Chebyshev coefficients for L1-preduals and for spaces with the extension property

Publicacions Matemütiques, Vol 34 (1990), 341-347 . A bstract CHEBYSHEV COEFFICIENTS FOR L 1 -PREDUALS AND FOR SPACES WITH THE EXTENSION PROPERTY JOSÉ M . BAYOD AND M . CONCEPCIÓN MASA We apply the Chebyshev coefficients A fand Ab, recently introduced by the authors, to obtain some results related to certain geometric properties of Banach spaces . We prove that a real normed space E is an Ll-predual if and only if \ f (E) = 1/2, and that if a (real or complex) normed space E is a 'P, space, then Ab(E) equals Ab(K), where oá is the ground field of E . In this note, IK will be the real or complex field, and E a normed space over IK ; when we want to state a result only for the real case or the complex case, we will indícate it specificaly . We will use the notations of [2], to which we refer for all concepts of the theory of normed spaces which may appear without defining them here . If S is a non empty subset of E, the number r(S) = inf sup lix - y¡¡ YEE-ES is called the Chebyshev radius of S, and b(S) denotes the diameter of S . Deflnition . We will call the finite Chebyshev coe ficient of E the real number A f(E) = sup{r(S)/b(S) : SC E, S finite, b(S) > 0}, and the bounded Chebyshev coeficient of E the real number Ab(E) = sup{r(S)/b(S) : SC E, 0 < b(S) < oo} . It is easy to prove that, in general, 1/2 < A f(E) < Ab(E) < 1 . 34 2  J .M . BAYOD, M .C . MASA Moreover, when E is finite dimensional, we have A f(E) = ab(E) . Specifically, the Chebyshev coefficients associated to the scalar fields are Af(R) = Ab(R) = 1/2 ,\f(C) = Ab(C) = 1/4 . Let us recall that a P,,(aá) space, where a is a real number greater than or equal to 1, is a Banach space E for which any of following equivalent condition holds : (i) Given two Banach spaces, F and G, a linear isometry into,  : F --> G, and a bounded linear operator, L : F -> E, there exists a bounded linear operator L : G -+ E, which extends L, in the sense of L o 0 = L, and such that JILII < aJILI1 (a-extension property) . (ii) Given a Banach space F, and a linear isometry,  : E -+ F, there exists a projection, P : F -+ O(E), such that JIPI¡ < a (a-projection property) . It is said that a Banach space E is a N a space, where a is a real number greater than or equal to 1, when there exists a collection (E . y ) ., E r of finite dimensional subspaces of E, which is upwards directed, their union is dense in E and every one of them is a Pa .(K) space . Note that a . Banach . .space is .a n L l -predual space if and only if it is aN,, space for every a > 1 ([2, theorem 2, pg . 232]) . Theorem 1 . If ¡he Banach space E is an L 1 -predual, then a f (E) = a f (K) . ProofWe fix an a > 1 . Given that E is an L l -predual, it is a N a space, and, so, there exists a collection (Ej-,Er of subspaces of E according to the definition above . We put F = U E,, which, because it is dense in E, satisfies y Er af(E) = af(F) . Let S be a finite subset of F with more than one point . There exists yE P such that S C E .y , and, if we indícate with subíndices the Chebyshev radii in subspaces of E, we have r(S) = r, (S) :5 re, (S) :5 b(S) .Af(E-,) :5 b(S) .a .a f(K), where the last inequality is due to E, E P,,(IK) . Therefore, A f(E) = a f(F) < a .A f(K), for every a > 1, so A f(E) < A f(K) . On the other hand, by the Hanh-Banach theorem, there exists a projection of norm 1 from E to K, so af(K) < af(E) . If *E is a non-standard enlargement of E, then, over the set fin*E = {x