Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous $$ x_i\to x\quad\Longrightarrow\quad ||\varphi(x_i)-\varphi(x)||\to 0, $$ and multiplicative $$ \varphi(x\cdot y)=\varphi(x)\cdot\varphi(y), \quad \varphi(1)=1. $$
Is every (norm-)continuous representation $\varphi:G\to A$ smooth?
If yes, I would be grateful for the references.
First an attempt at a counterexample. Theorem 5.2 of the paper
shows (in a simple way) that for a unitary representation $\rho: G\to L(H,H)$ there is the the Frechet subspace $H_\infty \subset C^\infty(G,H)$ consisting of those $v\in H$ such that $g\mapsto \rho(g)v$ is $C^\infty$ from $G$ to $H$, and, in general, $H_\infty\ne H$. For the regular representation of a compact Lie group $G$ on $L^2(G)$ we get $L^2(G)_\infty = C^\infty(G)$.
However, for a unitary representation as above, the map $\hat\rho:G\times H\to H$ is only separately continuous, thus $\rho: G\to (L(H,H), \|\;\|)$ is not continuous, and we do not get a counterexample.
For $ 1\le p\le\infty$, consider the space $W^{\infty,p}(G)$ of all $f\in C^\infty(G)$ such that $R_Xf\in L^p(G)$ for all $X\in\mathfrak g$, where $R_X$ is the right invariant vector field corresponding to $X$. Then $W^{\infty,2}(G)$ is a Frechet algebra under convolution, because $W^{\infty,2}\subset W^{\infty,1}$. Now we get an algebra homomorphism $W^{\infty,2}(G) \to A$, which is continuous by theorems about automatic continuity of algebra homomorphisms between Frechet algebras (Papers by Ernest Michael, if I remember correctly), thus smooth. We seem to be near smoothness of $\rho:G\to A$.
Moreover, $\rho: G \to L(W^{\infty,2}(G),W^{\infty,2}(G))$ is smooth since every vector is smooth (where $L$ denotes all bounded linear mappings).
Continuation of the proof attempt. Let $h_\epsilon\in W^{\infty,2}$ be an approximate unit: $h_{\epsilon} \star f\to f$ in $W^{\infty,2}$ for $\epsilon\to 0$ (a suitable sequence of smooth bunmp functions). Then $(\rho(g)h_\epsilon) \star f\to \rho(g)f$. Then $g\mapsto \rho(g)h_\epsilon$ is smooth $G\to W^{\infty,2}$ and thus smooth $G\to A$. Now it remains to prove that for $\epsilon\to 0$ this converges in $C^\infty(G,A)$.
Lots of things remain to be checked.
If $\rho: G\to A$ is norm continuous, and if $t\mapsto g(t)=\exp(tX)$ is a 1-parameter subgroup in $G$, than the usual proofs in books on Lie groups (see 4.21 of here) show that the continuous 1-parameter subgroup $t\mapsto \rho(g(t))$ in the Banach Lie group $A^{inv}$ of invertible elements in $A$ (which is open in $A$, with exponential mapping $\exp:A\to A^{inv}$ which is a local diffeomorphism) is smooth. But this proves, as usual (see 4.2 as cited above), that $\rho:G\to A$ is smooth.
