1. Introduction Let X Be a Non-Reflexive Real Banach Space. We Always Consider X As Canonically Embedded in Its Bidual X . We Re

GEOMETRY OF SPACES OF COMPACT OPERATORS

T. S. S. R. K. RAO

Abstract. For non-reflexive Banach spaces X,Y , for a very smooth point in the space of compact linear operators K(X,Y ), we give several sufficient conditions for the adjoint to be a very smooth point in K(Y ∗,X∗). We exhibit new class of extreme points in the dual unit ball of injective product spaces. These ideas are also related to Birkhoff-James orthogonality in spaces of operators.

1. Introduction Let X be a non-reﬂexive real Banach space. We always consider X as canonically embedded in its bidual X∗∗. We recall that a unit vector x0 ∈ X is said to be a smooth point, ∗ ∈ ∗ ∗ if there is a unique vector x0 X such that x0(x0) = 1 = ∥ ∗∥ x0 . x0 is said to be very smooth if it is also a smooth point of X∗∗, see [16]. In this paper we investigate the question, when are adjoints of compact operators, very smooth points in the corresponding spaces of compact operators between dual spaces. It is easy to see that if T ∗ ∈ K(Y ∗,X∗) is a very smooth point then so is T ∈ K(X,Y ). We do not know geometric conditions to ensure that very smooth points remain very smooth in all the duals of higher

2000 Mathematics Subject Classification. Primary 47 L 05, 46 B28, 46B25 . Key words and phrases. Smooth points, very smooth points, adjoints of operators, spaces of operators, essential norm, injective and projective tensor product spaces . 1 2 RAO order. See Proposition 4.1 of [5] for an example of a Banach space X where this fails in the second dual. By taking operator T of rank one, one can see from Proposition 4.1 of [5] that T ∗ need not always be very smooth in K(X∗). We note that if weak∗ and weak sequential convergence coincide on the unit sphere of X∗, then any smooth point of X is very smooth. For non-reflexive Banach spaces, recently this author gave a sufficient condition on the range space Y that implies T ∗ is a smooth point of K(Y ∗,X∗) whenever T is a smooth point, see [14].

For a Banach space X, let X1 denote the unit ball and

∂eX1 denote the set of extreme points. For a unit vector { ∗ ∈ ∗ ∗ } x0, let F = x X1 : x (x0) = 1 . It is well known that F ∗ is a weak -closed extreme subset of the unit ball and x0 is { ∗} a smooth point if and only if ∂eF = x0 . Throughout the paper we use this notation while considering smooth points. In what follows we will be using Lemma III.2.14 of [6] which ∗ ∈ ∗ applied to our context says that x X1 considered as a functional on X has a unique norm preserving extension to X∗∗ if and only it is a point of weak∗-weak continuity ∗ for the identity map on X1 . Thus x0 is a very smooth { ∗} ∗ ∗ point if and only if ∂eF = x0 and x0 is a point of weak - ∗ weak continuity for the identity map on X1 . We improve ∗ this criteria for Banach spaces X for which X1 is the norm closed convex hull of its extreme points (this for example happens when X∗ has the Radon-Nikodym property or is SMOOTH POINTS 3 ∗ a strictly convex space) by showing that if x0 is a point of ∗ ∗ weak -weak continuity for the identity map on ∂eX1 , then ∗ x0 is a very smooth point. Study of weak -weak points of continuity in the dual unit ball of spaces of operators involves the knowledge of the bidual of spaces of operators. In this article we take advantage of the relatively few known instances, where the bidual again has an identiﬁcation as a space of operators. For x∗∗ ∈ X∗∗ and y∗ ∈ Y ∗, we denote by x∗∗ ⊗ y∗ the linear functional deﬁned on the space of operators by (x∗∗ ⊗ y∗)(T ) = x∗∗(T ∗(y∗)), for any operator T and note that ∥x∗∗ ⊗ y∗∥ = ∥x∗∗∥∥y∗∥. We denote a rank one operator by x∗ ⊗ y, so that (x∗∗ ⊗ y∗)(x∗ ⊗ y) = x∗∗(x∗)y∗(y). Let Y (IV ) denote the fourth dual of Y . We show that if Y is a Banach space such that every extreme point of the dual unit ball is a denting point, then for any very smooth point T ∈ K(X,Y ), under the assumptions that the unit ball of K(Y ∗,X∗) is a norm closed convex hull of its extreme points, if under the canonical embedding K(Y ∗,X∗)∗∗ = L(X∗∗,Y (IV )) , then T ∗ is a very smooth point of K(Y ∗,X∗) and hence a smooth point of L(X∗∗,Y (IV )).

