ON α-NUCLEARITY AND TOTAL ACCESSIBILITY FOR SOME TENSOR NORMS α
Sten Kaijser, Oleg Reinov†
For a tensor norm α on the class of all Banach spaces is is a natural way to define the Banach operator ideal Nα associated with the tensor norm. We give sufficient conditions for an operator to be in Nα if its second adjoint possesses this property. We show one way to get from this the negative answers to some questions of A. Defant and K. Floret: their tensor norms g∞ , w1 and w∞ are not totally accessible.
The question goes back to A. Grothendieck [3] of whether an operator between Banach spaces is nuclear if its adjoint (or, what is here the same, its second adjoint) is nuclear (answered in negative by T. Figiel and W.B. Johnson [2]). The corresponding questions concerning some other operator ideals, such as the ideals of p-nuclear operators, or operators factoring through the spaces l p etc. have been considered, for example, in [5], [4], [6], [7], [8], [9]. In those works some sufficient conditions were given for the corresponding questions to have the positive answers, and in all of these conditions an assumption of the kind “either X ∗ or Y ∗∗∗ has some approximation property” appeared. Thus it is rather natural to state a general question of the above mentioned character. Namely, if α is a tensor norm on the class of all algebraic tensor product of Banach spaces and Nα is the operator ideal related to α of, let us say, α-nuclear operators (for the definitions see below), then in which cases it is true that for given Banach spaces X and Y and an operator T : X → Y it follows from the inclusion T ∗∗ ∈ Nα (X ∗∗ , Y ∗∗ ) the α-nuclearity of the operator T (from X to Y ) itself?
†
Supported by FCP “Integracija”, reg. No. 326.53, and by the Swedish Royal Academy of Sciences. AMS Subject Classification 2000: 46B28. Spaces of operators; tensor products; approximation properties Key words: α-nuclear operators, α-approximation property, totally accessible tensor norms, tensor products. 1
Below (see Theorem 1) we will give some general sufficient conditions (similar to those which were mentioned above) for a positive answer to the last question. The results of [4], [6], [7], [8], [9] show that, in general, the conditions are sharp and likely seem to be necessary. We use then our Theorem 1 to give negative answers to some questions, posed in the book [1] (we recall these questions after some preliminary definitions and notations). Recall [1; 12.1] that a tensor norm α on the class B of all Banach spaces assigns to each pair (E, F ) of Banach spaces E and F a norm α(·; E, F ) on the algebraic tensor e α F for the completion) such that the product E ⊗ F (shorthand: E ⊗α F and E ⊗
following two conditions are satisfied:
(1) α is reasonable, i.e.: ε ≤ α ≤ π, where ε and π, respectively, the smallest (injective) and the greatest (projective) Grothendieck tensor norms; (2) α satisfies the metric mapping property: if Ti : Ei → Fi , then ||T1 ⊗ T2 : E1 ⊗α E2 → F1 ⊗α F2 || ≤ ||T1 || ||T2 ||. If α is a tensor norm then αt denotes the transposed norm of α, i.e. αt ( P α( yj ⊗ xj ).
P
xj ⊗ yj ) :=
For a tensor norm α we denote by Nα the operator ideal generated by α: for every
pair of Banach spaces X, Y the space Nα (X, Y ) is a factor space of the tensor product e α Y via the canonical mapping X ∗ ⊗ e α Y → L(X, Y ) with the corresponding factor X ∗⊗ norm.
