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 ðŸ™‚*