Topological zero divisor

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

definition

Let be a Banach algebra over the field of complex numbers. An element other than 0 is called a left topological zero divisor if there is a sequence in A with: ${\displaystyle A}$ ${\displaystyle x\in A}$ ${\displaystyle (x_{n})_{n}}$

1. ${\displaystyle \|x_{n}\|=1}$for all , ${\displaystyle n}$
2. ${\displaystyle x\cdot x_{n}{\xrightarrow {n\to \infty }}0}$.

A right topological zero divisor is defined analogously, whereby the last point must of course be written. ${\displaystyle x_{n}\cdot x{\xrightarrow {n\to \infty }}0}$

A bilateral or bilateral topological zero divisor is a left and, at the same time, a right topological zero divider.

In commutative Banach algebras these three terms coincide and one speaks simply of topological zero divisors . Some authors also allow 0 as a topological zero divisor; here we have the same inconsistent situation as with the algebraic zero divisors.

Examples

• Left (right, bilateral) zero divisors are left (right, bilateral) topological zero dividers; one can choose a constant sequence in this case . ${\displaystyle (x_{n})_{n}}$

Sketch of the functions used

• In the function algebra of continuous functions on the unit interval [0,1] with the supremum is a topological zero divisor that is not a zero divisor is. is not a zero divisor, because is , then for must first apply, since on is not 0. The continuity of then provides for all the property and therefore must (the zero function on ) be and is not a zero divisor . ${\displaystyle C([0,1])}$ ${\displaystyle x=\mathrm {id} _{[0,1]}}$ ${\displaystyle x}$ ${\displaystyle xy=0}$ ${\displaystyle y(t)=0}$ ${\displaystyle 0<t\leq 1}$ ${\displaystyle x}$ ${\displaystyle ]0,1]}$ ${\displaystyle y}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle y(t)=0}$ ${\displaystyle y=0}$ ${\displaystyle C([0,1])}$ ${\displaystyle x}$

To see that is a topological zero divisor, consider the functions ${\displaystyle x}$

${\displaystyle x_{n}(t)={\begin{cases}1-nt&{\mbox{wenn }}0\leq t\leq {\frac {1}{n}}\\0,&{\mbox{sonst}}\end{cases}}}$

Then , and is proven to be a topological zero divisor. ${\displaystyle \|x_{n}\|=1}$ ${\displaystyle \|x\cdot x_{n}\|={\frac {1}{4n}}\rightarrow 0}$ ${\displaystyle x}$

• If a Banach algebra with unit 1, not a multiple of the unit and from the topological edge of the spectrum of , then is a topological zero divisor. With the Gelfand-Mazur theorem, this results in the following statement going back to W. Żelasko : Either is isomorphic to or has topological zero divisors. ${\displaystyle A}$ ${\displaystyle x\in A}$ ${\displaystyle \lambda }$ ${\displaystyle x}$ ${\displaystyle x-\lambda \cdot 1}$ ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$

Permanently singular elements

As is well known , an element of a Banach algebra is called singular if it is not invertible. An element is permanently singular , if there is no Banach are with (or is isometrically in embedded) so that it is invertible. The following theorem, proven by R. Arens , applies : ${\displaystyle A}$ ${\displaystyle {\tilde {A}}}$ ${\displaystyle A\subset {\tilde {A}}}$ ${\displaystyle A}$ ${\displaystyle {\tilde {A}}}$ ${\displaystyle {\tilde {A}}}$

• An element of a commutative -Banach algebra is permanently singular if and only if it is a topological zero divisor. ${\displaystyle \mathbb {C} }$

Zero divisor

Every topological zero divisor of a Banach algebra can be realized as a real (algebraic) zero divisor of a comprehensive Banach algebra. The following applies more precisely:

• For every Banach algebra there is a Banach algebra , so that the following applies: ${\displaystyle A}$ ${\displaystyle {\tilde {A}}}$

1. ${\displaystyle A}$is isometrically isomorphic to a sub-Banach algebra of . ${\displaystyle {\tilde {A}}}$
2. Every left (right, two-sided) topological zero divisor of is a left (right, two-sided) zero divisor in . ${\displaystyle A}$ ${\displaystyle {\tilde {A}}}$

For the construction of let the algebra of all bounded sequences in . For be . Then an ideal is in and the quotient is a Banach algebra with the quotient norm induced by . Constant sequences can be embedded isometrically isomorphic in . If there is now a left topological zero divisor, then by definition there is a sequence in with . Therefore , understood as an element in , is a left zero divisor. ${\displaystyle {\tilde {A}}}$ ${\displaystyle {\overline {A}}}$ ${\displaystyle A}$ ${\displaystyle (x_{n})_{n}\in {\overline {A}}}$ ${\displaystyle |(x_{n})_{n}|:=\limsup _{n\to \infty }\|x_{n}\|}$ ${\displaystyle N:=\{(x_{n})_{n}\in {\overline {A}};\,|(x_{n})_{n}|=0\}}$ ${\displaystyle {\overline {A}}}$ ${\displaystyle {\tilde {A}}={\overline {A}}/N}$ ${\displaystyle |\cdot |}$ ${\displaystyle A}$ ${\displaystyle {\tilde {A}}}$ ${\displaystyle x\in A}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle A}$ ${\displaystyle \limsup _{n\to \infty }\|x\cdot x_{n}\|=0}$ ${\displaystyle x}$ ${\displaystyle {\tilde {A}}}$

Individual evidence

1. ↑ Wiesław Żelazko: Banach Algebras , Elsevier (1973), ISBN 0-444-40991-2 , §14: Topological Divisors of Zero
2. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §1.12
3. ↑ Wiesław Żelazko: Banach Algebras , Elsevier (1973), ISBN 0-444-40991-2 , §14.4
4. ↑ Wiesław Żelazko: Banach Algebras , Elsevier (1973), ISBN 0-444-40991-2 , §14.7
5. ↑ Wiesław Żelazko: Banach Algebras , Elsevier (1973), ISBN 0-444-40991-2 , §14.8