By a stochastic matrix I mean a square real non-negative matrix with rows summing to one. Denote the set of all such matrices $\mathcal{S}$.
By matrix norm I mean a norm in the vector space of matrices which additionally has the property $\forall A B. \|AB\| < \|A\| \|B\|$.
Does there exist a matrix norm such that $\forall P \in \mathcal{S}. \| P \| = 1$ ?
The norm has to be the same for all choices of $P$.
Is there such a norm which has a name and is easily computable?