = 0 if and only if ϕ (1) = 1. (b) If ϕ ∈ H ( C k b ( M )), then ϕ is p ositiv e, i.e. ϕ ( f ) ≥ 0 for ev ery f ≥ 0. (c) If the C k Finsler manifold M is mo deled on a Banach space that admits a L ip sc hitz and C k smo oth b ump function, then C k b ( M ) is a unital algebr a that sep ar ates p oints an d close d sets of M . Let us briefly give th e p ro of for completeness. Let us tak e x ∈ M , and C ⊂ M a closed sub set of M with x 6∈ C . Let us tak e r > 0 small enough so that C ∩ B M ( x, r ) = ∅ and a c h art ϕ : B M ( x, r ) → X satisfying inequalit y ( 1.1 ). Let us tak e s > 0 sm all enough so th at ϕ ( x ) ∈ B ( ϕ ( x ) , s ) ⊂ ϕ ( B ( x, r / 2) ) ⊂ X and a Lipsc hitz and ALGEBRAS OF SMOOTH FUNCTIONS ON BANACH-FINSLER MANIFOLDS 7 C k smo oth bump function b : X → R with b ( ϕ ( x )) = 1 and b ( z ) = 0 for ev ery z 6∈ B ( ϕ ( x ) , s ). Let us defin e h : M → R as h ( p ) = b ( ϕ ( p )) for every p ∈ B M ( x, r ) and h ( p ) = 0 otherw ise. Th en h ∈ C k b ( M ), h ( x ) = 1 and h ( c ) = 0 for ev ery c ∈ C . (d) The sp ace H ( C k b ( M )) is closed as a top olo gical subspace of R C k b ( M ) with the pro du ct top ology . Moreo ver, since ev ery fun ction in C k b ( M ) is b ound ed, it can b e c h ec ked that H ( C k b ( M )) is compact in R C k b ( M ) . (e) If C k b ( M ) separates p oint s and closed subsets, then M can b e embedd ed as a top ological su b space of H ( C k b ( M )) by identifying ev ery x ∈ M with the p oint evaluation homo morphism δ x giv en by δ x ( f ) = f ( x ) for every f ∈ C k b ( M ). Also, it can b e c heck ed that the su bset δ ( M ) = { δ x : x ∈ M } is dense in H ( C k b ( M )). Therefore, it follo ws that H ( C k b ( M )) is a compactifica tion of M . (f ) Ev ery f ∈ C k b ( M ) admits a conti nuous extension b f to H ( C k b ( M )), where b f ( ϕ ) = ϕ ( f ) for ev ery ϕ ∈ H ( C k b ( M )). Notic e that this extension b f coincides in H ( C k b ( M )) w ith the pro jection π f : R C k b ( M ) → R , giv en by π f ( ϕ ) = ϕ ( f ), i.e. π f | H ( C k b ( M )) = b f . In the follo w ing, w e shall id en tify M with δ ( M ) in H ( C k b ( M )). The n ext p rop osition can b e p ro v ed in a s imilar wa y to the Riemannian case [ 12 ]. Prop osition 1.9. L et M b e a c omplete C k Finsler manifol d that is C k uniformly bump able. Then, ϕ ∈ H ( C k b ( M )) has a c ountable neighb orho o d b asis in H ( C k b ( M )) if and only if ϕ ∈ M . 2. A Mye rs-Nakai Theorem Our main result is the follo win g Banac h -Stone type theorem for a certain class of Finsler m an if olds . It states that th e algebra stru cture of C k b ( M ) determin es the C k Finsler manifold. Recall that t wo normed algebras ( A, || · || A ) and ( B , || · || B ) are e qui valent as norme d algebr as whenev er there exists an algebra isomorp hism T : A → B satisfying || T ( a ) || B = || a || A for every a ∈ A . Let us begin by d efining the class of Banac h spaces wh ere the Finsler