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Separating ideal-separation conditions in minimal tensor products when property (F) fails

Investigate, for C*-algebras A and B that do not satisfy Tomiyama’s property (F), whether there exist closed ideals I ⊆ A ⊗min B that (i) satisfy I = I ∩ (A ⊗π B) but not I = I ∩ (A ⊗alg B), or (ii) satisfy I = I ∩ (A ⊗alg B) but not that I equals the closure of the sum of all product ideals contained in I.

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Background

The authors compare three properties for closed ideals in A ⊗min B: (1) being the closure of the sum of all contained product ideals, (2) equality with the intersection with the algebraic tensor product A ⊗alg B, and (3) equality with the intersection with the projective tensor product A ⊗π B. They show (1) ⇒ (2) ⇒ (3), and that failure of property (F) guarantees existence of ideals failing (3). Whether strict separations exist between these properties remains explicitly unclear.

References

If the pair (A,B) does not satisfy Tomiyama's property (F), then there necessarily exists a closed ideal in A & B that does not satisfy (3) (hence neither (1), nor (2)). In this case, it remains unclear if there also exist closed ideals in A & B that satisfy (3) but not (2), or that satisfy (2) but not (1).

— The ideal separation property for reduced group $C^*$-algebras (2408.14880 - Austad et al., 27 Aug 2024) in Section 4, Remark 4.6