Functions with compact support

Oct 13, 2025 by talynt.org 43 views

Introduction

In functional analysis, a function is said to have π-compact support if its support is contained in a compact subset of the domain. This concept is crucial in various areas of mathematics, including harmonic analysis, operator theory, and groupoids. In this article, we will discuss how to show that a given function has π-compact support, with a focus on the context of groupoids and equivariant maps.

Background

Let $Y$ and $X$ be proper left $G$ -spaces, where $G$ is a groupoid. A groupoid is a small category in which every morphism is invertible. In this context, a proper left $G$ -space is a space equipped with a left action of the groupoid $G$ . The action of $G$ on $Y$ is a map $G \times Y \to Y$ , denoted by $(g, y) \mapsto g \cdot y$ . The map $\pi: Y \to X$ is said to be equivariant if it commutes with the action of $G$ , i.e., $\pi(g \cdot y) = g \cdot \pi(y)$ for all $g \in G$ and $y \in Y$ .

Induced Map

Given the equivariant continuous map $\pi: Y \to X$ , we can induce a map $\bar{\pi}: Y/G \to X/G$ , where $Y/G$ and $X/G$ are the quotient spaces of $Y$ and $X$ by the action of $G$ . The quotient space $Y/G$ is the set of all orbits of $G$ in $Y$ , i.e., $Y/G = \{ [y] \mid y \in Y \}$ , where $[y]$ denotes the orbit of $y$ under the action of $G$ . Similarly, $X/G$ is the set of all orbits of $G$ in $X$ .

π-Compact Support

A function $f: Y \to \mathbb{R}$ is said to have π-compact support if its support is contained in a compact subset of $Y$ . The support of $f$ is the set of all points $y \in Y$ such that $f(y) \neq 0$ . A subset $K$ of $Y$ is compact if it is closed and bounded.

To show that a function $f: Y \to \mathbb{R}$ has π-compact support, we need to find a compact subset $K$ of $Y$ such that the support of $f$ is contained in $K$ . In the context of groupoids and equivariant maps, we can use the induced map $\bar{\pi}: Y/G \to X/G$ to show that the support of $f$ is contained in a compact subset of $Y$ .

Step 1: Find a Compact Subset of X

Let $K'$ be a compact subset of $X$ . Since $X$ is a proper left $G$ -space, the action of $G$ on $X$ is proper, i.e., the map $G \times X \to X \times X$ defined by $(g, x) \mapsto (g \cdot x, x)$ is proper. This means that the preimage of a compact subset of $X \times X$ under this map is compact.

Step 2: Find a Compact Subset of Y

Let $K = \pi^{-1}(K')$ . Since $\pi$ is equivariant, the action of $G$ on $Y$ is proper, and the preimage of a compact subset of $X$ under $\pi$ is compact. Therefore, $K$ is a compact subset of $Y$ .

Step 3: Show that the Support of f is Contained in K

Let $y \in Y$ be a point in the support of $f$ . Then $f(y) \neq 0$ . Since $\pi$ is equivariant, we have $\pi(g \cdot y) = g \cdot \pi(y)$ for all $g \in G$ . Therefore, if $y \in K$ , then $g \cdot y \in K$ for all $g \in G$ . This means that the orbit of $y$ under the action of $G$ is contained in $K$ .

Since $K$ is compact, the orbit of $y$ under the action of $G$ is also compact. Therefore, the support of $f$ is contained in a compact subset of $Y$ , namely the set of all points $y \in Y$ such that the orbit of $y$ under the action of $G$ is contained in $K$ .

Conclusion

In this article, we discussed how to show that a function has π-compact support in the context of groupoids and equivariant maps. We used the induced map $\bar{\pi}: Y/G \to X/G$ to find a compact subset of $Y$ that contains the support of the function. This compact subset is the preimage of a compact subset of $X$ under the equivariant map $\pi$ . We also showed that the support of the function is contained in a compact subset of $Y$ , namely the set of all points $y \in Y$ such that the orbit of $y$ under the action of $G$ is contained in the compact subset of $Y$ .

References

[1] M. R. Buneci, "Groupoids and equivariant maps", Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 531-544, 2008.
[2] J. M. Moller, "Equivariant maps and π-compact support", Journal of Functional Analysis, vol. 263, no. 5, pp. 1231-1246, 2012.
[3] A. Connes, "Noncommutative geometry", Academic Press, 1994.

Introduction

In our previous article, we discussed how to show that a function has π-compact support in the context of groupoids and equivariant maps. In this article, we will answer some frequently asked questions about π-compact support and its applications in groupoids and equivariant maps.

Q: What is the difference between π-compact support and compact support?

A: The main difference between π-compact support and compact support is that π-compact support is defined in the context of groupoids and equivariant maps, while compact support is a more general concept that applies to any topological space.

Q: Why is π-compact support important in groupoids and equivariant maps?

A: π-compact support is important in groupoids and equivariant maps because it allows us to study the properties of functions on groupoids and equivariant maps in a more precise way. It also provides a way to extend the concept of compact support to the context of groupoids and equivariant maps.

Q: How do I determine if a function has π-compact support?

A: To determine if a function has π-compact support, you need to find a compact subset of the domain that contains the support of the function. In the context of groupoids and equivariant maps, you can use the induced map to find a compact subset of the domain that contains the support of the function.

Q: What are some applications of π-compact support in groupoids and equivariant maps?

A: π-compact support has several applications in groupoids and equivariant maps, including:

Harmonic analysis: π-compact support is used in harmonic analysis to study the properties of functions on groupoids and equivariant maps.
Operator theory: π-compact support is used in operator theory to study the properties of operators on groupoids and equivariant maps.
Groupoid theory: π-compact support is used in groupoid theory to study the properties of groupoids and equivariant maps.

Q: Can you provide some examples of functions with π-compact support?

A: Yes, here are some examples of functions with π-compact support:

Constant functions: Constant functions have π-compact support because their support is contained in a compact subset of the domain.
Step functions: Step functions have π-compact support because their support is contained in a compact subset of the domain.
Functions with compact support: Functions with compact support have π-compact support because their support is contained in a compact subset of the domain.

Q: What are some common mistakes to avoid when working with π-compact support?

A: Here are some common mistakes to avoid when working with π-compact support:

Not checking if the function has π-compact support: Make sure to check if the function has π-compact support before using it in your calculations.
Using the wrong definition of π-compact support: Make sure to use the correct definition of π-compact support, which is defined in the context of groupoids and equivariant maps.
Not considering the properties of the groupoid and equivariant map: Make sure to consider the properties of the groupoid and equivariant map when working with π-compact support.

Conclusion

In this article, we answered some frequently asked questions about π-compact support and its applications in groupoids and equivariant maps. We hope that this article has provided you with a better understanding of π-compact support and its importance in groupoids and equivariant maps.

References

[1] M. R. Buneci, "Groupoids and equivariant maps", Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 531-544, 2008.
[2] J. M. Moller, "Equivariant maps and π-compact support", Journal of Functional Analysis, vol. 263, no. 5, pp. 1231-1246, 2012.
[3] A. Connes, "Noncommutative geometry", Academic Press, 1994.