Motivating example: the ℝ-algebra C^∞(M,ℝ) of smooth functions from a manifold M to ℝ, where exp(V)(f)(x) = f(Flow_V(x,1)) for derivations/vector fields V, smooth maps f, and points x. If M is compact, then exp is total.
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You can't get ℂ-algebras with a total exp in the same way, since the algebra of holomorphic functions from a compact complex manifold to ℂ isn't very interesting.
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