manifolds shall b e mo deled. Definition 2.1. A Banach sp ac e ( X, || · || ) is said to b e k-admissib le if for every e quivalent norm | · | and ε > 0 , ther e ar e an op en subset B ⊃ { x ∈ X : | x | ≤ 1 } of X and a C k smo oth f unction g : B → R such that (i) | g ( x ) − | x || < ε for x ∈ B , and (ii) Lip( g ) ≤ (1 + ε ) for the norm | · | . It is easy to pro v e the follo wing lemma. Lemma 2.2. L et X b e a Banach sp ac e with one of the fol low ing pr op erties: (A.1) Density of the set of e quivalent C k smo oth norms: ev ery e quivalent norm on X c an b e appr oximate d in the Hausdorff metric by e quivalent C k smo oth norms [ 6 ] . (A.2) C k -fine appr oximation pr op e rty ( k ≥ 2 ) and d ensity of the set of e quivalent C 1 smo oth norms: F or every C 1 smo oth f unction f : X → R and every 8 J.A. JARAMILLO, M. JIM ´ ENEZ-SEVILLA AND L. S ´ ANCHEZ-GONZ ´ ALEZ ε > 0 , ther e is a C k smo oth f unction g : X → R satisfying | f ( x ) − g ( x ) | < ε and || f ′ ( x ) − g ′ ( x ) || < ε for al l x ∈ X (se e [ 16 ] , [ 2 ] and [ 20 ] ); also, every e quivalent norm define d on X c an b e appr oximate d i n the Hausdorff metric by e qu ivalent C 1 smo oth norms (se e [ 6 , T heorem I I 4.1] ). Then X is k-admissible. Banac h spaces satisfying condition (A.2) are, for instance, separable Banac h spaces with a Lipschitz C k smo oth bump fu n ction. Banach spaces satisfying condi- tion (A.1) for k = 1 are, for ins tance, W eakly Compactly Generated (W CG) Banac h spaces with a C 1 smo oth bump fun ction. Theorem 2.3. L et M and N b e c omplete C k Finsler manifolds that ar e C k uni- formly bump able and ar e mo dele d on k -adm issible Banach sp ac es. Then M and N ar e we akly C k e quivalent as Finsler manifolds if and only if C k b ( M ) and C k b ( N ) ar e e qu ivalent as norme d algebr as. Mor e over, every norme d algebr a isomorphism T : C k b ( N ) → C k b ( M ) is of the form T ( f ) = f ◦ h wh er e h : M → N is a we ak C k Finsler isometry. In p articular, h is a C k − 1 Finsler isometry whenever k ≥ 2 . In order to p ro v e Th eorem 2.3 , w e shall follo w the ideas of the Riemmanian case [ 12 ]. Let us divide the pro of in to seve ral prop ositio ns. Prop osition 2.4. L et M and N b e C k Finsler manifolds such that N is mo dele d on a k -admissible Banach sp ac e Y . L et h : M → N b e a ma p such that T : C k b ( N ) → C k b ( M ) given by T ( f ) = f ◦ h i s c ontinuous. Then h is || T || -Lisp chitz for the Finsler metrics. Pr o of. F or ev ery y ∈ N , let us tak e a c hart ψ y : V y → Y with ψ y ( y ) = 0. Let us consider the equiv alen t n orm on Y , ||| · ||| y := || dψ − 1 y (0)( · ) || y and fix ε > 0. Let us define the ball B |||·|| | y ( z , t ) := { w ∈ Y : ||| w − z ||| y < t } . F act. F or ev ery r > 0 suc h that B |||·|| | y (0 , r ) ⊂ ψ y ( V y ) and ev ery e ε > 0, th ere exists a C k smo oth and Lipschitz function f y : Y → R su c h that (1) f y (0) = r , (2) || f y || ∞ := su p {| f y ( z ) | : z ∈ Y } = r , (3) Lip( f y ) ≤ (1 + ε ) 2 for the norm ||| · ||| y , (4) f y ( z ) = 0 for every z ∈ Y with ||| z ||| y ≥ r , and (5) ||| z ||| y ≤ r − f y ( z ) + e ε for every ||| z ||| y ≤ r . Let us pr o ve the F act. First of all, let us tak e r > 0, e ε > 0 and 0 < α < min { 1 , ε 4 , 2 e ε 5 r } . Since N is a C k Finsler manifold mo deled on a k -admiss ib le Banac h space Y , there are an op en su bset B ⊃ { x ∈ Y : ||| x ||| y ≤ 1 } of Y and a C k smo oth function g : B → R su c h that (i) | g ( x ) − ||| x ||| y | < α/ 2 on B , and (ii) Lip( g ) ≤ (1 + α/ 2) for the norm ||| · ||| y . No w, let us tak e a C ∞ smo oth and Lipschitz function θ : R → [0 , 1] s u c h that (i) θ ( t ) = 0 whenever t ≤ α , (ii) θ ( t ) = 1 whenever t ≥ 1 − α , (iii) Lip( θ ) ≤ (1 + ε ), and (iv) | θ ( t ) − t | ≤ 2 α for every t ∈ [0 , 1 + α ]. ALGEBRAS OF SMOOTH FUNCTIONS ON BANACH-FINSLER MANIFOLDS 9 Let us d efine f ( x ) = ( θ ( g ( x )) if x ∈ B , 1 if x ∈ Y \ B . It is straigh tforward to v erify th at f is well -defined, C k smo oth, f ( x ) = 1 wheneve r ||| x ||| y ≥ 1 and f ( x ) = 0 wh enev er ||| x ||| y ≤ α/ 2 Let us no w consider f y : Y → [0 , r ] as f y ( z ) = r (1 − f ( z r )), which is C k smo oth, Lipsc hitz and s atisfies: (i) f y (0) = r , (ii) || f y || ∞ = r , (iii) | f y ( z ) − f y ( x ) | ≤ (1 + ε )(1 + α/ 2) ||| z − x ||| y ≤ (1 + ε ) 2 ||| z − x ||| y , (iv) f y ( z ) = 0 for every z ∈ Y with ||| z ||| y ≥ r , (v) ||| z r ||| y ≤ α 2 + g ( z r ) ≤ α 2 + 2 α + f ( z r ) f or eve ry ||| z ||| y ≤ r . Thus, ||| z ||| y ≤ r ( α 2 + 2 α ) + r − f y ( z ) ≤ e ε + r − f y ( z ) for every ||| z ||| y ≤ r . Let us no w p ro v e Prop osition 2.4 . Let us fi x p 1 , p 2 ∈ M and ε > 0. Let us consider σ : [0 , 1] → M a piecewise C 1 smo oth path in M joining p 1 and p 2 , with ℓ ( σ ) ≤ d M ( p 1 , p 2 ) + ε . S ince h : M → N is con tinuous, the p ath b σ := h ◦ σ : [0 , 1] → N , joining h ( p 1 ) and h ( p 2 ), is cont in uous as we ll. F or eve ry q ∈ b σ ([0 , 1]), there is 0 < r q < 1 and a chart ψ q : V q → Y suc h that ψ q ( q ) = 0, B N ( q , r q ) ⊂ V q and th e bijection ψ q : V q → ψ q ( V q ) is (1 + ε )-bi-Lipschitz for the norm || dψ − 1 q (0)( · ) || q in Y (see Lemma 1.3 ). Since b σ ([0 , 1]) is a compact set of N , there is a finite family of p oints 0 = t 1 < t 2 < ... < t m = 1 and a family of op en int erv als { I k } m k =1 co vering the in terv al [0 , 1] s o that, if we defi ne q k := b σ ( t k ) and r k := r q k , for ev ery k = 1 , ..., m , w e ha v e (a) b σ ( I k ) ⊂ B N ( q k , r k / (1 + ε )), (b) I j ∩ I k 6 = ∅ if, and only if, | j − k | ≤ 1. It is clear that b σ ([0 , 1]) ⊂ S m k =1 B N ( q k , r k 1+ ε ). No w, let us select a p oin t s k ∈ I k ∩ I k +1 suc h th at t k < s k < t k +1 , for every k = 1 , ..., m − 1. Let us write a k := b σ ( s k ), f or ev ery k = 1 , · · · , m − 1, ψ k := ψ q k , V k := V q k