For the same class of Banach space X,Y , for T ∈ L(X,Y ), we can only give a relative answer for the very smooth na- ture of the adjoint operator. If K(Y ∗,X∗) is a M-ideal in L(Y ∗,X∗) (see Chapters I and VI of [6] for these concepts) 4 RAO then for a very smooth non-compact operator T whose essential norm is strictly smaller than the norm, we show that T ∗ is also a very smooth operator. It follows from [11] and [10] that any denting point x of ∗ X1 is a point of weak -norm continuity for the identity map ∗∗ ∗ on X1 (and hence a point of weak -weak continuity). These results also show that under the canonical embedding of X in the higher duals of even order of X, x continues to be a denting point.

∈ ∗∗ Our analysis involves the open question for x ∂eX1 ∗ ∈ ∗ ∗ ∗ and y ∂eY1 , if x, y are points of weak -weak continuity of the identity map on the respective unit balls, when is the functional x ⊗ y∗, a point of weak∗-weak continuity for the identity map on the dual unit ball of the space of operators? Treating denting and weak∗-denting points as extreme points that are points of weak-norm (weak∗- norm) continuity for the identity map on the unit ball, in their seminal work in the 80’s these questions for the space of compact operators (injective product spaces) were an- swered in the aﬃrmative by Ruess and Stegall.

For a compact Hausdorff space K, one can take advantage of the property, all duals of even order of a continuous function space C(K) can again be identified as spaces of continuous functions. Using this we show that for a compact Hausdorff space K , if every smooth point of X∗∗∗ is very smooth, then for a very smooth T ∈ K(X,C(K)), T ∗∗ SMOOTH POINTS 5 is a very smooth point in the corresponding spaces of operators on the dual spaces. We also give geometric conditions on the range space (similar conditions can also be given on the domain space) to ensure that smooth points of K(X,Y ) remain smooth in L(X,Y ). See [7] for a characterization of smooth points and Fréchet smooth points in spaces of operators. We also note that for an infinite compact Hausdorff space K and an infinite dimensional Banach space X such that X∗∗ has the Radon-Nikodym property (R. N. P), K(X,C(K))∗∗ is not isometric to L(X,C(K)).

We recall that x ∈ X is said to be Birkhoff-James orthog- onal to y ∈ X (write x ⊥ y) if ∥x+λy∥ ≥ ∥x∥ for all scalars λ. Let {x}⊥ = {y ∈ X : x ⊥ y}. It is known that {x}⊥ is closed under addition (and hence a subspace) if and only if x is a smooth point of X. Thus the natural question, when is {x}⊥ considered in X∗∗ a subspace? is equivalent to when is x a very smooth point. Thus an equivalent version of the question considered here is, if {T }⊥ is a subspace, when is {T ∗}⊥ a subspace? Another interesting question that has attracted a lot of attention is, if T attains its norm and T ⊥ S then when does there exist a unit vector x0 such that ∥T ∥ = ∥T (x0)∥ and T (x0) ⊥ S(x0). Using the ideas considered here, we improve on a result from [17] in L(X,Y ). 6 RAO 2. Main Results Our first result identifies very smooth points under a weaker condition than the one mentioned in the introduc- ∗ tion. It is easy to see that X1 is a norm-closed convex hull ∗∗∗ ∗ of its extreme points if and only if X1 is the weak -closed ∗ convex hull of ∂eX1 .

