In Nestruev's (2000) Smooth Manifolds and Observables, the authors define an $\mathbb{R}$-algebra as a commutative, associative algebra with unit (p. 21). A natural generalization of this definition would drop the requirement of a unit. (For example, any self-adjoint, commutative, and non-unital C*-algebra defines such a "non-unital $\mathbb{R}$-algebra".) I am interested in how many of Nestruev's constructions carry over to the non-unital case (e.g., the notion of a smooth envelope of a geometric $\mathbb{R}$-algebra). Have folks investigated this matter? Preliminary searches have dead-ended.

P.S. Apologies if this question is too preliminary; I am new to posting on this site. Many thanks for reading :)

In Nestruev's (2000) Smooth Manifolds and Observables, the authors define an $\mathbb{R}$-algebra as a commutative, associative algebra with unit (p. 21). A natural generalization of this definition would drop the requirement of a unit. (For example, any self-adjoint, commutative, and non-unital C*-algebra defines such a "non-unital $\mathbb{R}$-algebra".) I am interested in how many of Nestruev's constructions carry over to the non-unital case. For example: is the notion of a smooth envelope of a geometric $\mathbb{R}$-algebra $\mathcal{F}$ well-defined if $\mathcal{F}$ lacks a unit?

P.S. Apologies if this question is too preliminary; I am new to posting on this site. Many thanks for reading :)