Editing Banach algebra (section)

==Examples==

The prototypical example of a Banach algebra is <math>C_0(X)</math>, the space of (complex-valued) continuous functions, defined on a [[locally compact Hausdorff space]] <math>X</math>, that [[vanish at infinity]]. <math>C_0(X)</math> is unital if and only if <math>X</math> is [[compactness|compact]]. The [[complex conjugation]] being an [[involution (mathematics)|involution]], <math>C_0(X)</math> is in fact a [[C*-algebra]]. More generally, every C*-algebra is a Banach algebra by definition.

* The set of real (or complex) numbers is a Banach algebra with norm given by the [[absolute value]].
* The set of all real or complex <math>n</math>-by-<math>n</math> [[matrix (mathematics)|matrices]] becomes a [[unital algebra|unital]] Banach algebra if we equip it with a sub-multiplicative [[matrix norm]].
* Take the Banach space <math>\R^n</math> (or <math>\Complex^n</math>) with norm <math>\|x\| = \max_{} |x_i|</math> and define multiplication componentwise: <math>\left(x_1, \ldots, x_n\right) \left(y_1, \ldots, y_n\right) = \left(x_1 y_1, \ldots, x_n y_n\right).</math>
* The [[quaternion]]s form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
* The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the [[supremum]] norm) is a unital Banach algebra.
* The algebra of all bounded [[continuous function (topology)|continuous]] real- or complex-valued functions on some [[locally compact space]] (again with pointwise operations and supremum norm) is a Banach algebra.
* The algebra of all [[continuous function (topology)|continuous]] [[linear transformation|linear]] operators on a Banach space <math>E</math> (with functional composition as multiplication and the [[operator norm]] as norm) is a unital Banach algebra. The set of all [[compact operator]]s on <math>E</math> is a Banach algebra and closed ideal. It is without identity if <math>\dim E = \infty.</math><ref>{{harvnb|Conway|1990|loc=Example VII.1.8.}}</ref>
* If <math>G</math> is a [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] and <math>\mu</math> is its [[Haar measure]], then the Banach space <math>L^1(G)</math> of all <math>\mu</math>-integrable functions on <math>G</math> becomes a Banach algebra under the [[convolution]] <math>x y(g) = \int x(h) y\left(h^{-1} g\right) d \mu(h)</math> for <math>x, y \in L^1(G).</math><ref name="harvnb conway 1990 example VII.1.9.">{{harvnb|Conway|1990|loc=Example VII.1.9.}}</ref>
* [[Uniform algebra]]: A Banach algebra that is a subalgebra of the complex algebra <math>C(X)</math> with the supremum norm and that contains the constants and separates the points of <math>X</math> (which must be a compact Hausdorff space).
* [[Uniform algebra|Natural Banach function algebra]]: A uniform algebra all of whose characters are evaluations at points of <math>X.</math>
* [[C*-algebra]]: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some [[Hilbert space]].
* [[Measure algebra]]: A Banach algebra consisting of all [[Radon measure]]s on some [[locally compact group]], where the product of two measures is given by [[Convolution#Measures|convolution of measures]].<ref name="harvnb conway 1990 example VII.1.9." />
* The algebra of the [[quaternion]]s <math>\H</math> is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex numbers.
* An [[affinoid algebra]] is a certain kind of Banach algebra over a nonarchimedean field. Affinoid algebras are the basic building blocks in [[Rigid analytic space|rigid analytic geometry]].