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Factorization of the coarse index pairing via the Paschke transformation

Establish whether, for general uniform bornological coarse spaces X and big families 𝒴, there exists a natural analytic coarse symbol pairing K(𝒴) × K^{an}(X) → K^{an}(𝒴) that factors the coarse symbol pairing through the Paschke transformation p_X: K^X(X) → K^{an}(X) and renders the corresponding comparison square commutative.

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Background

The paper introduces a coarse symbol pairing −∩{Xσ}−: K(𝒴) × KX(X) → KX(𝒴) and an index map a: KX → KX, with compatibility results linking corona pairings and symbol pairings. A natural question is whether this coarse symbol pairing can be factored through the analytic Paschke transformation p_X.

The authors explicitly state that such a factorization is unclear in general; they can construct it when 𝒴 consists of bounded subsets, but the existence of a general analytic pairing K(𝒴) × K{an}(X) → K{an}(𝒴) compatible with p_X remains unresolved.

References

Similarly it is not clear whether in general the coarse index pairing has a factorization over the Paschke transformation $p_{X}:K{X}(X)\to K{an}(X)$ from frefwrfregwreg, i.e., whether we can construct a pairing $ K(\mathcal Y)\times K{an}(X)\stackrel{-\cap{an \sigma}-}{\to} K{an}(\mathcal{Y})$ rendering the obvious comparison square commutative.

frefwrfregwreg:

pX:KX(X)→Kan(X)p_{X}:K^{X}(X)\to K^{an}(X)

— Coronas and Callias type operators in coarse geometry (2411.01646 - Bunke et al., 3 Nov 2024) in Remark in Subsection 'The coarse symbol pairing' (following Proposition erokgpwergrwefwerfwerfwer)