E *E : 3y E E, lix - y¡¡ is a finite hyperreal number} of the finite elements of *E, consider the equivalence relation "x is infinitely close to y", denoted by x - y, and defined by "l[x - y¡¡ is infinitesimal" . In the quotient set, denoted E, the norm ¡IxII = stlixil, í E E, is defined, and the resulting normed space is called an infinitesimal hull of E . CHEBYSHEV COEFFICIENTS FOR L'-PREDUALS  34 3 Lemma 2 . Af(É) = Af(E) . Proof . . Let S be a finite subset of E with more than one point . Then, S is a finite subset of fin*E without infinitely Glose points . It is obvious that 6(S) = 6(S) and r(S) < r(S) . We suppose that r(S) < r(S), and take a real number t such that r(S) < t < r(S) . Then, there exists c E fin*E such that S C B[c, t], and so, lix - cil < r(S) for every x E S . Since S is finite, *S = S, and we have a c E *E such that lix - cil < r(S) for every xE *S . Applying the Transfer Principle, there exists a standard element c E E such that lix - cil < r(S) for every x E S, and again because S is finite, this would imply SC B[c, p], with p = m S lix - cil < r(S) .  Therefore, it is true r(S) = r(S) and we conclude A f(E) < Af (E) . Let S be now a finite subset of fin*E with some points not infinitely Glose . Then, S is a *-finite subset of *E with some points not infinitely Glose and S is a finite subset of E such that 6(S) -*6(S) . Since the relation r(T) < 6(T) .Af(E) is true for every finite subset T of E, by the Transfer Principle, we have *r(T) <_*6(T) .Af(E) for every *-finite T, and, in particular, *r(S) <*6(S) .Af(E) - 6(S) .A f(E) . Let t be a hiperreal number such that t >*r(S), t - 6(S),Af(E) . There exists a c E *E such that SC B [c, t], and then, Iix - ¿Ii = stjjx - cil - lIx - cil < t'- 6(S) .X f(E), tlx E S . Since the first and last members are standard, we have 11 i-c11 < 6(S) .Af(E), for every xE S, so that r(S) < 6(S) .Af(E), and we conclude Af(E) < Af(E) . Theorem 3 . If E is a real Banach &pace such that A f(E) = 1/2, then E is an L l -predual . Proof : We consider an infinitesimal hull E of E . Then, E has the radial intersection property (2,4), that is, given four closed balls in E with the &ame radius, which intersect in pairs, the total intersection is non empty . Indeed, let p be a positive real number, and let xl, í2, i3, 24 E E, such that II .ii - xi il < 2p, for i, j = 1, 2, 3, 4 . We take S = {xi, i2, , i3, ¡J, a finite subset of E with 6(S) _< 2p, and, so, r(S) _< p . For every natural number p, there exists c p E E such that Iixi -Q¡ < p + (1/2p), for i = 1, 2, 3, 4, and, consequently, there exists c p E fin*E such that lixi - CP11 < p + (1/p), for i = 1, 2, 3, 4 . We consider now the sequence (cp)PEN in *E, which can be enlarged to an internal sequence (cp)pE*rN in *E . The set of index p E *h1 such that lixi -c p jj < p+(1/p), for i = 1, 2, 3,4 is an internal subset of *N containing all standard natural numbers, and, so, if we work in a suitably saturated model (cf . [3]), it contains an infinite index, wE *NI . Then, the element c,, E *E is 34 4  J .M . BAYOD, M .C . MASA finite, because Ilxi - c w ll <_ p, so that we can take c , E E thus verifying that j1x i - c u , 11< p, for i = 1, 2, 3, 4 . Therefore, E is an L 1 -predual ([2, theorem 6, pg . 212]), that is, E' is an L l space . Projecting E' over E' by means of the functiori T E É' -> TSE E E', we have that E' is also an L i space ([2, theorem 3, pg . 