I am aware that the definition (or one of several equivalent definitions) of the Lie algebra $\mathfrak g$ of a Lie group $G$ is as follows: $\mathfrak g$ is the set of left $G$-invariant derivations $D:C^\infty(G)\to C^\infty(G)$, equipped with the commutator as the Lie bracket. Here $C^\infty(G)$ is the space of smooth functions $G\to\mathbb R$. Similarly, for a (linear) algebraic group $G$ over a field $k$ (possibly must be characteristic 0), we can define its Lie algebra as the set of left-invariant derivations of $\mathcal O(G)$ equipped with the commutator bracket, where $\mathcal O(G)$ is the space of regular functions on $G$.
This leads me to the following question: What happens if we consider the set of left-invariant derivations of the space of continuous functions $G\to \mathbb R$ of a Lie group? Is this the same as the Lie algebra of $G$? If not, is there any relationship between the two?
