nLab dense subalgebra

Idea

A dense subalgebra of a topological algebra is a subalgebra? which is also a dense subspace.

For instance, the algebra of smooth functions on a given smooth manifold $M$ is a dense subalgebra of the C-star-algebra of continuous functions on $X$. Conversely, if we start with a commutative C-star-algebra $A$ and view it as the algebra of continuous maps on its spectrum $S$, then picking a dense subalgebra of $A$ may specify a smooth structure on $S$ (which may be any compact Hausdorff space). For noncommutative geometry, we can let $A$ be an arbitrary $C^*$-algebra. One also speaks of smooth C-star algebras if a $C^\ast$-algebra is equipped with an inverse system of inclusions of dense subalgebra (a noncommutative version of Frechet spaces).

Hence equipping a C-star-algebra with a choice of dense subalgebra serves may be thought of as refining from noncommutative topology to noncommutative geometry. This is for instance what happens in the definition of spectral triples and in smooth refinements of KK-theory.

Related concepts

• dense

• F-star-algebra

References

• E. Kissin, V. S. Shulman, Differential properties of some dense subalgebras of $C^\ast$-algebras, Proceedings of The Edinburgh Mathematical Society, vol. 37, no. 03 (1994)

• Larry Schweitzer, Dense nuclear Fréchet ideals in $C^\ast$-algebras (arXiv:1205.0089)