[Paper Review] A Schwartz type algebra for the Tangent Groupoid

Paulo Carrillo Rouse|ArXiv.org|Feb 25, 2008

Advanced Operator Algebra Research21 references13 citations

TL;DR

This paper constructs a Schwartz-type algebra, denoted $\mathscr{S}_{r,c}(\mathscr{G}^T)$, over the tangent groupoid $\mathscr{G}^T$ of a Lie groupoid $\mathscr{G}$, forming a continuous field of algebras over $[0,1]$. The fiber at $t=0$ is the Schwartz algebra $\mathscr{S}(A\mathscr{G})$ of the Lie algebroid, while fibers for $t>0$ are the convolution algebra $C^\infty_c(\mathscr{G})$. This intermediate algebra enables refined index morphisms compatible with cyclic cohomology and Chern-Connes character theory.

ABSTRACT

We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a further work, to define other analytic index morphisms as a sort of deformations.

Motivation & Objective

• To construct an intermediate algebra between the $C^\infty_c(\mathscr{G})$ and $C^*(\mathscr{G})$ levels for Lie groupoids.
• To overcome limitations of $C^\infty_c(\mathscr{G})$-indices, which lack homotopy invariance, while retaining compatibility with cyclic cohomology tools.
• To provide a smooth, geometrically natural algebraic structure on the tangent groupoid that interpolates between the Lie algebroid and groupoid at $t=0$ and $t>0$, respectively.
• To enable the definition of new analytic index morphisms in $K_0$-theory that are amenable to pairing with cyclic cocycles via Chern-Connes character theory.
• To establish a short exact sequence analogous to the $C^*$-algebraic case, facilitating index theory at the smooth level.

Proposed method

• Construct a sheaf of smooth functions over the tangent groupoid $\mathscr{G}^T = A\mathscr{G} \times \{0\} \sqcup \mathscr{G} \times (0,1]$, equipped with a Schwartz-type decay condition at $t=0$.
• Define the algebra $\mathscr{S}_{r,c}(\mathscr{G}^T)$ as the space of smooth functions on $\mathscr{G}^T$ that decay rapidly in the $t=0$ fiber (Lie algebroid) and are compactly supported in the $t>0$ fibers (groupoid).
• Use local trivializations and diffeomorphism invariance to define the algebra globally via transition maps, ensuring consistency across charts.
• Establish evaluation morphisms $e_t: \mathscr{S}_{r,c}(\mathscr{G}^T) \to \mathscr{S}(A\mathscr{G})$ for $t=0$ and $e_t: \mathscr{S}_{r,c}(\mathscr{G}^T) \to C^\infty_c(\mathscr{G})$ for $t>0$, making $\mathscr{S}_{r,c}(\mathscr{G}^T)$ a continuous field of algebras over $[0,1]$.
• Prove that the algebra is invariant under diffeomorphisms and well-defined across atlases via the use of partition of unity and transition maps.
• Construct a short exact sequence $0 \to J \to \mathscr{S}_{r,c}(\mathscr{G}^T) \xrightarrow{e_0} \mathscr{S}(A\mathscr{G}) \to 0$, analogous to the $C^*$-case, to support future index theory at the smooth level.

Experimental results

Research questions

• RQ1 Can a smooth, geometrically natural algebra be constructed on the tangent groupoid that interpolates between the Schwartz algebra of the Lie algebroid and the convolution algebra of the groupoid?
• RQ2 Does such an algebra allow for the definition of analytic index morphisms that are both refined (like $C^\infty_c$-indices) and compatible with cyclic cohomology tools (like Chern-Connes character)?
• RQ3 Is it possible to construct a short exact sequence of algebras analogous to the $C^*$-algebraic case, but at the smooth level, to support index theory without relying on $C^*$-homotopy invariance?
• RQ4 How can the algebraic structure of the tangent groupoid be used to define index morphisms that are more refined than $C^*$-indices but more stable than $C^\infty_c$-indices?
• RQ5 Can the resulting algebra support a deformation-theoretic approach to index theory that retains geometric and analytic control?

Key findings

• The algebra $\mathscr{S}_{r,c}(\mathscr{G}^T)$ is a continuous field of algebras over $[0,1]$, with fiber at $t=0$ isomorphic to $\mathscr{S}(A\mathscr{G})$, the Schwartz algebra of the Lie algebroid, and fiber at $t>0$ isomorphic to $C^\infty_c(\mathscr{G})$, the convolution algebra of compactly supported smooth functions on $\mathscr{G}$.
• The inclusion $C^\infty_c(\mathscr{G}^T) \subset \mathscr{S}_{r,c}(\mathscr{G}^T) \subset C^*(\mathscr{G}^T)$ holds, placing the new algebra as a natural intermediate object between the smooth and $C^*$-levels.
• The evaluation maps $e_t$ are well-defined algebra morphisms, ensuring that the algebra structure is compatible with the fibration over $[0,1]$, and that the fiber algebras are correctly realized.
• The construction is invariant under diffeomorphisms and glues consistently across charts, ensuring global well-definedness on the tangent groupoid.
• A short exact sequence $0 \to J \to \mathscr{S}_{r,c}(\mathscr{G}^T) \xrightarrow{e_0} \mathscr{S}(A\mathscr{G}) \to 0$ is established, analogous to the $C^*$-case, enabling future index-theoretic constructions.
• The algebra provides a framework for defining new analytic index morphisms in $K_0$-theory that are compatible with Chern-Connes character theory, overcoming limitations of both $C^\infty_c$ and $C^*$-indices.