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SUPER-IDEALS IN BANACH SPACES
Abstract
A natural class of ideals, super-ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observations that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces which are ideals in every superspace; we show that being a super-ideal in every superspace characterizes the class of Gurari˘ ı spaces.
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Basic theory of M-ideals.- Geometric properties of M-ideals.- Banach spaces which are M-ideals in their biduals.- Banach spaces which are L-summands in their biduals.- M-ideals in Banach algebras.- M-ideals in spaces of bounded operators.
which has become known as the Daugavet equation, holds for com-pact operators on C[0; 1]; shortly afterwards the same result for com-pact operators on L1[0; 1] was discovered by G. Ya. Lozanovskii [20]. Over the years, (1.1) has been extended to larger classes of opera-tors on various spaces; in particular, the Daugavet equation is known to hold for operators not fixing a copy of C[0; 1] defined on certain "large" subspaces of C(K), where K is a compact space without iso-lated points, and for operators not fixing a copy of L1[0; 1] defined on certain "large" subspaces of L1[0; 1] ([15], [22], [28]). See also the examples in the next section. Methods encountered in the investigation of (1.1) include Banach lattice techniques ([1], [2]), stochastic kernels and random measures ([13], [27], [29]) and other arguments from the geometry of Banach spaces ([14], [30]). The Daugavet equation has proved useful in approximation the-ory, where it was used to find the best constants in certain inequali-ties [26], and in the geometry of Banach spaces. In [13], G. Godefroy, N. Kalton and D. Li observed for an operator V with kV k < 2 and T := V ?Id satisfying (1.1) that V must be an isomorphism; indeed, (1.1) implies that kV ?Idk < 1, and hence the result follows from the Neumann series. They go on to apply this consequence to quotients by nicely placed subspaces of L1.V. Kadets [14] used the Daugavet equation to give a very simple argument that neither C[0; 1] nor L1[0; 1] have unconditional bases; see Proposition 3.1 below.
In this paper we study the notion of an ideal, which was introduced by Godefroy, Kalton and Saphar in [7] and was called "locally one complemented" in [11], for injective and projective tensor products of Banach spaces. For a Banach space X and an ideal Y in X, we show that the injective tensor product space Y ⊗ e Z is an ideal in X ⊗ e Z for any Banach space Z. This as a consequence gives us a way of proving some known results about intersection properties of balls and extensions of operators on injective tensor product spaces in a unified way that does not involve any vector-valued Choquet theory. We also exhibit classes of Banach spaces in which every ideal is the range of a norm one projection.
We show that every nonvoid relatively weakly open subset, in particular every
slice, of the unit ball of an infinite-dimensional uniform algebra has
diameter~2.
A Banach space X is said to have the Daugavet property if every operator
of rank~1 satisfies . We show that then every
weakly compact operator satisfies this equation as well and that X contains a
copy of . However, X need not contain a copy of . We also
study pairs of spaces and operators satisfying
, where is the natural embedding. This leads to
the result that a Banach space with the Daugavet property does not embed into a
space with an unconditional basis. In another direction, we investigate spaces
where the set of operators with is as small as possible and
give characterisations in terms of a smoothness condition.
1. Partially Ordered Banach Spaces.- 1. Vector Lattices.- 2. Partially Ordered Normed Linear Spaces.- 3. Normed Linear Lattices.- 2. Some Aspects of Topology and Regular Borel Measures.- 4. Existence Theorems for Continuous Functions.- 5. Dispersed Compact Hausdorff Spaces.- 6. The Cantor Set.- 7. Extremally Disconnected Compact Hausdorff Spaces.- 8. Regular Borel Measures.- 3. Characterizations of Banach Spaces of Continuous Functions.- 9. Lattice and Algebraic Characterizations of Banach Spaces of Continuous Functions.- 10. Banach Space Characterizations of Spaces of Continuous Functions.- 11. Banach Spaces with the Hahn-Banach Extension Property.- 4. Classical Sequence Spaces.- 12. Schauder Bases in Classical Sequence Spaces.- 13. Embedding of Classical Sequence Spaces into Continuous Function Spaces.- 5. Representation Theorems for Spaces of the Type Lp(T, ?, , ?).- 14. Measure Algebras and the Representation of Lp(T, ?, , ?) when a Finite Measure.- 15. Abstract Lp Spaces.- 6. Characterizations of Abstract M and Lp Spaces.- 16. Positive Contractive Projections in Abstract M and Lp Spaces.- 17. Contractive Projections in Abstract Lp Spaces.- 18. Geometric Properties of Abstract L1 Spaces and Some Dual Abstract L1 Spaces.- 7. L1-Predual Spaces.- 19. Partially Ordered L1-Predual Spaces.- 20. Compact Choquet Simplexes.- 21. Characterizations of Real L1-Predual Spaces.- 22. Some Selection and Embedding Theorems for Real L1-Predual Spaces.- 23. Characterizations of Complex L1-Predual Spaces.
