Bounding || zx - yz || given that || x - y || < M in a Banach algebra.

Question

Let $ X $ be an Banach algebra (not necessarily commutative), and let $ x, y, z \in X $. Suppose that $ \| x - y \| < M $.

I want to bound $ \| zx - yz \| $ in terms of $ M $ by writing $ zx - yz $ as a product of terms including the term $ x - y $, but I don't see how to do so (or if it is even possible).

If it is possible, I am assuming it is just an elementary factoring trick that I am blanking out on.

Any help would be appreciated. Thanks!

Answer 1

You can't in general. If you pick $y=x$ then you have $\|x-y\|<M$ for all $M$, while $\|zx-yz\|=\|[z,x]\|$ may be arbitrary (pick a ring of matrices for example).