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I have a few things I don't understand in your argument. I don't understand this step:

"[...] then, since q is a quotient homomorphism, there exists b \in A such that q(b)=q(a) and || b || < r. [...]"

sorry but why is ||b|| < r?

"Therefore 1-r < 2r, i.e. r > 1/3." agreed.

But:

"It follows that rA(a)\geq 1/3"

could you elaborate how this follows now, I don't see it.

About the origin of the problem: I don't know if this is a known problem, but I did a quite extensive search with no results. It came up in my research on Frechet algebras since I wanted to know if uniformity for Frechet algebras is a three space property (this would imply that the category of uniform Frechet algebras is exact in the sense of Quillen, but nvm that..). ...). Since every (uniform) Frechet algebra is the inductive limit of a sequence of (uniform) Banach algebras I looked at Banach algebras first.

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I have a few things I don't understand in your argument. I don't understand this step:

"[...] then, since q is a quotient homomorphism, there exists b \in A such that q(b)=q(a) and || b || < r. [...]"

sorry but why is ||b|| < r?

"Therefore 1-r < 2r, i.e. r > 1/3." agreed.

But:

"It follows that rA(a)\geq 1/3"

could you elaborate how this follows now, I don't see it.

About the origin of the problem: I don't know if this is a known problem, but I did a quite extensive search with no results. It came up in my research on Frechet algebras since I wanted to know if uniformity for Frechet algebras is a three space property (this would imply that the category of uniform Frechet algebras is exact in the sense of Quillen, but nvm that..). Since every (uniform) Frechet algebra is the inductive limit of a sequence of (uniform) Banach algebras I looked at Banach algebras first.