C*-algebra C_0(X) - Imported Knowlpedia Content

Definition

Let $X$ be a locally compact Hausdorff space. The $C^*$ -algebra $C_0(X)$ consists of the continuous functions $f:X\to\mathbb C$ that vanish at infinity: for every $\varepsilon>0$ , there is a compact set $K\subseteq X$ such that $|f(x)|<\varepsilon$ whenever $x\notin K$ . Addition, multiplication, and involution are pointwise, with $f^*(x)=\overline{f(x)}$ , and the norm is $\|f\|_\infty=\sup_{x\in X}|f(x)|$ . With these operations $C_0(X)$ is a commutative $C^*$ -algebra. It is unital exactly when $X$ is compact.

Relation to compact support

Every compactly supported continuous function belongs to $C_0(X)$ , and the subalgebra $C_c(X)$ of such functions is dense in $C_0(X)$ for the uniform norm. Vanishing at infinity does not require compact support: on $\mathbb R$ , the function $x\mapsto(1+x^2)^{-1}$ lies in $C_0(\mathbb R)$ but has noncompact support.

Spectrum and unitization

The characters of $C_0(X)$ are precisely the evaluation maps $f\mapsto f(x)$ for $x\in X$ . Thus locally compact Gelfand duality recovers $X$ as the character space of $C_0(X)$ . When $X$ is noncompact, the minimal unitization of $C_0(X)$ is naturally isomorphic to the continuous functions on the one-point compactification of $X$ .

Why it is canonical

The passage $X\mapsto C_0(X)$ converts proper continuous maps contravariantly into nondegenerate $*$ -homomorphisms. It is the locally compact, generally nonunital counterpart of the compact-space algebra $C(X)$ , and is the basic commutative model for ideals, spectra, and multiplier algebras.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. Publisher record. Relevant: Chapter 2 on commutative $C^*$ -algebras and their spectra.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, second edition edited by Søren Eilers and Dorte Olesen, Academic Press, 2018. Publisher record. Relevant: the opening chapters on locally compact function algebras.