Results 1  10
of
13
On Certain Extension Properties for the Space of Compact Operators
, 1999
"... Let Z be a fixed separable operator space, X ⊂ Y general separable operator spaces, and T: X → Z a completely bounded map. Z is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to Y; the Mixed Separable Extension Property (MSEP) i ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
Let Z be a fixed separable operator space, X ⊂ Y general separable operator spaces, and T: X → Z a completely bounded map. Z is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to Y; the Mixed Separable Extension Property (MSEP) if every such T admits a bounded extension to Y. Finally, Z is said to have the Complete Separable Complementation Property (CSCP) if Z is locally reflexive and T admits a completely bounded extension to Y provided Y is locally reflexive and T is a complete surjective isomorphism. Let K denote the space of compact operators on separable Hilbert space and K0 the c0 sum of Mn’s (the space of “small compact operators”). It is proved that K has the CSCP, using the second author’s previous result that K0 has this property. A new proof is given for the result (due to E. Kirchberg) that K0 (and hence K) fails the CSEP. It remains an open question if K has the MSEP; it is proved this is equivalent to whether K0 has this property. A new Banach space concept, Extendable Local Reflexivity (ELR), is
On Lindenstrauss–Pe̷lczyński spaces
 Studia Math
"... Abstract. In this work we shall be concerned with some stability aspects of the classical problem of extension of C(K)valued operators. We introduce the class LP of Banach spaces of LindenstraussPe̷lczyńsky type as those such that every operator from a subspace of c0 into them can be extended to c ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
Abstract. In this work we shall be concerned with some stability aspects of the classical problem of extension of C(K)valued operators. We introduce the class LP of Banach spaces of LindenstraussPe̷lczyńsky type as those such that every operator from a subspace of c0 into them can be extended to c0. We show that all LPspaces are of type L ∞ but not the converse. Moreover, L∞spaces will be characterized as those spaces E such that Evalued operators from w ∗ (l1, c0)closed subspaces of l1 extend to l1. Complemented subspaces of C(K) and separably injective spaces are subclasses of LPspaces and we show that the former does not contain the latter. It is established that L∞spaces not containing l1 are quotients of LPspaces, while L∞spaces not containing c0, quotients of an LPspace by a separably injective space and twisted sums of LPspaces are LPspaces.
MCOMPLETE APPROXIMATE IDENTITIES IN OPERATOR SPACES
, 1999
"... Abstract. This work introduces the concept of an Mcomplete approximate identity (Mcai) for a given operator subspace X of an operator space Y. Mcai’s generalize central approximate identities in ideals in C ∗algebras, for it is proved that if X admits an Mcai in Y, then X is a complete Mideal ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
Abstract. This work introduces the concept of an Mcomplete approximate identity (Mcai) for a given operator subspace X of an operator space Y. Mcai’s generalize central approximate identities in ideals in C ∗algebras, for it is proved that if X admits an Mcai in Y, then X is a complete Mideal in Y. It is proved, using “special ” Mcai’s, that if J is a nuclear ideal in a C ∗algebra A, then J is completely complemented in Y for any (isomorphically) locally reflexive operator space Y with J ⊂ Y ⊂ A and Y/J separable. (This generalizes the previously known special case where Y = A, due to EffrosHaagerup.) In turn, this yields a new proof of the OikhbergRosenthal Theorem that K is completely complemented in any separable locally reflexive operator superspace, K the C ∗algebra of compact operators on ℓ 2. Mcai’s are also used in obtaining some special affirmative answers to the open problem of whether K is Banachcomplemented in A for any separable C ∗algebra A with K ⊂ A ⊂ B(ℓ 2). It is shown that if conversely X is a complete M
The Complete Separable Extension Property
, 2000
"... This work introduces operator space analogues of the Separable Extension Property (SEP) for Banach spaces, the Complete Separable Extension Property (CSEP) and the Complete Separable Complemention Property (CSCP). The results use the technique of a new proof of Sobczyk’s Theorem, which also yields ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
This work introduces operator space analogues of the Separable Extension Property (SEP) for Banach spaces, the Complete Separable Extension Property (CSEP) and the Complete Separable Complemention Property (CSCP). The results use the technique of a new proof of Sobczyk’s Theorem, which also yields new results for the SEP in the nonseparable situation, e.g., n=1
