Editing Banach algebra (section)

==Properties==

Several [[list of functions|elementary functions]] that are defined via [[power series]] may be defined in any unital Banach algebra; examples include the [[exponential function]] and the [[trigonometric functions]], and more generally any [[entire function]]. (In particular, the exponential map can be used to define [[abstract index group]]s.) The formula for the [[geometric series]] remains valid in general unital Banach algebras. The [[binomial theorem]] also holds for two commuting elements of a Banach algebra.

The set of [[invertible element]]s in any unital Banach algebra is an [[open set]], and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a [[topological group]] under multiplication.<ref>{{harvnb|Conway|1990|loc=Theorem VII.2.2.}}</ref>

If a Banach algebra has unit <math>\mathbf{1},</math> then <math>\mathbf{1}</math> cannot be a [[commutator (ring theory)|commutator]]; that is, <math>xy - yx \neq \mathbf{1}</math>&thinsp; for any <math>x, y \in A.</math> This is because <math>x y</math> and <math>y x</math> have the same [[spectrum (functional analysis)|spectrum]] except possibly <math>0.</math>

The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:

* Every real Banach algebra that is a [[division algebra]] is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra that is a division algebra is the complexes. (This is known as the [[Gelfand–Mazur theorem]].)
* Every unital real Banach algebra with no [[zero divisor]]s, and in which every [[principal ideal]] is [[closed set|closed]], is isomorphic to the reals, the complexes, or the quaternions.<ref>{{Cite journal|last1=García|first1=Miguel Cabrera|last2=Palacios|first2=Angel Rodríguez|date=1995|title=A New Simple Proof of the Gelfand-Mazur-Kaplansky Theorem|url=https://www.jstor.org/stable/2160559|journal=Proceedings of the American Mathematical Society|volume=123|issue=9|pages=2663–2666|doi=10.2307/2160559|jstor=2160559|issn=0002-9939|url-access=subscription}}</ref>
* Every commutative real unital [[Noetherian ring|Noetherian]] Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
* Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
* Permanently singular elements in Banach algebras are [[topological divisor of zero|topological divisors of zero]], that is, considering extensions <math>B</math> of Banach algebras <math>A</math> some elements that are singular in the given algebra <math>A</math> have a multiplicative inverse element in a Banach algebra extension <math>B.</math>  Topological divisors of zero in <math>A</math> are permanently singular in any Banach extension <math>B</math> of <math>A.</math>