topological divisor of zero
Let $A$ be a normed ring^{}. An element $a\in A$ is said to be a left topological divisor of zero if there is a sequence ${a}_{n}$ with $\parallel {a}_{n}\parallel =1$ for all $n$ such that
$$\underset{n\to \mathrm{\infty}}{lim}\parallel a{a}_{n}\parallel =0.$$ 
Analogously, $a$ is a if
$$\underset{n\to \mathrm{\infty}}{lim}\parallel {b}_{n}a\parallel =0,$$ 
for some sequence ${b}_{n}$ with $\parallel {b}_{n}\parallel =1$. The element $a$ is a topological divisor of zero if it is both a left and a topological divisor of zero.
Remarks.

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Any zero divisor is a topological divisor of zero.

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If $a$ is a (left) topological divisor of zero, then $ba$ is a (left) topological divisor of zero. As a result, $a$ is never a unit, for if $b$ is its inverse, then $1=ba$ would be a topological divisor of zero, which is impossible.

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In a commutative Banach algebra $A$, an element is a topological divisor of zero if it lies on the boundary of $U(A)$, the group of units of $A$.
Title  topological divisor of zero 

Canonical name  TopologicalDivisorOfZero 
Date of creation  20130322 16:12:15 
Last modified on  20130322 16:12:15 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 46H05 
Synonym  generalized divisor of zero 