and ||| · ||| k := || dψ − 1 k (0)( · ) || q k , for ev ery k = 1 , ....m . Notice th at a k ∈ B N ( q k , r k 1+ ε ) ∩ B N ( q k +1 , r k +1 1+ ε ), for ev ery k = 1 , · · · , m − 1. Since ψ k : V k → ψ k ( V k ) is (1 + ε )-bi-Lipsc hitz for the norm ||| · ||| k in Y , we deduce that ψ k ( a k ) ∈ B |||·|| | k (0 , r k ), for ev er y k = 1 , · · · , m − 1. No w, let us we apply the ab o ve F act to r k , ε and e ε = ε/ 2 m to obtain fu nctions f k : Y → [0 , r k ] sat isfying prop ertie s (1)–(5), k = 1 , · · · , m . Let us define the C k smo oth and Lipsc hitz fu nctions g k : N → [0 , r k ] as g k ( z ) = f k ( ψ k ( z )) for every z ∈ V k and g k ( z ) = 0 for z 6∈ V k , k = 1 , · · · , m . Then, (i) g k ∈ C k b ( N ); (ii) g k ( q k ) = r k ; (iii) | g k ( z ) − g k ( x ) | ≤ (1 + ε ) 3 d N ( z , x ) for all z , x ∈ N ; (iv) If z ∈ ψ − 1 k ( B |||·|| | k (0 , r k )), then ||| ψ k ( z ) ||| k ≤ r k and from cond ition (5) on the F act, w e obtain d N ( z , q k ) ≤ (1+ ε ) ||| ψ k ( z ) − ψ k ( q k ) ||| k = (1+ ε ) ||| ψ k ( z ) ||| k ≤ (1+ ε )( r k − g k ( z )+ ε/ 2 m ) . The Lipsc hitz constan t of g k ◦ h , for k = 1 , · · · , m , is the f ollo wing Lip( g k ◦ h ) ≤ || g k ◦ h || C 1 b ( M ) = || T ( g k ) || C 1 b ( M ) ≤ || T |||| g k || C 1 b ( N ) = = || T || max {|| g k || ∞ , || dg k || ∞ } ≤ || T || (1 + ε ) 3 . 10 J.A. JARAMILLO, M. JIM ´ ENEZ-SEVILLA AND L. S ´ ANCHEZ-GONZ ´ ALEZ No w, since r k = g k ( q k ) = g k ( h ( σ ( t k ))) and ψ k ( h ( σ ( s k ))) ∈ B |||·|| | k (0 , r k ), we ha v e d N ( h ( p 1 ) ,h ( p 2 )) ≤ m − 1 X k =1 [ d N ( h ( σ ( t k )) , h ( σ ( s k ))) + d N ( h ( σ ( s k )) , h ( σ ( t k +1 )))] ≤ ≤ m − 1 X k =1 (1 + ε )[ g k ( q k ) − g k ( h ( σ ( s k )))+ + g k +1 ( q k +1 ) − g k +1 ( h ( σ ( s k ))) + ε/m ] ≤ ≤ m − 1 X k =1 (1 + ε )[Lip( g k ◦ h ) d M ( σ ( t k ) , σ ( s k ))+ + Lip( g k +1 ◦ h ) d M ( σ ( t k +1 ) , σ ( s k )) + ε/m ] ≤ ≤ m − 1 X k =1 || T || (1 + ε ) 4 [ d M ( σ ( t k ) , σ ( s k )) + d M ( σ ( t k +1 ) , σ ( s k ))] + ε (1 + ε ) ≤ ≤ m − 1 X k =1 || T || (1 + ε ) 4 ℓ ( σ | [ t k ,t k +1 ] ) + ε (1 + ε ) = || T || (1 + ε ) 4 ℓ ( σ ) + ε (1 + ε ) ≤ ≤ || T || (1 + ε ) 4 ( d M ( p 1 , p 2 ) + ε ) + ε (1 + ε ) for ev ery ε > 0. Thus, h is || T || -Lipsc hitz.  Lemma 2.5. L et M and N b e C k Finsler manifolds such that N is mo dele d on a Banach sp ac e with a Lipschitz C k smo oth bump function. L et h : M → N b e a home omorphism such that f ◦ h ∈ C k b ( M ) f or every f ∈ C k b ( N ) . Then, h is a we akly C k smo oth f unction on M . Pr o of. Let us fix x ∈ M and ε = 1. There are c h arts ϕ : U → X of M at x and ψ : V → Y of N at h ( x ) satisfying inequalities ( 1.1 ) and ( 1.2 ) on U and V , resp ectiv ely . W e can assume that h ( U ) ⊂ V . Since Y admits a Lipsc hitz and C k smo oth b ump fun ction and ψ ( h ( U )) is an op en n eighb orh o o d of ψ ( h ( x )) in Y , there are r eal n um b ers 0 < s < r suc h that B ( ψ ( h ( x )) , s ) ⊂ B ( ψ ( h ( x )) , r ) ⊂ ψ ( h ( U )) and a Lipsc hitz and C k smo oth function α : Y → R su ch that α ( y ) = 1 for y ∈ B ( ψ ( h ( x )) , s ) and α ( y ) = 0 for y 6∈ B ( ψ ( h ( x )) , r ). Let u s define U 0 := h − 1 ( ψ − 1 ( B ( ψ ( h ( x )) , s ))) ⊂ U , wh ic h is an op en neigh b orho o d of x in M . Let us chec k that y ∗ ◦ ( ψ ◦ h ◦ ϕ − 1 ) is C k smo oth on ϕ ( U 0 ) ⊂ X for all y ∗ ∈ Y ∗ . F ollo wing the pr o of of [ 9 , Th eorem 4], w e define g : N → R as g ( y ) = 0 whenev er y 6∈ V and g ( y ) = α ( ψ ( y )) · y ∗ ( ψ ( y )) w henev er y ∈ V . It is clear that g ∈ C k b ( N ) and , b y assumption, g ◦ h ∈ C k b ( M ). No w, it follo ws that ψ ( h ( ϕ − 1 ( z ))) ∈ B ( ψ ( h ( x )) , s ) for ev ery z ∈ ϕ ( U 0 ). Thus y ∗ ◦ ( ψ ◦ h ◦ ϕ − 1 )( z ) = y ∗ ( ψ ( h ( ϕ − 1 ( z )))) = α ( ψ ( h ( ϕ − 1 ( z )))) y ∗ ( ψ ( h ( ϕ − 1 ( z )))) = = g ( h ( ϕ − 1 ( z ))) = g ◦ h ◦ ϕ − 1 ( z ) , for ev ery z ∈ ϕ ( U 0 ). Since ( g ◦ h ) ◦ ϕ − 1 is C k smo oth on ϕ ( U 0 ), we ha v e that y ∗ ◦ ( ψ ◦ h ◦ ϕ − 1 ) is C k smo oth on ϕ ( U 0 ). Th us ψ ◦ h ◦ ϕ − 1 is w eakly C k smo oth on ϕ ( U 0 ) and h is weakly C k smo oth on M .  ALGEBRAS OF SMOOTH FUNCTIONS ON BANACH-FINSLER MANIFOLDS 11 Pr o of of The or em 2.3 . If h : M → N is a wea k C k Finsler isometry , we can define the op erato r T : C k b ( N ) → C k b ( M ) by T ( f ) = f ◦ h . Let us c hec k that T is w ell d efined. F or eve ry x ∈ M , there are charts ϕ : U → X of M at x and ψ : V → Y of N at h ( x ), suc h that h ( U ) ⊂ V and ψ ◦ h ◦ ϕ − 1 is we akly C k smo oth on ϕ ( U ) ⊂ X . Also, f ◦ ψ − 1 is C k smo oth on ψ ( V ) ⊂ Y . Th us, by [ 15 , Pr op osition 4.2], ( f ◦ ψ − 1 ) ◦ ( ψ ◦ h ◦ ϕ − 1 ) = f ◦ h ◦ ϕ − 1 is C k smo oth on ϕ ( U ). Th erefore, f ◦ h is C k smo oth on U . Since this holds for eve ry x ∈ M , w e deduce that f ◦ h is C k smo oth on M . Moreo v er, T is an algebra isomorphism with || T ( f ) || C 1 b ( M ) = || f ◦ h || C 1 b ( M ) = || f || C 1 b ( N ) for ev ery f ∈ C k b ( N ). Con v ersely , let T : C k b ( N ) → C k b ( M ) b e a normed algebra isometry . T hen, w e can define the fu n ction h : H ( C k b ( M )) → H ( C k b ( N )) by h ( ϕ ) = ϕ ◦ T for ev ery ϕ ∈ H ( C k b ( M )). The function h is a bijection. Moreo v er, h is an homeomorphism. Recall that w e identify x ∈ M w ith δ x ∈ C k b ( M ). Th us, h ( x ) = h ( δ x ) = δ x ◦ T . Since h is an homeomorphism , b y Prop osition 1.9 , we obtain for ev ery p ∈ N a unique p oint x ∈ M suc h that h ( δ x ) = δ p . Let us chec k that T ( f ) = f ◦ h for all f ∈ C k b ( N ). Indeed, for every x ∈ M and ev ery f ∈ C k b ( N ), T ( f )( x ) = δ x ( T ( f )) = ( δ x ◦ T )( f ) = h ( δ x )( f ) = δ h ( x ) ( f ) = f ( h ( x )) = f ◦ h ( x ) . No w, from Prop osition 2.4 and L emm a 2.5 we deduce that h is a weak C k Finsler isometry .  