topological divisor of zero


Let A be a normed ringMathworldPlanetmath.  An element aA is said to be a left topological divisor of zero if there is a sequence an with  an=1  for all n such that

limnaan=0.

Analogously, a is a if

limnbna=0,

for some sequence bn with  bn=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
Canonical name TopologicalDivisorOfZero
Date of creation 2013-03-22 16:12:15
Last modified on 2013-03-22 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