Let us define the following matrix:

$C=A*B$

where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$, …, $B_N$, each of dimensions $M×M$. Hence, $B$ is a matrix of dimensions $(MN x MN)$. I know that the matrix 2 norm of each one of the blocks is smaller than 1. Moreover, I know that the matrix $(MN x MN)$ $A$ is right stochastic, i.e., the sum of the elements in each row equals one. Can I say that the spectral radius of C is smaller than one? If so, how could I prove that? I have tried doing the proof with the block maximum norms and 2-norm. However, it did not work out.