The universal property of graded KK^G-theory

2026/03/24 by Bernhard Burgstaller · 1 voice
Mathematics · #math.KT #math.OA

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arxiv published 2026/03/24 · arxiv updated 2026/04/06

Abstract

A universal category-theoretical characterization of groupoid equivariant KK^G-theory for ℤ₂-graded C^*-algebras is established, by observing the ``KK-axiom'' that for each [s,\cal E ⊕ B, \mathbbF] ∈ KK^G(A,B), the `corner-embedding' *-homomorphism \bf j: B → \sf cl (\cal K_B(\cal E ⊕ B) + s(A) + \mathbbF ⋅ s(A) ) is invertible in KK^G. This KK-axiom and homotopy-invariance characterize graded KK^G-theory universally and completely, thus directly extending the well-known characterization of KK-theory for ungraded C^*-algebras via stability, homotopy invariance and splitexactness by Higson.

Discussions

• Bernhard Burgstaller: The universal property of graded $KK^G$-theory https://arxiv.org/abs/2603.23157 https://arxiv.org/pdf/2603.23157 https://arxiv.org/html/2603.23157 [bsky, 0 points, 0 comments]

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