# 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  $\|a_{n}\|=1$  for all $n$ such that

 $\lim_{n\to\infty}\|aa_{n}\|=0.$

Analogously, $a$ is a if

 $\lim_{n\to\infty}\|b_{n}a\|=0,$

for some sequence $b_{n}$ with  $\|b_{n}\|=1$.  The element $a$ is a topological divisor of zero if it is both a left and a topological divisor of zero.

Remarks.

• Any zero divisor is a topological divisor of zero.

• 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.

• 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 TopologicalDivisorOfZero 2013-03-22 16:12:15 2013-03-22 16:12:15 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 46H05 generalized divisor of zero