On Arazy's problem concerning isomorphic embeddings of ideals of compact operators

The primary objective of this paper is to study the problem concerning isomorphic embeddings of ideals of compact operators due to Arazy (see [5, Problem C] and [6, Problem (B)]), which is closely related to Pełczyński's problem of uniqueness of symmetric structure in such ideals. In particular, we show that for a wide class of symmetric sequence spaces E, if C_F↪S_E, then F=ℓ₂ and E has a subspace isomorphic to ℓ₂, where C_X is the symmetric ideal of compact operators on the Hilbert space ℓ₂ induced by a symmetric sequence space X and S_X=(⊕_n=1^∞M_n)_CX. This extends results due to Arazy [5,6]. As a consequence, we contribute to the Arazy–Pełczyński problem on the uniqueness of symmetric structures [5,6]. In particular, we can deal with the case when E has a subspace isomorphic to ℓ₂ or c₀, which was left untreated in [5,6].