To formulate the above mentioned questions of A. Defant and K. Floret, we need some additional information. Given a tensor norm α on the class B, we define (see [1; 12.4]) for Banach spaces E and F and for z ∈ E ⊗ F →
α(z; E, F ) := inf{α(z; M, N ) | M ⊂ E, N ⊂ F, z ∈ M ⊗ N }
where the infimum is taken over all finite dimensional subspaces M ⊂ E and N ⊂ F, and 2
←
F α(z; E, F ) := sup{α(QE K ⊗ QL (z); E/K, F/L) | K ⊂ E, L ⊂ F }
where the supremum is taken over all closed finite codimensional subspaces K ⊂ E and L ⊂ F (here QE K : E → E/K is the canonical mapping). →
←
It is known (cf. [1; Prop. 12.4]) that α and α are tensor norms on B with ←
→
ε ≤ α ≤ α ≤ α ≤ π. →
←
A tensor norm α is called finitely generated if α = α and cofinitely generated if α = α. Examples. The tensor norms gp and wp , where 1 ≤ p ≤ ∞, are finely generated (cf. [1; 12.5 and 12.7]). For our purposes it is enough only to recall that for z ∈ E ⊗ F, where E and F are Banach spaces, and for 1 ≤ p ≤ ∞ wp (z; E, F ) = inf sup
||x0 ||≤1
n X
|hx0 , xi i|p
i=1
!1/p
sup ||y 0 ||≤1
n X i=1
!1/p0 n X 0 p z= xi ⊗ y i |hy , yi i| 0
i=1
(in the cases where p = 1 or p = ∞ evident modifications must be made). Note that, by definitions from [1], g∞ = w∞ and wp0 = wpt for all p ∈ [1, ∞]. Let us remark that g∞ is the tensor norm associated with ∞-nuclear operators, w∞ — with compactly c0 -factored operators and w1 — with compactly l1 -factored operators. →
←
A tensor norm α is called totally accessible if α = α, i.e. if α is finitely and cofinitely generated. Some of the questions of A. Defant and K. Floret [1; 21.12]: are the last three norms totally accessible? Bellow we answer the questions in negative. Let α be a finitely generated tensor norm. A Banach space E is said to have the α-approximation property (cf. [1; 21.7]) if for all Banach spaces F the natural mapping e αE → F ⊗ e ε E is injective. F⊗
We shall use Proposition 21.7,(2) from [1]: (∗) If α is totally accessible, then each Banach space has the α-approximation property. 3
Theorem 1. Let α be a finitely generated tensor norm. If (1) the space X ∗ has the αt -approximation property, or (2) the space Y ∗∗∗∗ has the α-approximation property, or (3) the space Y ∗∗ has the α-approximation property and the space Y ∗∗∗ has the αt approximation property, then any operator from X into Y, whose second adjoint is a Nα -operator, itself belongs to the space Nα (X, Y ). Remark. The idea to consider the third conjugate space Y ∗∗∗ belongs to Eve Oja (it was used in some of the above mentioned papers). The appearance in the formulation of the theorem of a fourth conjugate space seems to be interesting but perhaps confusing. Proof. Suppose that there exists an operator T ∈ L(X, Y ) such that T ∈ / Nα (X, Y ), but πY T ∈ Nα (X, Y ∗∗ ). Since either X ∗ has the αt -approximation property, or Y ∗∗ has e α Y ∗∗ . Therefore, the operator the α-approximation property, then Nα (X, Y ∗∗ ) = X ∗ ⊗
e α Y ∗∗ ; in addition, by the choice πY T can be identified with a tensor element t ∈ X ∗ ⊗
e α Y (here X ∗ ⊗ e α Y is considered as a subspace of the space X ∗ ⊗ e α Y ∗∗ ). of T, t ∈ / X ∗⊗ e α Y ∗∗ ∗ ⊂ L(Y ∗∗ , X ∗∗ ) such that trace U ◦t = Hence, there exists an operator U ∈ X ∗ ⊗
e α Y. From the last trace (t∗ ◦ (U ∗ |X ∗ )) = 1 and trace U ◦ πY ◦ z = 0 for every z ∈ X ∗ ⊗
it follows, in particular, that (1)
U |πY (Y ) = 0 and πY∗ U ∗ |X ∗ = 0.
