I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and extending this to a map $\phi^k: V^{\otimes k} \rightarrow U^{\otimes k}$ defined by
$$ \phi^k(v_1 \otimes \ldots \otimes v_k) = \phi(v_1) \otimes \ldots \otimes \phi(v_k). $$
I aim to extend this linearly to an arbitrary tensor which might be a sum of pure tensors, i.e.,
$$ v = \sum_{i=1}^n v_{1i} \otimes \ldots \otimes v_{ki}. $$
Upon trying to prove the continuity of $\phi^k$, assuming a "subtensorial" inequality of the norms on both tensor product spaces:
$$ \|v_1 \otimes \ldots \otimes v_k\| \leq \|v_1\| \cdot \ldots \cdot \|v_k\|, $$
I arrived at the inequality:
$$ \|\phi^k(v)\| \leq M^k \sum_{i=1}^n \|v_{1i}\| \cdot \ldots \cdot \|v_{ki}\| $$
where $M$ is a bound on the operator norm of $\phi$.
I realized that the right-hand side doesn't represent the norm of $v$ directly. However, considering this inequality holds for any representation of $v$ as a sum of pure tensors, we can see that the right-hand side is bounded by the projective norm of $v$ in $V^{\otimes k}$, i.e.
$$\|v\|_{\text{projective}} := \text{inf}\,\Big\{\sum_{i=1}^n \|v_{1i}\| \cdot \ldots \cdot \|v_{ki}\| \\ \text{ : } v = \sum_{i=1}^n v_{1i} \otimes \ldots \otimes v_{ki} \Big\}.$$
So, it appears $\phi^k$ would be continuous if $V^{\otimes k}$ is endowed with the projective norm, while $U^{\otimes k}$ retains the submultiplicative property.
It seems like I need the projective norm on $V^{\otimes k}$ in order to make $\phi^k$ continuous. That my subtensorial inequality isn’t enough for the norm on $V^{\otimes k}$. Is this correct? Would continuity hold perhaps instead if we endowed $V^{\otimes k}$ with the injective norm instead?
PS: In fact, I am trying to prove an extended version of a known property of the free tensor algebra. Traditionally, it is known that for any linear map $\phi:V \to U$, there exists a unique algebra homomorphism $\tilde{\phi}$ that extends it as a map $T(V) \to T(U)$, where for $a \in \bigoplus_{k=0}^\infty V^{\otimes k} = T(V)$, $\tilde{\phi}(a) = (\phi^k(a^k))_{k\geq 0} \in T(U)$.
The question I am trying to elucidate is whether it can be proven that a linear and continuous $\phi$ is extendable to a continuous algebra homomorphism. This seems to necessitate imposing the projective norm on the tensor powers of $V$, ensuring that the $\phi^k$'s are continuous homomorphisms. It seems that the subtensorial inequality is not a sufficient requirement for the norm on $V^{\otimes k}$.
Thank you.