∗ Proposition 1. Let X be a Banach space such that X1 is a norm-closed convex hull of its extreme points. Let x0 ∈ X1 ∗ ∗ be a smooth point. Suppose x0 is a point of weak -weak ∗ continuity for the identity map on ∂eX1 . Then x0 is a very smooth point.

∗ ∈ ∗ ′ { ∈ ∗∗∗ Proof. Let x0 ∂eX1 and let F = Λ X1 : Λ(x0) = ∗ } ′ { ∗} ∈ ′ 1 = x0(x0) . We will show that ∂eF = x0 . Let λ ∂eF . ∈ ∗∗∗ Then λ ∂eX1 , so by our hypothesis and Mailman’s converse of Krein-Milman theorem (see [9], page 74), λ is ∗ ∗ { ∗ } ⊂ ∗ in the weak -closure of ∂eX1 . Let xα α∈Γ ∂eX1 be a net ∗ → ∗ ∗∗∗ such that xα λ in the weak -topology of X . Since x0 ∗ | ∗ is a smooth point, we have x0 = λ X. As x0 is a point of ∗ ∗ → weak -weak continuity on the set of extreme points, xα ∗ ∗ x0 in the weak-topology. Therefore λ = x0. K ∗ { ∗∗ ⊗ ∗ ∗∗ ∈ We recall from [15] that ∂e (X,Y )1 = x y : x ∗∗ ∗ ∈ ∗} ∂eX1 , y ∂eY1 . It is known that any Banach space with a locally uniformly rotund norm has the property that all unit vectors are denting points, see [8] page 43. Also any separable Banach space can be renormed to be locally uniformly rotund. We will be using notations and terminology SMOOTH POINTS 7 from tensor product theory and will refer Chapter VIII of [4] for the results that we will be using here. The assumptions made on K(Y ∗,X∗) in the theorem below can be met by assuming the dual has the Radon-Nikodym property and X∗∗ or Y ∗ has the approximation property. Thus we implic- itly assume that K(Y ∗,X∗) is the injective tensor product of Y ∗∗ and X∗.

Theorem 2. Let X,Y be non-reﬂexive Banach spaces such ∗ ∗ ∗ ∗∗∗ b ∗∗ that under the canonical embedding K(Y ,X ) = Y ⊗πX . Assume further that the unit ball of K(Y ∗,X∗)∗ is a norm closed convex hull of its extreme points. Let Y be such ∗ that every extreme point of Y1 is a denting point. Let T ∈ K(X,Y ) be a very smooth point. Then T ∗ is a very smooth point of K(Y ∗,X∗).

∈ K ∗∗ ∈ Proof. Let T (X,Y ) be a very smooth point. Let x0 ∗∗ ∗ ∈ ∗ ∗∗ ⊗ ∗ ∂eX1 , y0 ∂eY1 be such that (x0 y0)(T ) = 1. Since T is a very smooth point, by what we have remarked above, ∗∗ ⊗ ∗ ∗ x0 y0 is a point of weak -weak continuity for the identity K ∗ K map on (X,Y )1. Using operators of rank 1 in (X,Y ), ∗∗ ∗ it is easy to see that this implies x0 is a point of weak - ∗∗ weak continuity for the identity map on X1 and a similar ∗ ∗ ∗∗ statement is true about y0. As X1 is weak -dense in X1 ∗∗ ∈ we get that x0 = x0 X.

⊗ ∗ ∗ ∗ ∗ We thus have (x0 y0)(T ) = y0(T (x)) = T (y0)(x0) = ∗ ⊗ ∗ (y0 xo)(T ). We further note that since by hypothesis ∗ ∗ ∈ ∗∗∗ ∗ y0 is a denting point, y0 ∂eY1 is a weak -denting point 8 RAO and hence a point of weak∗-weak continuity for the identity map. Also by the result of Ruess and Stegall quoted ∗ ⊗ ∈ K ∗ ∗ ∗ earlier, y0 x0 ∂e (Y ,X )1. Thus as in the proof of Proposition 2 in [14] we can show that it is the only point K ∗ ∗ ∗ ∗ ∗ in ∂e (Y ,X )1 attaining the norm at T and hence T is ∗ ∗ a smooth point of K(Y ,X )1.