The Banach space c0
 Extracta Math
, 2001
"... We survey in these notes some recent progress on the understanding of the Banach space c0 an of its subspaces. We did not try to complete the (quite ambitious) task of writing a comprehensive survey. A few topics have been selected, in order to display the variety of techniques which are required in ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
We survey in these notes some recent progress on the understanding of the Banach space c0 an of its subspaces. We did not try to complete the (quite ambitious) task of writing a comprehensive survey. A few topics have been selected, in order to display the variety of techniques which are required in
Subspaces Of Maximal Operator Spaces
"... . We explore subspaces of maximal operator spaces (submaximal spaces) and give a new characterization of such spaces. We show that the set of ndimensional submaximal spaces is closed in the topology of c.b. distance, but not compact. We also investigate subspaces of MAX(L# ) and prove that any ho ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
. We explore subspaces of maximal operator spaces (submaximal spaces) and give a new characterization of such spaces. We show that the set of ndimensional submaximal spaces is closed in the topology of c.b. distance, but not compact. We also investigate subspaces of MAX(L# ) and prove that any homogeneous Hilbertian subspace of MAX(L1 ) is completely isomorphic to R + C. 1. Introduction In this paper we consider Banach spaces equipped with the maximal operator space structure, and their subspaces. Recall that an operator space is a subspace X of B(H), where H is a Hilbert space. If X and Y are operator spaces, embedded into B(H) and B(K), respectively, then their minimal (or injective) tensor product X# min Y is defined as the norm closure of X# Y in B(H# 2 K). The space Mn of nn matrices plays an important role in the theory of operator spaces. If X is a subspace of B(H), then Mn (X) def = Mn# min X can be identified with a subspace of B(# n 2 (H)). The sequence of ma...
SOME APPROXIMATION PROPERTIES OF BANACH SPACES AND BANACH LATTICES
"... Abstract. The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is an L∞−space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a s ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is an L∞−space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a surjection, X is an L1−space and Z is an Lp−space (1 ≤ p ≤ ∞) then ker Q has the UAP. A complemented subspace of a weakly sequentially complete Banach lattice has the separable complementation property = SCP. A criterion for a space with GLl.u.st. to have the SCP is given. Spaces which are quotients of weakly sequentially complete lattices and are uncomplemented in their second duals are studied. Examples are given of spaces with the SCP which have subspaces that fail the SCP. The results are applied to spaces of measures on a compact Abelian group orthogonal to a fixed Sidon set and to Sobolev spaces of functions
COMPLEMENTABLY UNIVERSAL BANACH SPACES, II
"... Abstract. The two main results are: A. If a Banach space X is complementably universal for all subspaces of c0 which have the bounded approximation property, then X ∗ is non separable (and hence X does not embed into c0), B. There is no separable Banach space X such that every compact operator (betw ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. The two main results are: A. If a Banach space X is complementably universal for all subspaces of c0 which have the bounded approximation property, then X ∗ is non separable (and hence X does not embed into c0), B. There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X. Theorem B solves a problem that dates from the 1970s. 1.
THE WEAK METRIC APPROXIMATION PROPERTY
"... Abstract. We introduce and investigate the weak metric approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property. Among others, we show that if a Banach space has the approximation propert ..."
Abstract
 Add to MetaCart
Abstract. We introduce and investigate the weak metric approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property. Among others, we show that if a Banach space has the approximation property and is 1complemented in its bidual, then it has the weak metric approximation property. We also study the lifting of the weak metric approximation property from Banach spaces to their dual spaces. This enables us, in particular, to show that the subspace of c0, constructed by Johnson and Schechtman, does not have the weak metric approximation property. 1.