nLab vanishing at infinity

Idea

A continuous map between topological spaces vanishes at infinity if $f(x)$ gets arbitrarily close to zero as $x$ gets sufficiently close to infinity.

In order to make sense of this for a map $f\colon X \longrightarrow A$, one:

1. assumes that $A$ is pointed by an element $0 \in A$,

   then getting arbitrarily close to $0$ means entering any neighbourhood of this basepoint.

2. assumes that $X$ is locally compact Hausdorff,

   then getting sufficiently close to infinity means entering the exterior $Ext(K)$ of some compact subspace $K \subset X$.

Finally “getting close” is understood in terms of nets.

Definition

Consider:

• $X$ a locally compact Hausdorff space,

• $A$ a point topological spaces, with basepoint to be denoted $0 \in A$,

• $f \,\colon\, X \longrightarrow A$ a continuous map.

Definition

The map $f\colon X \to A$ vanishes at infinity if for every neighbourhood $N$ of the basepoint $0 \in A$, there is a compact subspace $K$ of $X$ such that for all $x \in Ext(K)$ in the exterior of $K$, we have $f(x) \in N$.

Remark

If $A$ is a pointed metric space (such as a Banach space, in particular the real line), then Def. is equivalent to:

• For every positive real number $\epsilon$, there is a compact subspace $K$ of $X$ such that ${\|f(x)\|} \lt \epsilon$ whenever $x$ lies in the exterior of $K$ in $X$.

Here, ${\|{-}\|}$ denotes the distance from the basepoint in the metric space, such as the norm in the case of Banach spaces.

Properties

Relation to one-point compactification

One may neatly understand Def. as saying that a function “vanishes at infinity” if it literally sends “$\infty \mapsto 0$”, where “$\infty$” denotes the point adjoined to $X$ when passing to its “one-point compactification” $X^{cpt}$ (and demanding that $f$ extends to a continuous function on $X^{cpt}$).

This is achieved by the fact that in $X^{cpt}$ — whose underlying set is $X \sqcup \{\infty\}$ — the open neighborhoods of $\infty$ are subsets of the form $\{\infty\} \cup Ext(K)$ for compact $K \subset X$.

Precisely:

Proposition

A continuous function $f \colon X \to A$ vanishes at infinity according to Def. iff it extends along $X \hookrightarrow X^{cpt}$ to a continuous map $\widehat{f} \,\colon\, X^{cpt} \longrightarrow A$ such that $\widehat{f}(\infty) = 0$.

Remark

For applications of this equivalence to physics see:

• at one-point compactification – Relevance for Monopoles and Instantons

• at Yang-Mills instanton – SU(2)-instantons from the correct maths to the traditional physics story.

The $C^\ast$-algebra of functions vanishing at infinity

In view of the equivalence of Prop. , we may say that for locally compact $X$, the vector space of continuous maps $X \longrightarrow \mathbb{C}$ that vanish at $\infty$, denoted $C_0(X)$, is isomorphic to the kernel of the evaluation map

$$ev_\infty \,\colon\, [X^{cpt}, \mathbb{C}] \longrightarrow \mathbb{C}$$

from the function space on $X^{cpt}$, with pointwise defined $C^\ast$-algebra structure and the sup-norm topology.

Proposition

$C_0(X)$ is a (nonunital) $C^\ast$-algebra.

Proof

Since the function space is a $C^\ast$-algebra and $C_0(X) \cong \ker(ev_\infty)$ is a closed subspace, and thus a Banach space, and since the $C^\ast$-algebra structure on $[X^{cpt}, \mathbb{C}]$ clearly restricts to a $C^\ast$-algebra structure on the kernel, the result is clear.

Remark

$C_0(X)$ is nonunital unless $X$ is already compact. Its unitalization is $C_0(X^{cpt})$.

(cf. Warner 2010 Ex 1.15)

References

The algebra $C_0(X)$ is discussed in most monographs on $C^\ast$-algebras, for instance:

• Garth Warner Ex. 1.15 in: $C^\ast$-Algebras, EPrint Collection, University of Washington (2010) [hdl:1773/16302, pdf, pdf]

• Bruce Blackadar Ex. II.1.1.3(ii) in: Operator Algebras – Theory of $C^\ast$-Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences 122, Springer (2006) [doi:10.1007/3-540-28517-2]

  (which however does not dwell on the definition of “vanishing at infinity”)

• Ian Putnam, Ex. 1.2.4 in: Lecture notes on $C^\ast$-algebras (2019) [pdf, pdf]

See also:

• Wikipedia: Vanish at infinity

• Wikipedia: Locally compact space