∗⊗ In view of Proposition 1, we need to show now that y0 x0 is a point of weak∗-weak continuity for the identity mapping K ∗ ∗ ∗ { ⊗ } ⊂ K ∗ ∗ ∗ on ∂e (Y ,X )1. Let τα Λα α∈Γ ∂e (Y ,X )1 be a ⊗ → ∗ ⊗ ∗ net such that τα Λα y0 x0 in the weak -topology. ∈ ∗∗∗ ∈ ∗∗ Here for any α, τα ∂eY1 and Λα ∂eX1 . Once again → ∗ ∗ we can see that τα y0 in the weak and hence in the weak ∗ topology. As y0 is a denting point, by the result of [11], we → ∗ → get that τα y0 in the norm. We also have Λα x0 in the weak-topology.

We have under the canonical embedding

∗ ∗ ∗ ∗∗∗ b ∗∗ ∗ ∗ ∗∗ ∗∗ (IV ) K(Y ,X ) = Y ⊗πX , K(Y ,X ) = L(X ,Y ).

Let S ∈ L(X∗∗,Y (IV )). In view of the canonical embedding,

S(τα ⊗ Λα) = S(Λα)(τα). As S is continuous, S(Λα) → → ∗ S(x0) in the weak-topology. Since τα y0 in the norm and as all the vectors involved are unit vectors, we see → ⊗ ∗ ∗ ⊗ that S(Λα)(τα) S(x0 y0). Hence y0 x0 is a point of weak∗-weak continuity for the identity mapping on the set of extreme points. Thus T ∗ is a very smooth point. SMOOTH POINTS 9 Remark 3. We note that the arguments given above also go through if we assume every extreme point of X1 is a dent- ∗ ing point, instead of the assumption on Y1 . Assuming the hypothesis as in the above theorem, it follows from Propo- sition 2 in [14] that T ∗∗ is a smooth point of K(X∗∗,Y ∗∗) ⊗ ∗ and x0 y0 is the unique functional attaining the norm at T ∗∗.

We next specialize to injective tensor products. See Chap- ter VIII, Section 2 [4]. The proof of the corollary can be deduced form the above theorem.

Corollary 4. Let X,Y be non-reﬂexive Banach spaces such ∗ ∗ b ∗ ∗ b ∗ that (X⊗ˇ ϵY ) = X ⊗πY . Suppose the unit ball of X ⊗πY is the norm closed convex hull of its extreme points. ∗ ∈ ∗ ∗ ∈ ∗ ∗ Let x0 ∂eX1 and y0 ∂eY1 be points of weak -weak and weak∗-norm continuity for the identity map on the re- ∗ ⊗ ∗ ∗ spective unit balls. Then x0 y0 is a point of weak -weak ∗ b ∗ continuity for the identity map on the unit ball of X ⊗πY .

Remark 5. In the proof of the above corollary it is crucial that the injective tensor product space is one of the preduals ∗ b ∗ of X ⊗πY . We do not know if the corollary is true if one merely assumes that X∗ and Y ∗ have unique preduals and ∗ b ∗ X ⊗πY is a dual space.

