Pre-C*-algebra - Knowlpedia

A pre- $C^*$ -algebra is a complex normed $*$ -algebra $A$ , not assumed complete, whose norm is submultiplicative and satisfies the $C^*$ -identity

\lVert a^*a\rVert=\lVert a\rVert^2 \qquad (a\in A).

The involution and multiplication are therefore compatible with the norm, but incompleteness distinguishes $A$ from a $C^*$ -algebra. Unitality is not part of the definition; when $A\neq0$ has an identity, the $C^*$ -identity forces that identity to have norm $1$ . Some authors say “pre- $C^*$ -normed algebra” to emphasize that the displayed identity, rather than completeness, is the decisive condition.

Completion

The involution is isometric and therefore extends continuously to the Banach-space completion $\overline A$ . Multiplication extends as well, and the $C^*$ -identity passes to limits, so $\overline A$ is a $C^*$ -algebra in which $A$ is a dense $*$ -subalgebra. Conversely, every $*$ -subalgebra of a $C^*$ -algebra, equipped with the inherited norm, is a pre- $C^*$ -algebra.

Examples and non-examples

For a compact smooth manifold $M$ , $C^\infty(M)$ with pointwise operations, complex conjugation, and the supremum norm is a pre- $C^*$ -algebra; its completion is $C(M)$ . It is not a $C^*$ -algebra unless it is already complete in that norm. By contrast, an arbitrary dense $*$ -subalgebra carrying a stronger norm need not be pre- $C^*$ : the stronger norm may fail $\lVert a^*a\rVert=\lVert a\rVert^2$ .

Conventions and scope

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §2.1 on $C^*$ -norms and completion.
Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. Chapter record. Relevant: Chapter II, §3.1 on local Banach $*$ -algebras and pre- $C^*$ -norms.