Countability and countable-sum continuity for finite-dimensional commutative groupoid algebras

From papers

Let HH be a finite groupoid, let AA be a finite-dimensional commutative HH-algebra, and let BB be an object of KKHKK^H. The group KKH(A,B)KK^H(A,B) is countable, and the functor KKH(A,)KK^H(A,-) commutes with countable direct sums in the variable BB.

Countability and continuity conjecture. For every finite groupoid HH and finite-dimensional commutative HH-algebra AA, KKH(A,B)KK^H(A,B) is countable for all BKKHB\in KK^H and commutes with countable direct sums in the variable BB.

This property concerns the size and continuity of equivariant Kasparov groups for finite groupoids. The supplied text does not state whether the assertion is known or resolved.

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Sources & referencesView supporting material

Primary source

Bernhard Burgstaller, “Attempts to define a Baum–Connes map via localization of categories for inverse semigroups”, arXiv:1506.08412 (2017).

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