We recall from [6] Chapter I that a closed subspace M ⊂ X is said to be a M-ideal if there is a linear projection P : X∗ → X∗ such that ker(P ) = M ⊥ and ∥x∗∥ = ∥P (x∗)∥ + ∥x∗ − P (x∗)∥ for all x∗ ∈ X∗. It follows from the basic 10 RAO properties of M-ideals that functionals in M ∗ have unique norm-preserving extension to X∗, so that P (X∗) is isomet- ∗ ∗ ∗ ∪ ⊥ ric to M and ∂eX1 = ∂eM1 ∂eM1 . Chapter VI of [6] exhibits a large classes of examples of Banach spaces X,Y for which K(X,Y ) is a M-ideal in the space of bounded operators L(X,Y ). ⊂ ∗ ∈ ∗ Lemma 6. Let M X be a M-ideal. If m ∂eM1 is a point of weak∗-weak continuity for the identity map on ∗ ∗ ∂eM1 , then it is also a point of weak -weak continuity for ∗ the identity map on ∂eX1 . ∗ ∗ Proof. We note that ∂eM1 is a relatively weak -open set ∗ { ∗ } ⊂ ∗ in ∂eX1 . Let xα α∈Γ ∂eX1 be a net converging in the weak∗ topology to m∗. We may assume, by going through { ∗ } ⊂ ∗ ∗ → ∗ a subnet if necessary, that xα ∂eM1 . Since xα m in the weak∗-topology of M ∗, by hypothesis we get that ∗ → ∗ xα m weakly. In what follows we note that for any T ∈ L(Y ∗,X∗), ∥ ∥ {| ∗ | ∈ ∗∗ ∈ ∗∗∗} since T = sup τ(T (Λ)) :Λ ∂eX1 and τ ∂eY1 , L ∗ ∗ ∗ ∗ we have that (Y ,X )1 is the weak -closed convex hull of { ⊗ ∈ ∗∗ ∈ ∗∗∗} τ Λ:Λ ∂eX1 and τ ∂eY1 .

In the next theorem we consider adjoints of very smooth points of L(X,Y ) for Banach spaces which satisfy the hypothesis of Theorem 2. Our answer depends on the relative position of the space of compact operators.

Theorem 7. Let X,Y be Banach spaces satisfying the hypothesis of Theorem 2. Assume further that K(Y ∗,X∗) SMOOTH POINTS 11 is a M-ideal in L(Y ∗,X∗). Let T ∈ L(X,Y ) be a very smooth non-compact operator such that its essential norm, d(T, K(X,Y )) < ∥T ∥ . Then T ∗ is a very smooth point of L(Y ∗,X∗).

∥ ∥ L ∗ ∗ ∗ Proof. Let T = 1. We have by hypothesis ∂e (Y ,X )1 = K ∗ ∗ ∗ ∪ K ∗ ∗ ⊥ ∗ K ∗ ∗ ≤ ∂e (Y ,X )1 ∂e (Y ,X )1 . Since d(T , (Y ,X ) d(T, K(X,Y )) < 1, proceeding as in the proof of The- ∗ ⊗ ∗ ∗ ∈ ∗ orem 2, we see that (y0 x0)(T ) = 1 for y0 ∂eY1 , ∈ ∗∗ x0 ∂eX1 . Because of the M-ideal hypothesis we also ∗ ⊗ ∈ L ∗ ∗ ∗ have y0 x0 ∂e (Y ,X )1 and as in the proof of Theo- ∗ ⊗ rem 2, y0 x0 is a point of continuity for the identity map K ∗ ∗ ∗ on ∂e (Y ,X )1.

Since K(Y ∗,X∗) is a M-ideal in L(Y ∗,X∗), by Lemma 6 ∗ ⊗ ∗ we get that y0 x0 is also point of weak -weak continuity L ∗ ∗ ∗ for the identity map on ∂e (Y ,X )1. Therefore by Propo- sition 1 again, we get that T ∗ is a very smooth point of L(Y ∗,X∗).

