Definition Weakly smooth space [efr-4T1Z]

Let A = C_0(X) be a commutative C^*-algebra. A weakly smooth structure on A consists of a sub-*-algebra S \subseteq A satisfying the following properties:

1. S is dense in the norm topology of A.
2. If f: \mathbb {R}^n \to \mathbb {R} is a smooth function, and a_1, \dots a_n \in S, then f(a_1,\dots a_n) \in S, (in the sense of the continuous functions calculus).
3. S is equipped with a Frechet topology.

Given two weakly smooth structure (A,S), (B,S'), a C^*-homomorphism A \to B is said to preserve the smooth structure if it carries elements of S to S' and the restricted map S \to S' is continuous for the given topologies.