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For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V):=B(V,V)$ are linear bounded endomorphisms with operator norm? If so, is it also smooth (wrt Michal/Bastiani-differentiability in locally convex spaces)? Does it induce a continuous (smooth) map $GL(V)\times_\pi GL(W) \to GL(V \hat{\otimes}_\pi W)$, where $GL(V)=B(V)^{\times}$.

Edit: The answer to the very first question is yes

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