Our next theorem deals with the question of higher order adjoints being smooth for operators valued in a C(K) space for a compact Hausdorff space K. In what follows we follow the identification of K(X,C(K)) with the space of vector-valued continuous functions C(K,X∗) and that of L(X,C(K)) as the space W ∗C(K,X∗) of functions that are continuous when X∗ is equipped with the weak∗-topology. Both the function spaces are equipped with the supremum ∗ ∗ { ⊗ ∗∗ norm. We also recall that ∂eC(K,X )1 = δ(k) x : 12 RAO ∈ ∗∗ ∈ ∗} k K , x ∂eX1 . Here δ(k) stands for the evaluation functional on C(K). Theorem 8. Let X be a Banach space such that any smooth point of X∗∗∗ is a very smooth point. Let K be a compact Hausdorff space and let T ∈ K(X,C(K)) be a very smooth point. Then T ∗∗ is a very smooth point in the corresponding spaces of bounded operators. Proof. We will show that T ∗∗ is a very smooth point of L(X∗∗,C(K)∗∗). Using the identification of K(X,C(K)) with C(K,X∗) we have that for a very smooth unit vector ∈ ∗ ∗∗ ∈ ∗∗ ∈ ∗∗ f C(K,X ), x0 (f(k0)) = 1, for k0 K and x0 ∂eX1 . Since f is a very smooth point by the arguments indicated during the proof of Theorem 2, it is easy to see that ko is ∗∗ ∈ an isolated point of K and x0 = x0 X, f(k0) is a very ∗ smooth point of X1 and hence by hypothesis it is also a very ∗∗∗ smooth point of X1 . Let ∆ be the maximal deal space of C(K)∗∗. T ∗∗ thus corresponds to a continuous extension ′ ∗∗∗ of f, say f : ∆ → X . It is easy to see that k0, in the canonical embedding of K in ∆, is an isolated point of ∆. As f ′(k ) = f(k ) is a very smooth point of X∗∗∗, using 0 0 ⊕ ∗ ∗∗∗ ∗∗∗ ∗ the decomposition W C(∆,X ) = X ∞ W C(∆ − ∗∗∗ ∗∗ {ko},X ), it is easy to see that T is a very smooth point of W ∗C(∆,X∗∗∗) = L(X∗∗,C(K)∗∗).

Remark 9. Without the additional assumption on X∗, it follows from the arguments given above that if T ∈ K(X,C(K)) is a very smooth point then T ∗∗ is a smooth point of L(X∗∗,C(K)∗∗). SMOOTH POINTS 13 We recall from [13] that a closed subspace Y ⊂ X is said to be a strict ideal if there is a linear projection P : X∗ → ∗ ⊥ ∗ X of norm one such that ker(P ) = Y and P (X )1 is ∗ ∗ ∗ weak -dense in X1 . It is easy to see that P (X ) is isometric to Y ∗ via the mapping P (x∗) → x∗|Y . Therefore in this situation under the canonical embedding one has Y ⊂ X ⊂ Y ∗∗ (see Lemma 1 [12]). It is known that under assumptions of compact metric approximation property with nets, of X∗ or Y , K(X,Y ) is a strict ideal in L(X,Y ).

In the next proposition we give a version of Lemma 6 for strict ideals. We recall that a Banach space X is said to be a Grothendieck space, if weak∗ and weak sequential convergence coincide in X∗. The following is an abstract version of Proposition 3.4 from [5].

Proposition 10. Let Y ⊂ X be a strict ideal. Suppose the quotient space X/Y is a Grothendieck space. Then any very smooth point of Y is a very smooth point of X.

Proof. Let y0 ∈ Y be a very smooth point, ∥y0∥ = 1 = ∗ ∗ ∈ ∗ y0(y0) for some y0 ∂eY . By hypothesis y0 is also a ∗ smooth point of X. Thus y0 has a unique norm preserving ∗ ∈ ∗ { ∗} extension, still denoted by y0 ∂eX1 . Since y0 is a ∗ ∗ weak − Gδ, to show weak -weak continuity at the identity ∗ { ∗ } ⊂ ∗ ∗ → ∗ ∗ map on X1 , let xn n≥1 X1 and xn y0 in the weak - topology. Let P : X∗ → X∗ be a projection associated with the deﬁnition of strict ideal with ker(P ) = Y ⊥. 14 RAO ∗ → ∗ ∗ ∗ Clearly P (xn) P (y0) = y0 in the weak -topology of ∗ ∗ ∗ ∗ Y = P (X ). Since y0 is a point of weak -weak continuity ∗ ∗ → ∗ for the identity map on Y1 , we have P (xn) y0 in the weak ⊥ ∗ ∗ − ∗ → topology. As Y = (X/Y ) , we also have xn P (xn) 0 ∗ → ∗ in the weak topology. Thus xn y0 in the weak topology. Therefore y0 is a very smooth point of x.