We present selected known results and some of their improvements, involving
Gurarii spaces. A Banach space is Gurarii if it has certain natural extension
property for almost isometric embeddings of finite-dimensional spaces. Deleting
the word "almost", we get the notion of a strong Gurarii space. There exists a
unique (up to isometry) separable Gurarii space, however strong Gurarii spaces
cannot be separable. The structure of the class of non-separable Gurarii spaces
seems to be not very well understood. We discuss some of their properties and
state some open questions. In particular, we characterize non-separable Gurarii
spaces in terms of skeletons of separable subspaces, we construct a
non-separable Gurarii space with a projectional resolution of the identity and
we show that no strong Gurarii space can be weakly Lindel\"of determined.
Let A be a closed subalgebra of the complex Banach algebra C(S), containing the constant functions. We assume that one has found a probability measureμ on S and a function F from L∞(μ) such that: 1)¦F¦= 1 a.e. relative to μ; 2) Fμ
ε A1; 3) F is a limit point of the unit ball of the algebra A in the topology δ(L∞(μ), L1(μ)). One proves in the paper that under these conditions the space A** contains a complement space, isometric to H∞. The measure μ and the function F, satisfying the conditions l)-3) indeed exist if the maximal ideal space of the algebra A contains a non-one-point part (and it is very likely that such aμ. and F exist whenever the algebra A is not self-adjoint). Thus, the above-formulated result allows us to extend A. Pelczynski's theorem (Ref, Zh. Mat., 1975, 1B894) regarding the space H∞ to a very broad class of uniform algebras.
We study strict u-ideals in Banach spaces. A Banach space X is a strict u-ideal in its bidual when the canonical decomposition X = X X? is unconditional. We characterize Banach spaces which are strict u-ideals in their bidual and show that if X is a strict u-ideal in a Banach space Y then X contains c0. We also show that '1 is not a u-ideal.
A Banach space is said to have the diameter two property if every non-empty relatively weakly open subset of its unit ball has diameter two. We prove that the projective tensor product of two Banach spaces whose centralizer is infinite-dimensional has the diameter two property. The same statement also holds for X⊗ˆπY if the centralizer of X is infinite-dimensional and the unit sphere of Y⁎ contains an element of numerical index one. We provide examples of classical Banach spaces satisfying the assumptions of the results. If K is any infinite compact Hausdorff topological space, then C(K)⊗ˆπY has the diameter two property for any nonzero Banach space Y. We also provide a result on the diameter two property for the injective tensor product.
We give, departing from Grothendieck's description of the dual of the space of weak*-weak continuous finite-rank operators, a clear proof for the principle of local reflexivity in a general form.
We show that every Banach space X whose centralizer is infinite-dimensional satisfies that every non-empty weakly open set in BY has diameter 2, where Y=⊗ˆN,s,πX (N-fold symmetric projective tensor product of X, endowed with the symmetric projective norm), for every natural number N. We provide examples where the above conclusion holds that includes some spaces of operators and infinite-dimensional C∗-algebras. We also prove that every non-empty weak∗ open set in the unit ball of the space of N-homogeneous and integral polynomials on X has diameter two, for every natural number N, whenever the Cunningham algebra of X is infinite-dimensional. Here we consider the space of N-homogeneous integral polynomials as the dual of the space ⊗ˆN,s,εX (N-fold symmetric injective tensor product of X, endowed with the symmetric injective norm). For instance, every infinite-dimensional L1(μ) satisfies that its Cunningham algebra is infinite-dimensional. We obtain the same result for every non-reflexive L-embedded space, and so for every predual of an infinite-dimensional von Neumann algebra.