Indeed, if x0 ∈ X ∗ and y ∈ Y, then hU πY y, x0 i = hy, πY∗ U ∗ |X ∗ x0 i = trace (U πY ) ◦ (x0 ⊗ y) = 0. Evidently, the tensor element U ◦ t generates the operator U πY T which is, by the previous reasoning, equal identically to zero. e α X ∗∗ = Nα (X, X ∗∗ ) If the space X ∗ has the αt -approximation property then X ∗ ⊗
and so the tensor element U ◦t is zero, in contradiction with the equality trace U ◦t = 1. 4
Let now Y ∗∗∗∗ has the α-approximation property, or Y ∗∗∗ has the αt -approximation property. In this case V := (U ∗ |X ∗ ) ◦ T ∗ ◦ πY∗ : π∗
U ∗ |X ∗
T∗
Y ∗∗∗ −−−Y−→ Y ∗ −−−−→ X ∗ −−−−→ Y ∗∗∗ e αt Y ∗∗∗ . Let us take any representation uniquely determines a tensor element t0 ∈ Y ∗∗∗∗ ⊗ P e α Y ∗∗ . Denoting for the simplicity t= αn x0n ⊗ yn00 for t as an element of the space X ∗ ⊗ by U∗ the operator U ∗ |X ∗ and recalling that πY T = t, we get:
V y 000 = U∗ (T ∗ πY∗ y 000 ) = U∗ ((T ∗ πY∗ πY ∗ ) πY∗ y 000 ) = X 00 0 ∗ 000 ∗ ∗ 000 ∗ ∗ α n y n ⊗ x n πY πY y = = U∗ ((πY T ) πY ) πY y ) = U∗ X X = U∗ αn hyn00 , πY∗ y 000 i x0n = αn hπY∗∗ yn00 , y 000 i U∗ x0n . e αt Y ∗∗∗ the repreThus, the operator V (or the element t0 ) has in the space Y ∗∗∗∗ ⊗
sentation
V = Therefore (see (1)), trace t0 = trace V =
X
X
αn πY∗∗ (yn00 ) ⊗ U∗ (x0n ).
αn hπY∗∗ (yn00 ), U∗ (x0n )i =
X
αn hyn00 , πY∗ U∗ x0n i =
X
0 = 0.
On the other hand, ∗
V y 000 = U∗ (πY T ) y 000 = U∗ ◦ t∗ (y 000 ) = = U∗ whence V =
P
X
X αn hyn00 , y 000 i U∗ x0n , αn hyn00 , y 000 i x0n =
yn00 ⊗ U∗ (x0n ). Therefore
trace t0 = trace V =
X
αn hyn00 , U∗ x0n i =
X
αn hU yn00 , x0n i = trace U ◦ t = 1.
The obtained contradiction completes the proof of the theorem. 5
Corollary 1. If α is a finitely generated tensor norm. and if there exists a non-Nα operator in Banach spaces whose second adjoint is a Nα -operator, then there exists a Banach space both without the αt -approximation property, and without the α-approximation property. Proof. The assertion follows immediately from the previous theorem: to get one space without α- and αt -approximation properties it is enough to sum two obtained (with the help of the theorem) spaces by type l 2 . It follows from Corollary 1 and from (∗) Corollary 2. If α is a finitely generated tensor norm. and if there exists a non-Nα operator in Banach spaces whose second adjoint is a Nα -operator, then neither the tensor norm α, nor the tensor norm αt is totally accessible. Because there are operators, whose second adjoints factor compactly through l 1 (respectively, through c0 ), but which do not factor through l 1 (respectively, through c0 ) themselves (see, [5], [6], [9]), we get from Corollary 2 Corollary 3. The tensor norms g∞ , w1 and w∞ are not totally accessible.
6
References 1. Defant A. and Floret F., Tensor norms and operator ideals, North-Holland, Amsterdam, London, New York, Tokio, 1993. 2. Figiel T., Johnson W. B., The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197–200. 3. Grothendieck A., Produits tensoriels topologiques et espaces nucl´ eaires, Mem. Amer. Math. Soc. 16 (1955), pp. 196+140. 4. Oja E., Reinov O. I., Un contre-exemple a ` une affirmation de A.Grothendieck, C. R. Acad. Sc. Paris. — Serie I 305 (1987), 121-122. 5. Reinov O. I., Approximation properties of order p and the existence of non-p-nuclear operators with p-nuclear second adjoints, Math. Nachr. 109 (1982), 125-134. 6. Reinov O. I., On factorization of operators through the spaces lp , Vestnik SPb GU. Ser.1 (2000), vyp. 2 (No 8), 27-32, (in Russian). 7. Reinov O. I., On linear operators with p-nuclear adjoints, Vestnik SPb GU. Ser.1 (2000), vyp. 4, 24-27, (in Russian). 8. Reinov O. I., Approximation properties APs and p-nuclear operators (the case where 0 < s ≤ 1), Zapiski nauchn. sem. POMI 270 (2000), 277-291, (in Russian). 9. Reinov O.I., Approximation properties and some classes of operators, Problemy matematicheskogo analiza 23 (2001), 147-205, (in Russian).
!"# $&% ')(* + ", - $ . /0 . 2134 E-mail address:
[email protected]
5 !/ 6879 3:; ? @6 . 213 4 !"# $&% ')(* + ", - $ . /0 . 2134 E-mail address:
[email protected] [email protected]
7
