Banach spaces of \mathcal I-convergent sequences

2023/09/15 by Rincón-Villamizar, Michael A., Aylwin, Carlos Uzcátegui
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2309.08076

Abstract

We study the space c_0,I of all bounded sequences (x_n) that I-converge to 0, endowed with the sup norm, where I is an ideal of subsets of ℕ. We show that two such spaces, c_0,I and c_0,J, are isometric exactly when the ideals I and J are isomorphic. Additionally, we analyze the connection of the well-known Katětov pre-order ≤_K on ideals with some properties of the space c_0,I. For instance, we show that I≤_KJ exactly when there is a (not necessarily onto) Banach lattice isometry from c_0,I to c_0,J, satisfying some additional conditions. We present some lattice-theoretic properties of c_0,I, particularly demonstrating that every closed ideal of ℓ_∞ is equal to c_0,I for some ideal I on ℕ. We also show that certain classical Banach spaces are isometric to c_0,I for some ideal I, such as the spaces ℓ_∞(c₀) and c₀(ℓ_∞). Finally, we provide several examples of ideals for which c_0,I is not a Grothendieck space.

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