# Neumann series in an incomplete normed algebra

Let $\mathcal{A} \equiv (A, \|\cdot\|_A)$ be a unital (associative) normed algebra over the real or complex field, and assume that $\mathcal{A}$ is not complete. Provided $\mathcal{B}_\mathcal{A}$ is the open unit ball of $\mathcal{A}$, define $N$ to be the set of all $a \in \mathcal{B}_\mathcal{A}$ such that the Neumann series $\sum_{n=0}^\infty a^n$ does not converge in $\mathcal{A}$.

Questions. 1) Is $N$ dense in $\mathcal{B}_\mathcal{A}$? 2) And what about $\mathcal{B}_\mathcal{A}\setminus N$?

Edit (11 Dec 2011). Following Davide's comment below, let $\mathcal{C}^0([0,1],\mathbb{R})$ be the usual Banach algebra (over the real field) of all continuous functions $[0,1] \to \mathbb{R}$ endowed with the uniform norm $\|\cdot\|_\infty$. Define $A$ to be the subalgebra of $\mathcal{C}^0([0,1],\mathbb{R})$ of all polynomial functions. For each $\phi \in A$ such that $\|\phi\|_\infty < 1$, the Neumann series $\sum_{n=0}^\infty \phi^n$ converges in $\mathcal{C}^0([0,1],\mathbb{R})$ to $(1 - \phi)^{-1}$, but it does not in $\mathcal{A} \equiv (A,\|\cdot\|_\infty)$ so far as $\phi$ is not a constant. Thus, $N$ is dense in $\mathcal{B}_\mathcal{A}$, and indeed in the unit ball of $\mathcal{C}^0([0,1],\mathbb{R})$ (by the Stone-Weierstrass theorem).

• As far as I can understand, there still exists no tag for normed algebras, and I'm not yet enabled to create a new one by myself. Dec 10, 2011 at 18:38
• What example(s) of $\mathcal A$ make(s) you think that $N$ can be dense in $\mathcal B_{\mathcal A}$? Dec 10, 2011 at 23:10
• I've retagged question - in my opinion, it's better to think twice before creating a new tag whether it will be useful. But if you think that the new tag would be useful, fell free to retag the question again, of course. Dec 11, 2011 at 14:20
• @Martin. I don't care much about labels but still think that normed ring/algebras would deserve their own tag. :) Dec 11, 2011 at 15:01
• Speaking as a Banach algebraist, to act as if that area is subsumed by "operator-algebras" and "c-star-algebras" seems quite mistaken in my view. (Just try applying continuous functional calculus to self-adjoint elements in $\ell^1$-group algebras and watch your norm estimates blow up in your face...)
– user16299
Jan 6, 2012 at 6:52