Smooth subalgebra of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. In this knowl, a smooth subalgebra is a dense $*$ -subalgebra $\mathcal A\subseteq A$ with a complete Fréchet topology finer than the $C^*$ -norm topology, continuous multiplication and involution, and spectral invariance: after adjoining identities when necessary,

x\in\widetilde{\mathcal A}\text{ is invertible in }\widetilde A \quad\Longrightarrow\quad x^{-1}\in\widetilde{\mathcal A}.

Under the standard complete locally convex hypotheses, this is equivalent to closure under holomorphic functional calculus in $\widetilde A$ .

Why spectral invariance matters

Holomorphic closure keeps spectral constructions inside the regular algebra. In particular, idempotents and invertibles used in $K$ -theory can be deformed without leaving $\mathcal A$ , and the inclusion $\mathcal A\hookrightarrow A$ induces the expected $K$ -theory isomorphisms under the usual matrix-stable formulation. This is why a spectrally invariant pre- $C^*$ -algebra is often the preferred coordinate algebra in noncommutative geometry.

Standard constructions

If a finite-dimensional Lie group acts strongly continuously on $A$ by $*$ -automorphisms, the elements whose orbit maps are smooth form a dense Fréchet $*$ -subalgebra $A^\infty$ ; it is spectrally invariant. Classical examples include $C^\infty(M)\subset C(M)$ for a compact smooth manifold and smooth noncommutative tori inside their $C^*$ -completions. Schwartz-type convolution algebras provide further examples only when the relevant decay and spectral-invariance theorems are verified.

Conventions and scope

References

Alain Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted book. Relevant: Part IV on smooth algebras and spectral triples.
Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. Author-hosted text. Relevant: §III.5 on dense subalgebras stable under holomorphic functional calculus.