We develop a theory of ℒp-spaces for 0 < p < 1, basing our definition on the concept of a locally complemented subspace of a quasi-BANACH space. Among the topics we consider are the existence of basis in ℒp-spaces, and lifting and extension properties for operators. We also give a simple construction of uncountably many separable ℒp-spaces of the form ℒp(X) where X is not a ℒp-space. We also give some applications of our theory to the spaces Hp, 0 < p < 1.
This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved,
e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures
and finite dimensional Schauder decompositions. The basic results are then applied to show that every separableℒ
p
space has a basis.
Let X be a closed subspace of a Banach space Y and J be the inclusion map. We say that the pair (X, Y) has the Daugavet property if for every rank one bounded linear operator T from X to Y the equality ‖J+T‖=1+‖T‖ (1)holds. A new characterization of the Daugavet property in terms of weak open sets is given. It is shown that the operators not fixing copies of l1 on a Daugavet pair satisfy (1). Some hereditary properties are found: if X is a Daugavet space and Y is its subspace, then Y is also a Daugavet space provided X/Y has the Radon–Nikodým property; if Y is reflexive, then X/Y is a Daugavet space. Besides, we prove that if (X, Y) has the Daugavet property and Y⊂Z, then Z can be renormed so that (X, Z) possesses the Daugavet property and the equivalent norm coincides with the original one on Y.
Topics in Banach space theory, Graduate Texts in Mathematics
- F Albiac
- N J Kalton
F. Albiac and N. J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298 (2006h:46005)
Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas Function spaces
- J Lindenstrauss
- L Tzafriri
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin, 1979, Function spaces. MR 540367 (81c:46001)
Unconditional ideals in Banach spaces MR 1208038 (94k:46024) [Gur66] V. I. Gurari˘ ı, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces
- G Godefroy
- N J Kalton
- P D Saphar
G. Godefroy, N. J. Kalton, and P. D. Saphar, Unconditional ideals in Banach
spaces, Studia Math. 104 (1993), no. 1, 13-59. MR 1208038 (94k:46024)
[Gur66]
V. I. Gurari˘ ı, Spaces of universal placement, isotropic spaces and a problem of
Mazur on rotations of Banach spaces, Sibirsk. Mat. Ž. 7 (1966), 1002-1013.
MR 0200697 (34 #585)
[HLP]
R. Haller, J. Langemets, and M. Põldvere, Private communication.
[HM82]
S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform
and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), no. 3,
225-251. MR 675426 (84h:46026)
Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete Function spaces The Gurarij spaces are unique
- J Lindenstrauss
- L Tzafriri
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Ergebnisse der
Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related
Areas], vol. 97, Springer-Verlag, Berlin, 1979, Function spaces. MR 540367
(81c:46001)
[Lus76]
W. Lusky, The Gurarij spaces are unique, Arch. Math. (Basel) 27 (1976),
no. 6, 627-635. MR 0433198 (55 #6177)
[NW01]
The isometric theory of classical Banach spaces Die Grundlehren der mathematischen Wissenschaften
- H E Lacey
- V Lima
- Å Lima
H. E. Lacey, The isometric theory of classical Banach spaces, Springer-Verlag,
New York, 1974, Die Grundlehren der mathematischen Wissenschaften, Band
208. MR 0493279 (58 #12308)
[LL09]
V. Lima and Å. Lima, Strict u-ideals in Banach spaces, Studia Math. 195
(2009), no. 3, 275-285. MR 2559177 (2010g:46014)
[LT79]
Die Grundlehren der mathematischen Wissenschaften
- H E Lacey
H. E. Lacey, The isometric theory of classical Banach spaces, Springer-Verlag,
New York, 1974, Die Grundlehren der mathematischen Wissenschaften, Band
208. MR 0493279 (58 #12308)