Editing Banach space (section)

===Banach algebras===

A ''[[Banach algebra]]'' is a Banach space <math>A</math> over <math>\mathbb{K} = \R</math> or <math>\Complex,</math> together with a structure of [[Algebra over a field|algebra over <math>\mathbb{K}</math>]], such that the product map <math>A \times A \ni (a, b) \mapsto ab \in A</math> is continuous. An equivalent norm on <math>A</math> can be found so that <math>\|ab\| \leq \|a\| \|b\|</math> for all <math>a, b \in A.</math>

====Examples====

* The Banach space <math>C(K)</math> with the pointwise product, is a Banach algebra.
* The [[disk algebra]] <math>A(\mathbf{D})</math> consists of functions [[Holomorphic function|holomorphic]] in the open unit disk <math>D \subseteq \Complex</math> and continuous on its [[Closure (topology)|closure]]: <math>\overline{\mathbf{D}}.</math> Equipped with the max norm on <math>\overline{\mathbf{D}},</math> the disk algebra <math>A(\mathbf{D})</math> is a closed subalgebra of <math>C\left(\overline{\mathbf{D}}\right).</math>
* The [[Wiener algebra]] <math>A(\mathbf{T})</math> is the algebra of functions on the unit circle <math>\mathbf{T}</math> with absolutely convergent Fourier series. Via the map associating a function on <math>\mathbf{T}</math> to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra <math>\ell^1(Z),</math> where the product is the [[Convolution#Discrete convolution|convolution]] of sequences.
* For every Banach space <math>X,</math> the space <math>B(X)</math> of bounded linear operators on <math>X,</math> with the composition of maps as product, is a Banach algebra.
* A [[C*-algebra]] is a complex Banach algebra <math>A</math> with an [[Antilinear map|antilinear]] [[Involution (mathematics)|involution]] <math>a \mapsto a^*</math> such that <math>\|a^* a\| = \|a\|^2.</math> The space <math>B(H)</math> of bounded linear operators on a Hilbert space <math>H</math> is a fundamental example of C*-algebra. The [[Gelfand–Naimark theorem]] states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some <math>B(H).</math> The space <math>C(K)</math> of complex continuous functions on a compact Hausdorff space <math>K</math> is an example of commutative C*-algebra, where the involution associates to every function <math>f</math> its [[complex conjugate]] <math>\overline{f}.</math>