Remark 2.6. It is worth mentioning that, for Riemannian manifolds, every metric isometry is a C 1 Finsler isometry. Th is r esult was pr ove d by S. Myers and N . Ste enr o d [ 21 ] in the finite- dimensional c ase and by I. Garrido, J.A. Jar amil lo and Y.C. R angel [ 12 ] in the ge ne r al c ase. Also, S. Deng and Z. Hou [ 5 ] obtaine d a version for finite- dimensional R iemannian-Finsler manifolds. Nevertheless, ther e is no a gener alization, up to our know le dge, of the Myers-Ste enr o d the or em for al l Finsler manifolds. Thus, for k = 1 we c an only assur e that the metric isometry obtaine d in The or e m 2.3 is we akly C 1 smo oth. Let u s fi nish this note w ith some int eresting corollarie s of Theorem 2.3 . First, recall that ev ery s ep arable Banac h sp ace with a Lipschitz C k smo oth b ump fu nction satisfies condition (A.2) and ev ery WCG Banac h space with a C 1 smo oth bump function satisfies condition (A.1) for k = 1. Corollary 2.7. L et M and N b e c omplete, C 1 Finsler manifolds that ar e C 1 uniformly bump able and ar e mo dele d on WCG Banach sp ac es. Then M and N ar e we akly C 1 e quivalent as Finsler manifolds if, and only if, C 1 b ( M ) and C 1 b ( N ) ar e e qu ivalent as norme d algebr as. Mor e over, every norme d algebr a isomorphism T : C 1 b ( N ) → C 1 b ( M ) is of the form T ( f ) = f ◦ h wher e h : M → N is a we ak C 1 Finsler isometry. Corollary 2.8. L e t M and N b e c omplete, sep ar able C k Finsler manifolds that ar e mo dele d on Banach sp ac es with a Lipschitz and C k smo oth bump function. Then M and N ar e we akly C k e quivalent as Finsler manifolds if and only if C k b ( M ) and C k b ( N ) ar e e quivalent as norme d algebr as. Mor e over, every norme d al gebr a isomor- phism T : C k b ( N ) → C k b ( M ) is of the form T ( f ) = f ◦ h wher e h : M → N is a we ak C k Finsler isometry. In p articular, h is a C k − 1 Finsler isometry whenever k ≥ 2 . 12 J.A. JARAMILLO, M. JIM ´ ENEZ-SEVILLA AND L. S ´ ANCHEZ-GONZ ´ ALEZ Since every w eakly C k smo oth function with v alues in a fin ite-dimensional n ormed space is C k smo oth and every fin ite-dimensional C k Finsler manifold is C k uniformly bumpable [ 18 ], we obtain the follo w ing My ers -Nak ai result for fin ite-dimensional C k Finsler manifolds. Corollary 2.9. L et M and N b e c omplete and finite dimensional C k Finsler mani- folds. Then M and N ar e C k e quivalent as Finsler manifolds if, and only if, C k b ( M ) and C k b ( N ) ar e e quiv alent as norme d algebr as. M or e over, every norme d algebr a iso- morphism T : C k b ( N ) → C k b ( M ) is of the form T ( f ) = f ◦ h wher e h : M → N is a C k Finsler isometry. W e obtain an in teresting application of Finsler manifolds to Banac h spaces. Re- call the we ll-kno wn Mazur-Ulam Theorem establishin g th at ev ery sur jectiv e isometry b et w een tw o Banac h spaces is affine. Corollary 2.10. L et X and Y b e WCG Banach sp ac es with C 1 smo oth bump fu nc- tions. 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