1620, and then E is an L l -predual . Lemma 4 . ab(E) is the infmum of the positive real numbers r such that for every -y > 0, whenever (xcJaEI C E is a y -Cauchy net (that is, given e > 0, there exists ceo E I such that lixa - xpll < -y + e, for every pair of subíndices a, ,p E I greater ¡han or equal to ao), (xcjcrEI has some ry-limit x in E (that is, given e > 0, there exists ao E I such that Iix a - xjj <_ ry -Ie, for every subindex a E I greater than or equal to ao) . Proof . . Let S be a bounded subset of E, with more than one point . For every natural númber n, we consider S(n) = S x {n} and the bijection x r-r x(n) = (x, n) . We take now I = U S(n) with the order relation nEN a <p  a = / 0 V (a = x(n), ,Q = y(-), n < m), which makes I a directed set . Over it, we build the net (xa)aC-I defined by x, = x if a E I is such that a = x(n), for some nE N . Thus, (xa)aEI ls a 6(S)-Cauchy net in E, with range S, and such that for every y E S and every a E I, there exists a /3 E I, /i >_ a, which satisfies xp = y, that is, for every y E S there exists and infinite index 0 which satisfies xp = y . We call A'(E) the infimum of the positive real numbers r such that for every y> 0, every y-Cauchy net has an ry-limit in E, and let t be greater than a'(E) . The previously builded net (xa)aEI has a t6(S)-limit x E E, and, so, Ilx a - xil ;5 tó(S) for every infinite índex a . Hence, we have ¡¡y - xjj S t6(S) for every y E S, and, because both members are standard, lix - y¡¡ < t6(S) for every y E S, that is, SC E[s,tb(S)j . Thus, r(S) <_ t6(S) for every t > A' (E), and ab(E) < A'(E) . Conversely, if (xa)arj is a ,. -y-Cauchy net in E for some y > 0, we put S « _ {xp : 3 E I, fl >_ a}, for every a E I . Thus, every set S I is bounded and we can suppose that it has more than one point (otherwise the proof is trivial) ; therefore, given e > 0, there exists a E I such that ó(S a ) < y+e/Ab(E) . Then, r(Sa) :5 ó(Sa) . Aj(E) < (y + e/Ab(E))-ME) = - Y- , \b(E) + e, and, so, there exists, c E E E such that Iixp-ce¡¡ <_ y,Ab(E)+e for every ,Q > a . Hence, c E is a (y .Ab(E) -Ie)-limit of (xa)aEI, and, since this is true for every e > 0 and for every y-Cauchy net in E, it follows that A'(E) < Ab(E) . CHEBYSHEV COEFFICIENTS FOR L'-PREDUALS  34 5 Theorem 5 . If E is a P1(K) space, then Ab(E) = Ab(K) . Proof . If we embed K into E by means of a linear isometry, identifying it with a onedimensional subspace of E, the Hahn-Banach theorem assures the existence of a projection of norm 1, P : E ---> K, whence we deduce Ab(K) < Ab(E) . We will prove the reciproca¡ inequality in several stages : (I) In the first place, we observe that if E E Pl(K) and E is an infinitesimal hull of E, then Ab(E) <_ Ab(E), because, considering the canonical linear isometry E --+ E, there exists a contractive projection E -+ E . (II) Let F be a non empty set . We denote by P0(I', K) the set of all bounded functions from F to K, with the uniform norm . Giving to r the discrete topology, we know that 1°°(r, K) is linearly isometric to the space C(f I', K) of continuous functions with values in K defined over the Stone-Cech compactification of r ; so, l 00 (17,K) E PI(K) ([2]), and A b (100(r , K)) < ab(¡-(r,K)), by (I) . (III) We suppose that F is a finite set . We will prove that, in this case, Ab(1'(17, K)) < Ab(K) . Because 100(17, K) is a finite dimensional, we know that its bounded and finite Chebyshev coefficients are equal, as are those of K . Let p be a real number, p > Af(K), and let S = {xl . . . . . . xx,,} be a finite subset of 1 0° (17, K) . Fixed ^y E F, we consider the finite subset of K S 7 = {xl (y), . . . , x .