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PreprintCC BY-NC-ND 4.0arXiv

On the isomorphism of the spaces k and 𝒦

  • Tomasz Kiwerski
  • Jakub Tomaszewski
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Abstract

Let 1 leqslant p leqslant q < ∞, let X be a separable β„’p-space and let Y be a separable β„’q-space. We show that the space L(X,Y) of bounded linear operators from X into Y is isomorphic to L(β„“p,β„“q) if, and only if, Y is complemented in its bidual. We also show that L(Z,β„“βˆž) is isomorphic to L(c0,β„“βˆž) for every separable β„’βˆž-space Z. This extends and complements the earlier result of Arias and Farmer [On the structure of tensor products of β„“p-spaces, Pacific J. Math.]. Based on this general result, we give a completely different proof of the main result of Astashkin, LeΕ›nik and Maligranda [Isomorphic structure of CesΓ ro and Tandori spaces, Canad. J. Math.], which states that the space 𝒦 and its sequence counterpart k are isomorphic. Furthermore, we complete the picture by showing that the spaces k and 𝒦, although isomorphic, are not isometric. Along the way, we show that for every infinite compact metrizable space K the β„“βˆž-sum of copies of the space β„³(K) of regular Borel measures on K is isomorphic to ( ⨁n=1∞ L1[0,1])β„“βˆž β‰ˆ L(β„“1,L1[0,1]). This provides an abstract blueprint for another result of Astashkin, LeΕ›nik and Maligranda, namely that 𝒦 is isomorphic to ( ⨁n=1∞ β„³[0,1])β„“βˆž. Finally, we show that there are models of set theory without the Axiom of Choice in which the spaces k and 𝒦 are no longer isomorphic. This last part was inspired by VΓ€th's paper [The dual space of L∞ is L1, Indag. Math. (N.S.)].

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Cite

BibTeXRIS
BibTeX
@misc{kiwerski2026isomorphism,
  title = {{On the isomorphism of the spaces $\mathcalk$ and $\mathcalK$}},
  author = {Tomasz Kiwerski and Jakub Tomaszewski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34421},
  eprint = {2610.34421},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34421},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On the isomorphism of the spaces k and 𝒦
AU  - Tomasz Kiwerski
AU  - Jakub Tomaszewski
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34421
UR  - https://arxiv.org/abs/2610.34421
AB  - Let 1 leqslant p leqslant q < ∞, let X be a separable β„’_p-space and let Y be a separable β„’_q-space. We show that the space L(X,Y) of bounded linear operators from X into Y is isomorphic to L(β„“_p,β„“_q) if, and only if, Y is complemented in its bidual. We also show that L(Z,β„“_∞) is isomorphic to L(c_0,β„“_∞) for every separable β„’_∞-space Z. This extends and complements the earlier result of Arias and Farmer [On the structure of tensor products of β„“_p-spaces, Pacific J. Math.]. Based on this general result, we give a completely different proof of the main result of Astashkin, LeΕ›nik and Maligranda [Isomorphic structure of CesΓ ro and Tandori spaces, Canad. J. Math.], which states that the space 𝒦 and its sequence counterpart k are isomorphic. Furthermore, we complete the picture by showing that the spaces k and 𝒦, although isomorphic, are not isometric. Along the way, we show that for every infinite compact metrizable space K the β„“_∞-sum of copies of the space β„³(K) of regular Borel measures on K is isomorphic to ( ⨁_n=1^∞ L_1[0,1])_β„“_∞ β‰ˆ L(β„“_1,L_1[0,1]). This provides an abstract blueprint for another result of Astashkin, LeΕ›nik and Maligranda, namely that 𝒦 is isomorphic to ( ⨁_n=1^∞ β„³[0,1])_β„“_∞. Finally, we show that there are models of set theory without the Axiom of Choice in which the spaces k and 𝒦 are no longer isomorphic. This last part was inspired by VΓ€th's paper [The dual space of L_∞ is L_1, Indag. Math. (N.S.)].
ER  -