Topological divisor of zero

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In mathematics, an element z of a Banach algebra A is called a topological divisor of zero if there exists a sequence x1x2x3, ... of elements of A such that

  1. The sequence zxn converges to the zero element, but
  2. The sequence xn does not converge to the zero element.

If such a sequence exists, then one may assume that ||xn|| = 1 for all n.

If A is not commutative, then z is called a left topological divisor of zero, and one may define right topological divisors of zero similarly.

Examples[edit]

  • If A has a unit element, then the invertible elements of A form an open subset of A, while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero.
  • In particular, any quasinilpotent element is a topological divisor 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.

Generalization[edit]

The notion of a topological divisor of zero may be generalized to any topological algebra. If the algebra in question is not first-countable, one must substitute nets for the sequences used in the definition.