(-y)} ; then, r(Sy)/S(S7) < A f(K) < p, and r(S . y ) < p .ó(S .,) = p .  max  I xi(y) - xi(y)j <p .  max  11xi - xj1l = p .b(S) . 1<i, j<n  1 :5i, ,1<n Then, there exists a centre c, E K such that S 7 C B[c 7 , p .b(S)] CK . We define thus a function from F in K which associates c ., to every -y, and which satisfies Ilxi - cil = sup I xi( - y) - cl C p .b(S), i = 1, . . . , n, YEr so that S C B[c, p .b(S)] C 1 00 (17, K) . Therefore, r(S) < p .b(S), and we conclude A f(l'(r K)) < A f (K) . (IV) The inequality A b (100(F , K)) <Ab(K) is valid also when F is an infinite set . Indeed, let (xa)aEI be a y-Cauchy net in 1 00 (F,K), for y > 0 . We take X o =K U F U I in order to build a superstructure X with base X o and over it a polysaturated nonstandard model satisfying the Ro-isomorphism property (cf . [3, sec . 0 .4 .]) . In this case, we can identify ¡ 00 (17, K) to ¡ 00 (w, K), for every infinite natural number w, since ¡'(w, K) is isometrically isomorphic to ¡ 00 (INI,K), and this is so to ¡ 00 (I',K) ([4, theorem 2 .11]) . Let p be a natural number . There exists an index a p E I such that Ilxa - xp11 < y + 1/(2p), when a 0 E `I, a,/l > a p . We consider the set 34 6  J .M . BAYOD, M .C . MASA S = {x,, : a E *I,a >_ a p }, an internal *-bounded subset of 1°° (w, K), where we can apply (III), and so * r(S)l *6 (S) < I\b(K) *r(S) < a b (K)*6(S) < ab(K)(-Y+ 1/(2p)) < t, where t = Ab(K)(-y -F1/p) .  Since *r(S) < t, there exists c p E 1 °° (w, IK) such that S C B[c p , t] and cp E l-(w, K) = l-(I', K) satisfies  cp ll < t, for aEI,a>a p . Bearing in mind that 1°°(I',IK) E Pj(IK), consider the natural embedding 1°°(I', IK) --> Í' (I', VK) ; then, there exists a projection of norm 1, P : (I', IK) -> 1 - (r, K), and x = P(c p ) is an element of 1'(I',OK)`which satisfies lix« - xjj = IIP(xa) T P(cjj C j1xa - Ql < t, a E I, a > ap . Thus, for every p > Ab(K), taking a p E N greater than the real number Ab(K)/y(P - Ab(K)), we Nave IIx« - XII < ab(K)(y +p) < P7, a E I, a >_ ap, that is, x is ap - y - limit of (xj,, in 1°°(I', Bá) .  Since this is valid for every yCauchy net in 100(I',IK) and for any -y > 0 we conclude by Lemma 4that ab(1'(r,K)) _< ab(o-c) . (V) Now, we can embed E linearly and isometrically into 1°°(E, IK) by means of the application 0 :E -> 1 00 (E, K) x->¢(x) :E-~6á y  O(x)(y) = fy(x) where f y is a continuous linear functional from E to K which satisfies jjf y jj = 1 and f y (y) = 11yII, the existente of which is guaranteed by the Hahn-Banach theorem . So, there exists a projection of norm 1, 1°°(E, IK) -> E, which permits us to deduce the inequality Ab(E) < Ab(1°°(E,IK)), and, from the result in the preceding paragraph, Ab(E) < Ab(K) . Referentes [1] BAYOD, J .M ., MASA, M .C ., Coeficientes de Chebyshev en espacios de funciones continuas, to appear in the Actas de las XIV Jornadas Hispano-lusas de Matemáticas, La Laguna, Spain (1989) . [2J LACEY . H .E ., "The Isometrictheory of classicál Banach spaces," Springer-Verlag, Berlin, Heidelberg, New York, 1974 . CHEBYSHEV COEFFICIENTS FOR L 1 -PREDUALS  347 [3] STROYAN, K .D ., BAYOD, J .M ., "Foundations of infinitesimal stochastic analysis," North-Holland, Amsterdam, 1986 . [4] WARD HENSON, C ., The isomorfism property in nonstandard analysis and its use in the theory of Banach spaces, The Journal of Symbolic Logic 39, 4 (1974) . 1980 Mathematics subject classificaiions : 46B20, 46B25 Departamento de Matemáticas, Estadística y Computación Universidad de Cantabria 39071Santander SPAIN Rebut el 18 de Desembre de 1989