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Mayer--Vietoris and Twisted Čech Spectral Sequences for C$^*$-Algebras with Free Quantum Group Coefficients

Takao Inoué

Abstract

We formulate a Mayer--Vietoris/Čech viewpoint on $K$-theory for crossed products by discrete quantum groups, emphasizing how local-to-global gluing data interacts with quantum-group coefficients. Starting from a $G$-equivariant ideal cover and the associated Mayer--Vietoris six-term exact sequence, we package the resulting $K$-theoretic computation into a Čech-type spectral sequence whose $E^1$-page is explicitly described by iterated intersections. We then introduce a minimal ``twisted'' gluing mechanism controlled by a $\mathbb Z/2$-valued Čech $2$-cocycle and an involutive automorphism of the coefficient algebra. Under a Kirchberg--UCT hypothesis on the quantum-group crossed-product coefficient, the twist produces a nontrivial differential $d_2$ identified as $\varphi_*-\mathrm{id}$ on coefficient $K$-theory. In a concrete regime where the coefficient $K$-groups are cyclic (e.g.\ order $3$), the differential becomes an isomorphism and forces a $K$-theoretic obstruction to Morita triviality. This yields a conceptual mechanism for producing non-Morita-trivial twisted C$^*$-algebras with quantum-group crossed-product fibers, detected purely by Mayer--Vietoris/Čech data.

Mayer--Vietoris and Twisted Čech Spectral Sequences for C$^*$-Algebras with Free Quantum Group Coefficients

Abstract

We formulate a Mayer--Vietoris/Čech viewpoint on -theory for crossed products by discrete quantum groups, emphasizing how local-to-global gluing data interacts with quantum-group coefficients. Starting from a -equivariant ideal cover and the associated Mayer--Vietoris six-term exact sequence, we package the resulting -theoretic computation into a Čech-type spectral sequence whose -page is explicitly described by iterated intersections. We then introduce a minimal ``twisted'' gluing mechanism controlled by a -valued Čech -cocycle and an involutive automorphism of the coefficient algebra. Under a Kirchberg--UCT hypothesis on the quantum-group crossed-product coefficient, the twist produces a nontrivial differential identified as on coefficient -theory. In a concrete regime where the coefficient -groups are cyclic (e.g.\ order ), the differential becomes an isomorphism and forces a -theoretic obstruction to Morita triviality. This yields a conceptual mechanism for producing non-Morita-trivial twisted C-algebras with quantum-group crossed-product fibers, detected purely by Mayer--Vietoris/Čech data.
Paper Structure (21 sections, 3 theorems, 28 equations)

This paper contains 21 sections, 3 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Motivation: MV gluing with quantum-group coefficients
  3. What is new in this note
  4. Comparison with prior work
  5. Preliminaries
  6. Equivariant ideal covers and the MV short exact sequence
  7. Čech viewpoint: $d_1$ as alternating restriction
  8. Free orthogonal quantum group and a coefficient crossed product
  9. MV and a Čech spectral sequence with quantum-group coefficients
  10. A basic model: trivial base action
  11. Explicit $E^1$-page under contractibility hypotheses
  12. Example: interval covers and immediate degeneration
  13. A minimal twist: a $\mathbb Z/2$ Čech 2-cocycle and an involution
  14. Twisting data
  15. Kirchberg--UCT hypothesis (assumption)
  16. ...and 6 more sections

Key Result

Theorem 4.4

Let $X=S^2$ with a good cover $\mathfrak U$ and let $A_c$ be the twisted algebra determined by $(\varphi,c)$ as above. Assume the coefficient reduction hypotheses so that the associated Čech--MV spectral sequence has If $c$ represents the nontrivial element of $\check H^2(S^2;\mathbb Z/2)$, then the differential is given on coefficient groups by up to the conventional identifications of the spe

Theorems & Definitions (10)

  • Remark 2.1: Coefficient reduction principle
  • Remark 4.1: C$^*$-algebraic model
  • Definition 4.2: UCT Kirchberg assumption
  • Remark 4.3
  • Theorem 4.4: Identification of the first higher differential
  • Remark 4.5: Interpretation
  • Corollary 5.2: Nontriviality of $d_2$
  • Proposition 5.3: Morita obstruction
  • proof : Proof sketch
  • Remark 5.4: Groupoid reformulation