[Paper Review] Induction for Banach algebras, groupoids and KK^ban

Walther Paravicini|ArXiv.org|Feb 24, 2009

Advanced Operator Algebra Research15 references4 citations

TL;DR

This paper establishes the functoriality of Lafforgue's $<math xmlns='http://www.w3.org/1998/Math/MathML'>\operatorname{KK}^{\operatorname{ban}}</math>$-theory for Banach algebras and groupoids under generalised morphisms, proving that equivalent locally compact Hausdorff groupoids have Morita equivalent $L^1$-algebras and that the Bost conjecture with Banach algebra coefficients is invariant under groupoid equivalence. The key result is a commutative diagram linking $\operatorname{K}$-theory of induced Banach algebras, showing the Bost conjecture holds for one groupoid if and only if it holds for any equivalent groupoid.

ABSTRACT

Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*-coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L^1-algebras (with Banach algebra coefficients).

Motivation & Objective

• To establish functoriality of $\operatorname{KK}^{\operatorname{ban}}$-theory for Banach algebras and groupoids under generalised morphisms.
• To generalize Morita equivalence from $L^1$-algebras to arbitrary unconditional completions of groupoid algebras.
• To show that the Bost conjecture with Banach algebra coefficients is invariant under equivalence of locally compact Hausdorff groupoids.
• To construct a commutative diagram linking topological $K$-theory and $K$-theory of Banach algebras via induction and descent.
• To extend known results on the Bost conjecture to Banach algebra coefficients, filling a gap in the literature.

Proposed method

• Develops a systematic theory of groupoid Banach algebras and their unconditional completions $\mathcal{A}(\mathcal{G})$.
• Introduces the induction functor $\operatorname{Ind}_\mathcal{H}^\mathcal{G} B$ for $\mathcal{H}$-Banach algebras $B$ to construct $\mathcal{G}$-Banach algebras.
• Proves that $\mathcal{A}(\mathcal{H}, B) \sim_{\text{M}} \mathcal{A}(\mathcal{G}, \operatorname{Ind}_\mathcal{H}^\mathcal{G} B)$ via Morita equivalence.
• Uses cut-off functions and projections to relate $\mathcal{A}(\mathcal{H}, \mathcal{C}_0(X|_{\mathcal{H}^{(0)}}))$ and $\mathcal{A}(\mathcal{L}, \mathcal{C}_0(X))$ in the proof of diagram commutativity.
• Applies functoriality of $\operatorname{KK}^{\operatorname{ban}}$-theory in both variables to verify commutativity of the key diagram.
• Leverages the fact that the Bost assembly map for $C^*$-algebras factors through $\operatorname{K}^{\operatorname{top}2,\operatorname{ban}}$ to extend results to $C^*$-coefficients.

Experimental results

Research questions

• RQ1 Does the Bost conjecture with Banach algebra coefficients remain invariant under equivalence of groupoids?
• RQ2 Can the $\operatorname{KK}^{\operatorname{ban}}$-theory for Banach algebras be made functorial under generalised morphisms of groupoids?
• RQ3 Is the Morita equivalence between $L^1$-algebras of equivalent groupoids extendable to general unconditional completions?
• RQ4 How do induction and descent interact in the context of Banach algebraic $K$-theory for groupoids?
• RQ5 Can the commutativity of the $K$-theory diagram be established using cut-off functions and projection liftings?

Key findings

• The Bost conjecture with Banach algebra coefficients holds for a groupoid $\mathcal{G}$ if and only if it holds for any equivalent groupoid $\mathcal{H}$, due to the invariance of $\operatorname{KK}^{\operatorname{ban}}$-theory under equivalence.
• The induced $\mathcal{G}$-Banach algebra $\operatorname{Ind}_\mathcal{H}^\mathcal{G} B$ yields a Morita equivalence between $\mathcal{A}(\mathcal{H}, B)$ and $\mathcal{A}(\mathcal{G}, \operatorname{Ind}_\mathcal{H}^\mathcal{G} B)$, implying isomorphic $K$-theory.
• The commutative diagram linking $\operatorname{K}^{\operatorname{top}2,\operatorname{ban}}_*(\mathcal{H}, B)$, $\operatorname{K}_*(\mathcal{A}(\mathcal{H}, B))$, and $\operatorname{K}_*(\mathcal{A}(\mathcal{G}, \operatorname{Ind}_\mathcal{H}^\mathcal{G} B))$ is established via pullback and pushforward along Morita equivalences.
• The left vertical arrow in the diagram is an isomorphism due to the invariance of $\operatorname{KK}^{\operatorname{ban}}$-theory under groupoid equivalences.
• The Bost conjecture with $C^*$-algebra coefficients is also invariant under groupoid equivalence, as it factors through the Banach algebraic version.
• The proof relies on extending cut-off functions from $X|_{\mathcal{H}^{(0)}}$ to $X$ and showing that the associated projections are preserved under the inclusion maps $\varphi_{{\mathcal{C}}_0(X)}$.