Let $\mathcal A \subset B(H)$ be an operator algebra and $\varphi: \mathcal A \rightarrow B(K)$ a completely bounded homomorphism. Suppose $\mathcal M \subset \mathcal A$ is an operator space such that $\overline{\rm Alg}(\mathcal M) = \mathcal A$.



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> > Does there exist a universal constant $C$ such that $$C \|\varphi|\_\mathcal M\|\_{\rm cb} \geq \|\varphi\|\_{\rm cb}? $$
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Norm of a cb-homomorphism restricted to a generating operator space
No, not even when $\cal A$ is commutative and $\phi $ is a multiplicative linear functional. For example, given $0<a<1$ let ${\cal A} =C[a,1]$, let $\cal M$ be the linear span of the function $f(x) = x$, and let $\phi $ be pointwise evaluation at $a$.


