Topological divisor of zero

A topological zero divisor is a term from the mathematical theory of Banach algebras. Taking advantage of the topology of the algebraic concept of a zero divisor is generalized.

Definition

Let be a Banach algebra over the field of complex numbers. A different from 0 element is called left topological zero divisor if there exists a sequence in A with:

A right topological divisor of zero is defined analogously, with the last point of course is to write.

One -sided or two-sided topological divisor of zero is a left and right at the same topological zero divisor.

In commutative Banach algebras these three concepts coincide and simply speaks of topological zero divisors. Some authors have also 0 as a topological divisor of zero to; So here the same inconsistent situation is as in the algebraic zero divisors.

Examples

  • Left (right, two-sided ) zero divisor are left (right, two-sided ) topological zero divisor; you can select a constant sequence in this case.
  • In the function algebra of continuous functions on the unit interval [0,1] with the supremum norm is a topological divisor of zero, which is not a zero divisor. is not a zero divisor, as is, so must first apply for, since it is not on 0. The continuity of supplies then for all the property and as a consequence (the zero function ) be and is not a zero divisor.
  • If a Banach algebra with unit element 1, not a multiple of the identity and of the topological boundary of the spectrum of so is a topological zero divisor. As a result the set of Gelfand - Mazur following going back to W. Żelasko statement: either is isomorphic to or has topological zero divisor.

Permanent singular elements

Is known to be an element of a Banach singular if it is not inverted. An element is permanently singular, if there is no Banach with (or is embedded isometrically in ), so that it is inverted. The following applies to proven R. Arens theorem:

  • An element of a commutative Banach algebra if and only permanently - singular if it is a topological zero divisor.

Zero divisor

One can realize any topological zero divisor of a Banach algebra as real ( algebraic ) zero divisor of a comprehensive Banach algebra. More precisely:

  • For every Banach algebra is a Banach algebra, so that the following holds:

To construct is the algebra of all bounded sequences in. For his. Then, an ideal is in and the quotient is the quotient norm induced by a Banach algebra. Means constant episodes you can embed isometrically isomorphic. Is now a left topological divisor of zero, then there is by definition a sequence in with. Therefore, considered as an element in a left zero divisor.

Function space Uniform norm Gelfand–Mazur theorem Isometry Quotient ring
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