Topological divisor of zero

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In mathematics, in a topological algebra A, $z\in A$ is a topological divisor of zero if there exists a neighbourhood U of zero and a net $(x_i)_{i\in I}$ with $\forall i\in I, x_i \in A\setminus U$ and $zx_i \longrightarrow O\in A.$ If the topological algebra is not commutative use left resp. right topological divisor of zero.

They are not invertible, otherwise multiplying by the inverse would give $x_i\longrightarrow 0$ , contradicting $x_i\in A\setminus U$ .

[edit] Example

In a Banach algebra $(A,\|\cdot\|)$ with a norm $\|\cdot\|$ an element z is a topological divisor of zero if and only if it there exists a sequence $(x n)$ in A such that $\|x_n\|=1$ for all $n$ while $\lim_{n\rightarrow \infty} \|z\cdot x_n\| = 0$

An element of a Banach algebra with unity, which is at the boundary of the closed set of non-invertible elements and the open set of invertible ones, is a left- and right topological divisor of zero. Thus, quasinilpotents are topological divisors of zero (e.g. the Volterra operator).

An operator on a Banach space $X$ , which is injective, not surjective, but whose image is dense in $X$ , is a left topological divisor of zero.