A dual method of constructing hereditarily indecomposable Banach spaces

Argyros, Spiros A.; Motakis, Pavlos

doi:10.1007/s11117-015-0378-9

Cited by 7 publications

(21 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[AM1], [ABM]). The version appearing here is similar to the one used in [AM2]. The difference from [AM2] is that here we deal with families (A n j ) j instead of (S n ) n , which makes the definitions and the proofs easier.…”

Section: The Following Holdsmentioning

confidence: 99%

“…The difference from [AM2] is that here we deal with families (A n j ) j instead of (S n ) n , which makes the definitions and the proofs easier. It is also worth pointing out that, as in [AM2], the conditional structure of the space X nr is imposed by certain α c -averages and not by special sequences (γ k ) n k=1 . The space X nr satisfies the scalar-plus-compact property.…”

Section: The Following Holdsmentioning

confidence: 99%

“…In this section we define the space X nr , combining the method from [AH] with that form [AM2]. We start by recalling the definition of the BourgainDelbaen mixed-Tsirelson space B mT from [AH], in fact a slight variation of it, and then we define the space X nr as a quotient of B mT by selecting an appropriate self-determined subset of the setΓ associated to B mT .…”

Section: The Definition Of the Space X Nrmentioning

confidence: 99%

“…A Tsirelson version of this method and its consequences in a classical setting appeared in [AM2]. It is worth pointing out that the new method leads to a unified approach for constructing HI spaces that are either reflexive or do not contain a reflexive subspace.…”

Section: Introductionmentioning

confidence: 99%

“…A typical and known example is Schlumprecht space S[(A n , 1/ log(n + 1)) n ] [Sch] that serves as a preexisting reflexive space with an unconditional basis for Gowers-Maurey space [GM]. Furthermore, in [AM2] the norming set W is a subset of W T , the norming set of Tsirelson space.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The scalar-plus-compact property in spaces without reflexive subspaces

Argyros

Motakis

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. A hereditarily indecomposable Banach space Xnr is constructed that is the first known example of a L∞-space not containing c0, ℓ1, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space Xnr satisfies the "scalar-plus-compact" property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the space Xnr has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.

show abstract

Section: The Following Holdsmentioning

confidence: 99%

Section: The Following Holdsmentioning

confidence: 99%

Section: The Definition Of the Space X Nrmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The scalar-plus-compact property in spaces without reflexive subspaces

Argyros

Motakis

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

On the complete separation of asymptotic structures in Banach spaces

Argyros

Motakis

2020

Advances in Mathematics

View full text Add to dashboard Cite

Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

Motakis¹,

Puglisi²,

Tolias³

2020

Michigan Math. J.

Self Cite

View full text Add to dashboard Cite

For every Banach space X with a Schauder basis consider the Banach algebra RI ⊕ K diag (X) of all diagonal operators that are of the form λI + K. We prove that RI ⊕ K diag (X) is a Calkin algbra i.e., there exists a Banach space Y X so that the Calkin algebra of Y X is isomorphic as a Banach algebra to RI ⊕ K diag (X). Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces Jp and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras, and that sums of reflexive spaces with unconditional bases with certain James-Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.where the supremum is taken over all disjoint collections of intervals of N, is finite.The spaces J p are quasi-reflexive of order one. It was for observed by A. D. Andrew and W. L. Green in [AG] that J p , after appropriate renorming, becomes a non-unital Banach algebra when endowed with coordinate-wise multiplication with respect to the basis e 1 , e 2 − e 1 , e 3 − e 2 , . . .. We denote the unitization of James space by Re ω ⊕ J p , for 1 < p < ∞.Theorem III. The spaces Re ω ⊕ J p , 1 < p < ∞ are Calkin algebras.S. F. Bellenot, R. G. Haydon, and E. Odell in [BHO] extended the definition of James, based on the unit vector basis of ℓ p , to an arbitrary space X with an unconditional basis to define the space J(X), the "jamesification" of X. The space J(X) is quasi-reflexive of order one whenever X is reflexive. As it so happens this space is a non-unital Banach algebra as well the unitization of which we denote by Re ω ⊕ J(X). Furthermore, a special subspace of J * (X) which we denote by J * (X), and coincides with J(X) * when X does not contain ℓ 1 , is a separable unital Banach algebra.Theorem IV. For any space X with a normalized unconditional basis the spaces Re ω ⊕ J(X) and J * (X) are Calkin algebras.Next we turn our attention to spaces with unconditional bases endowed with coordinate-wise multiplication. We study these spaces themselves as Banach algebras. We do not prove that they are Calkin algebras but they always embed in them as complemented ideals. In what follows "⊕" denotes the direct sum of Banach spaces.Theorem V. Let A be Banach space with a normalized unconditional basis endowed with coordinate-wise multiplication and B be a Banach algebra. In the cases described in the list bellow there exists a Banach space Y so that the Calkin algebra Cal(Y) contains an idealÃ isomorphic as a Banach algebra to A and a subalgebrã B isomorphic as a Banach algebra to B so that Cal(Y) =Ã ⊕B.(a) The space B is C(ω), the space of convergent scalar sequences with pointwise multiplication.

show abstract

A dual method of constructing hereditarily indecomposable Banach spaces

Cited by 7 publications

References 20 publications

The scalar-plus-compact property in spaces without reflexive subspaces

The scalar-plus-compact property in spaces without reflexive subspaces

On the complete separation of asymptotic structures in Banach spaces

Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

Contact Info

Product

Resources

About