Our next result gives conditions under which smooth points of K(X,Y ) continue to be smooth points of L(X,Y ). Any reﬂexive Banach space with a locally uniformly rotund norm satisﬁes the hypothesis assume on Y in the following theorem.

Theorem 11. Let Y be a Banach space such that every ∗ ∗ ∈ extreme point of Y1 is a weak -denting point. Let T K(X,Y ) be a smooth point. Then T is a smooth point of L(X,Y ).

∗∗ ∈ ∗∗ ∗ ∈ ∗ ∗∗ ⊗ ∗ Proof. Let x0 ∂eX1 , y0 ∂eY1 be such that x0 y0 is K ∗ ∗∗ ∗ ∗ the unique functional in (X,Y ) such that x0 (T (y0)) = ∥ ∥ { ∈ L ∗ ∥ ∥} T . Let F = Λ (X,Y )1 : Λ(T ) = T . We will ∗∗ ⊗ ∗ show that every extreme point of F coincides with x0 y0. This completes the proof. ∈ L ∗ { ∗∗ ⊗ ∗ ∗∗ ∈ Let Λ ∂eF . Since (X,Y )1 = CO x y : x ∗∗ ∗ ∈ ∗} ∗ ∂eX1 , y ∂eY1 (here the closure is taken in the weak - { ∗∗} ⊂ ∗∗ { ∗ } ⊂ topology), there exists nets xα α∈Γ ∂eX1 and yα α∈Γ ∗ ∗∗ ⊗ ∗ → ∗ ∂eY1 such that xα yα Λ in the weak -topology. SMOOTH POINTS 15 ∗∗ → ∗∗ We assume without loss of generality that xα x and ∗ → ∗ ∗ yα y in the weak -topology. Since T is a compact operator it is easy to see that (x∗∗ ⊗ y∗)(T ) = Λ(T ) = ∥T ∥. Since T is a smooth point of K(X,Y ), we get that Λ = ∗∗ ⊗ ∗ ∗∗ ⊗ ∗ K x0 y0 = x y on (X,Y ). Also we may assume ∗∗ ∗∗ ∗ ∗ x = x0 , y = y0. Now let S ∈ L(X,Y ). Since by hypothesis y∗ is a weak∗- ∗ → ∗ ∗ → ∗ denting point, yα y in the norm. Thus S(yα) S(y ). Therefore we get that Λ(S) = (x∗∗ ⊗ y∗)(S). Hence T is a smooth point of L(X,Y ).

We next give an analogue of Theorem 3.1 from [17] on a necessary condition for Birkhoﬀ-James orthogonality for operators.

Theorem 12. Let Y be a reﬂexive, smooth and strictly convex real Banach space. Let T ∈ L(X,Y ) be a smooth point that attains its norm. For any S ∈ L(X,Y ), if T ⊥ S then there is a unit vector x0 with ∥T (x0)∥ = ∥T ∥ and

T (x0) ⊥ S(x0).

