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On split exact sequences and KK-equivalences of amplified graph C*-algebras

Jesse Reimann, Sophie Emma Zegers

Abstract

We give a general methodology for constructing split exact sequences of amplified graph C*-algebras with sinks. This in turn allows us to construct explicit KK-equivalences with $\mathbb{C}^N$ for a large class of C*-algebras, including the quantum Grassmannian $\mathrm{Gr}_q(2,4)$. We discuss compatibility with known (quantum) CW-constructions and give an explicit KK-equivalence between the classical and quantum projective spaces $\mathbb{C}P^1$ and $\mathbb{C}P_q^1$.

On split exact sequences and KK-equivalences of amplified graph C*-algebras

Abstract

We give a general methodology for constructing split exact sequences of amplified graph C*-algebras with sinks. This in turn allows us to construct explicit KK-equivalences with for a large class of C*-algebras, including the quantum Grassmannian . We discuss compatibility with known (quantum) CW-constructions and give an explicit KK-equivalence between the classical and quantum projective spaces and .

Paper Structure

This paper contains 10 sections, 9 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. Acknowledgements
  5. Graph C*-algebras of amplified directed graphs
  6. KK-theory
  7. KK-equivalence from split exact sequences
  8. Equivariant KK-equivalences from split exact sequences
  9. Explicit splittings for amplified graph C*-algebras
  10. Application: KK-equivalences and CW-decomposition of $Gr_q(2,4)$

Key Result

Theorem 2.2

Let $\Gamma$ be an amplified directed graph. Then there is a one-to-one correspondence between ideals $I\subseteq C^*(\Gamma)$ and hereditaryIf $\Gamma$ is not amplified, the set $H$ is furthermore required to be saturated, i.e. a non-sink vertex emitting finitely many edges must belong to $H$ if al $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Graph descriptions of selected C*-algebras. Here, $\tilde{\mathbb{K}}$ denotes the minimal unitisation of the compact operators, and $(\infty)$ denotes countably infinitely many edges from one vertex to another.
  • Figure 2: Construction of the graph description of $\mathbb{C}P_q^3$ and $Gr_q(2,4)$ from their tagged Dynkin diagrams. Here, the tagged nodes are denoted by a cross and the untagged nodes by a circle. The graphs are understood to be amplified; this has been suppressed from the notation to improve readability. Their vertices are labelled by the corresponding element in $W/W_S$.
  • Figure 3: Amplified graphs appearing in the CW-decomposition of $C(Gr_q(2,4))$, yielding graph C*-algebras isomorphic to the skeleta $C(X_q^6)$ and $C(\mathbb{C}P_q^2\sqcup_{\mathbb{C}P_q^1}\mathbb{C}P_q^2)$. The infinite multiplicity of the edges has been suppressed from the notation.

Theorems & Definitions (26)

  • Definition 2.1: Graph C*-algebras
  • Theorem 2.2: Ideal structure of amplified graphs, cf. bates_hong_raeburn_szymanski_2002
  • Theorem 2.3: eilers_ruiz_sørensen_2012
  • Definition 3.1: Hilbert C*-modules
  • Definition 3.2: $\mathbb{K}(E)$
  • Remark 3.3
  • Definition 3.4: KK via Kasparov modules
  • Definition 3.5: KK via quasihomomorphisms
  • Definition 3.6: continuous $G$-action on Hilbert C*-modules
  • Definition 3.7: Kasparov $G$-modules, $\mathrm{KK}^G$
  • ...and 16 more