Banach spaces with a unique nontrivial decomposition

Argyros, Spiros A.; Raikoftsalis, Theocharis

doi:10.1090/s0002-9939-08-09368-4

Cited by 9 publications

(14 citation statements)

References 15 publications

(17 reference statements)

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“…This answers a question raised in [27], and together with [5], [34] and [8] it completes the list of solutions to the problems left open in [27].…”

Section: Introductionsupporting

confidence: 61%

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Koszmider¹,

Laustsen²

2020

Preprint

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We consider the closed subspace of ℓ∞ generated by c 0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N. This Banach space has the form C 0 (K A ) for a locally compact Hausdorff space K A that is known under many names, such as Ψ-space and Isbell-Mrówka space.We construct an uncountable, almost disjoint family A such that the Banach algebra of all bounded linear operators on C 0 (K A ) is as small as possible in the precise sense that every bounded linear operator on C 0 (K A ) is the sum of a scalar multiple of the identity and an operator that factors through c 0 (which in this case is equivalent to having separable range). This implies that C 0 (K A ) has the fewest possible decompositions: whenever C 0 (K A ) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C 0 (K A ) and Y to c 0 , or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required.To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family of Mrówka we need to deal with N×N-matrices rather than with the usual partitioners. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to F. van Engelen, K. Kunen and A. Miller concerning Borel subsets of the square.

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“…This answers a question raised in [27], and together with [5], [34] and [8] it completes the list of solutions to the problems left open in [27].…”

Section: Introductionsupporting

confidence: 61%

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Koszmider¹,

Laustsen²

2020

Preprint

View full text Add to dashboard Cite

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“…Theorem 2 answers a question raised in [34], and together with results of Argyros and Raikoftsalis [5], Marciszewski and Pol [42], and Cabello Sánchez, Castillo, Marciszewski, Plebanek and Salguero-Alarcón [9], it completes the list of solutions to the problems left open in [34].…”

Section: Introductionsupporting

confidence: 55%

“…A and go straight to Step (5). Otherwise let {M ξ : ξ < c} be an enumeration of M with each matrix repeated continuum many times, and note that the second alternative of the dichotomy for acceptance and rejection (Lemma 35) holds for each M ξ and A.…”

Section: Proofmentioning

confidence: 99%

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Koszmider

Laustsen

2021

Advances in Mathematics

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We consider the closed subspace of ∞ generated by c 0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N. This Banach space has the form C 0 (K A ) for a locally compact Hausdorff space K A that is known under many names, including Ψ-space and Isbell-Mrówka space.We construct an uncountable, almost disjoint family A such that the algebra of all bounded linear operators on C 0 (K A ) is as small as possible in the precise sense that every bounded linear operator on C 0 (K A ) is the sum of a scalar multiple of the identity and an operator that factors through c 0 (which in this case is equivalent to having separable range). This implies that C 0 (K A ) has the fewest possible decompositions: whenever C 0 (K A ) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C 0 (K A ) and Y to c 0 , or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space C 0 (K A ) concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms.To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family by Mrówka, we need to deal with N × N matrices rather than with the usual partitioners of an almost disjoint family. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to van Engelen, Kunen and Miller concerning Borel subsets of the square.

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“…2 An example of a scattered K (of height three, but with C (K ) which is not Lindelöf) where all nontrivial decompositions of C (K ) have one factor isomorphic to the C (K ) and the other to c 0 was obtained under CH in [12]. Also Argyros and Raikoftsalis have constructed in [4] a separable Banach space X (necessarily not of the form C (K )) where all nontrivial decompositions are of the form c 0 ⊕ X .…”

Section: Claimmentioning

confidence: 99%

“…As shown in 4.11, one factor is isomorphic to c 0 or C 0 (ω ω ) and the other is isomorphic to C (K 0 ). This is related to few decompositions as in [12] and [4].…”

Section: Introductionmentioning

confidence: 99%

Complementation and decompositions in some weakly Lindelöf Banach spaces

Koszmider

Zieliński

2011

Journal of Mathematical Analysis and Applications

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Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C (K ) of a given class (1) has a complemented copy of c 0 (Γ ) or (2) for every c 0 (Γ ) ⊆ X has a complemented c 0 (E) for an uncountable E ⊆ Γ or (3) has a decomposition X = A ⊕ B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely K s such that C (K ) is Lindelöf in the weak topology and we restrict our attention to K s scattered of countable height. We show that the answers to all these questions for these C (K )s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P -ideal dichotomy, for every c 0 (Γ ) ⊆ C (K ) there is a complemented c 0 (E) for an uncountable E ⊆ Γ , which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C (K ) for K of height ω + 1 where every operator is of the form cI + S for c ∈ R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions.Since, in the case of a scattered compact K , the weak topology on C (K ) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space C p (K ).

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Banach spaces with a unique nontrivial decomposition

Cited by 9 publications

References 15 publications

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Complementation and decompositions in some weakly Lindelöf Banach spaces

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