nLab vanishing at infinity

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

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

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

  1. assumes that AA is pointed by an element 0A0 \in A,

    then getting arbitrarily close to 00 means entering any neighbourhood of this basepoint.

  2. assumes that XX is locally compact Hausdorff,

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

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

Definition

Consider:

Definition

The map f:XAf\colon X \to A vanishes at infinity if for every neighbourhood NN of the basepoint 0A0 \in A, there is a compact subspace KK of XX such that for all xExt(K)x \in Ext(K) in the exterior of KK, we have f(x)Nf(x) \in N.

Remark

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

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 “0\infty \mapsto 0”, where “\infty” denotes the point adjoined to XX when passing to its “one-point compactificationX cptX^{cpt} (and demanding that ff extends to a continuous function on X cptX^{cpt}).

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

Precisely:

Proposition

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

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

In view of the equivalence of Prop. , we may say that for locally compact XX, the vector space of continuous maps XX \longrightarrow \mathbb{C} that vanish at \infty, denoted C 0(X)C_0(X), is isomorphic to the kernel of the evaluation map

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

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

Proposition

C 0(X)C_0(X) is a (nonunital) C * C^\ast -algebra.

Proof

Since the function space is a C *C^\ast-algebra and C 0(X)ker(ev )C_0(X) \cong \ker(ev_\infty) is a closed subspace, and thus a Banach space, and since the C *C^\ast-algebra structure on [X cpt,][X^{cpt}, \mathbb{C}] clearly restricts to a C *C^\ast-algebra structure on the kernel, the result is clear.

Remark

C 0(X)C_0(X) is nonunital unless XX is already compact. Its unitalization is C 0(X cpt)C_0(X^{cpt}).

(cf. Warner 2010 Ex 1.15)

References

The algebra C 0(X)C_0(X) is discussed in most monographs on C * C^\ast -algebras, for instance:

  • Garth Warner Ex. 1.15 in: C *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 *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 *C^\ast-algebras (2019) [pdf, pdf]

See also:

Last revised on June 27, 2025 at 08:26:43. See the history of this page for a list of all contributions to it.