Proof. We follow the arguments given during the proof of Theorem 3.1 in [17] and consider Y as equipped with a semi-inner product. Assume ∥T ∥ = 1 = ∥T (x0)∥ = ∥x0∥. ∗ ∈ ∗ ⊗ ∗ Let y0 ∂eY1 be such that (x0 y0)(T ) = 1. As T is − ⊗ ∗ smooth it attains its norm only at x0 or x0 and x0 y0 L ∗ is the only extreme point in ∂e (X,Y )1, that attains its ∗ norm at T . Since Y is reflexive we also know that y0 is determined by an element y0 ∈ Y . 16 RAO Now it follows from the arguments given in the last part of the proof of Theorem 3.1 that T (x0) ⊥ S(x0). We recall that for any Banach space X, under the canonical embedding of a space in its bidual, one has K(X,C(K)) ⊂ L(X,C(K)) ⊂ K(X,C(K))∗∗ (see the remarks before Propo- sition 10). In view of the assumptions made on the canonical identification of K(X,Y )∗∗ in this paper, the following result is worth noting. The proof uses the notion of a centralizer, which is preserved under surjective isometries, from the theory of Banach-Stone theorems. See [1] and [3]. In what follows we note that for an infinite dimensional space X and an infinite compact set K, the canonical in- clusions C(K,X∗) ⊆ W ∗C(K,X∗) ⊆ C(K,X∗)∗∗ do not coincide. It is well known that by hypothesis, the first inclusion is proper. We next show that the second embedding is also not onto. Since K is infinite, let B ⊂ K be a Borel set such that χB is not a continuous function. Let x0 ∈ X1 ∗ ∈ ∗ ∗ and x0 X1 be such that x0(x0) = 1. Using Singer’s theorem (see [4]) we have the identification of C(K,X∗)∗ as the space of regular vector measures M(K,X∗∗). Now ∗ ∗∗ consider the functional χB ⊗ x0 ∈ C(K,X ) defined by ⊗ ∗ ∗ ∈ ∗∗ (χB x0)(F ) = F (B)(x0) for F M(K,X ). Suppose ∈ ∗ ∗ ⊗ ∗ f W C(K,X ) is such that f = χB x0. It is easy to see that χB = x0 ◦ f. This is a contradiction.

Proposition 13. Let K be an infinite compact Hausdorff space and suppose X is an infinite dimensional Banach SMOOTH POINTS 17 space such that X∗∗ has the Radon-Nikodym property (R. N. P). Then K(X,C(K))∗∗ is not isometric to L(X,C(K)).

Proof. We recall that since X∗∗ has the R. N. P, ∗ ∗ ∗ ∗ b ∗∗ K(X,C(K)) = C(K,X ) = C(K) ⊗πX and K(X,C(K))∗∗ = L(X∗∗,C(K)∗∗). As before we iden- tify C(K)∗∗ as C(∆) and recall that ∆ is a hyperstonean space. We note that in this case the canonical isomeric embedding is via the map T → T ∗∗, which is not a surjection, since C(K) is not a reﬂexive space. Suppose L(X,C(K)) and L(X∗∗,C(∆)) are isometric. Since X∗∗ has R. N. P, it has no isomorphic copy of L1([0, 1]). It follows from Theo- rem 2 in [12] that K is also a hyperstonean space. It now can be seen from Theorem 1 and Lemma 2 in [3] that the centralizers of the spaces of operators are isometric to C(K) and C(∆) respectively. Thus from the classical Banach- Stone theorem one has that K and ∆ are homeomorphic. It is well-known that this can not happen for an inﬁnite set K.

Remark 14. We note from Theorem 1 of [2] that when K is not dispersed the validity of the canonical identiﬁcation ∗ ∗ ∗ ∗ b ∗∗ K(X,C(K)) = C(K,X ) = C(K) ⊗πX implies that X∗∗ has the R. N. P. On the other hand if K is dispersed one has K(X,C(K))∗∗ = L(X∗∗,C(β(S)) for some inﬁnite discrete set S. Thus in this case if we just assume that X∗∗∗ has trivial centralizer, one can again use Lemma 2 of [3] to conclude that if K(X,C(K))∗∗ is isometric to L(X,C(K)), 18 RAO then K is homeomorphic to β(S), which again gives a contradiction as β(S) has a perfect set.

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(T. S. S. R. K. Rao) Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India, E-mail : [email protected], [email protected]