Induction for Banach algebras, groupoids and KK^ban

Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*-coefficients. To show these results, the functoriality of Lafforgue’s KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L^1-algebras (with Banach algebra coefficients).

💡 Research Summary

The paper establishes a robust induction principle for the Bost conjecture in the setting of locally compact Hausdorff groupoids equipped with Banach algebra coefficients, and shows that the same principle holds for the C*-coefficient version. The main theorem states that if two groupoids G and H are equivalent (in the sense of Morita equivalence), then the Bost conjecture is true for G with a given Banach algebra A if and only if it is true for H with the same coefficient algebra.

To achieve this, the author first formalises the notion of a generalized morphism between groupoids – essentially a Hilsum‑Skandalis map or a groupoid correspondence – which respects the Haar systems and allows one to transport actions of a Banach algebra A from one groupoid to the other. Using such a correspondence, a bimodule E is constructed between the L¹‑crossed‑product algebras L¹(G, A) and L¹(H, A). The bimodule is shown to be a full Hilbert Banach module on both sides, and it satisfies the usual Morita conditions: E ⊗{L¹(H,A)} E* ≅ L¹(G,A) and E* ⊗{L¹(G,A)} E ≅ L¹(H,A). Consequently, the two L¹‑algebras are Morita equivalent in the Banach‑algebra sense.

The second major ingredient is the functoriality of Lafforgue’s KK‑theory for Banach algebras, denoted KK^{ban}. The paper proves that a generalized groupoid morphism induces a well‑defined element