Spectral Mapping Theorem

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra $A$ over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy
$\, x \, y\, \ \leq \, x\, \, \, y\, \quad \text x, y \in A.$

This ensures that the multiplication operation is continuous.

A Banach algebra is called ''unital'' if it has an identity element for the multiplication whose norm is $1,$ and ''commutative'' if its multiplication is commutative.
Any Banach algebra $A$ (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra $A_e$ so as to form a closed ideal of $A_e$. Often one assumes ''a priori'' that the algebra under consideration is unital: for one can develop much of the theory by considering $A_e$ and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.

Banach algebras can also be defined over fields of $p$-adic numbers. This is part of $p$-adic analysis.

Examples

The prototypical example of a Banach algebra is $C_0(X)$, the space of (complex-valued) continuous functions on a locally compact ( Hausdorff) space that vanish at infinity. $C_0(X)$ is unital if and only if $X$ is compact. The complex conjugation being an involution, $C_0(X)$ 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 $n$-by-$n$ matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.
* Take the Banach space $\R^n$ (or $\Complex^n$) with norm $\, x\, = \max_ , x_i,$ and define multiplication componentwise: $\left(x_1, \ldots, x_n\right) \left(y_1, \ldots, y_n\right) = \left(x_1 y_1, \ldots, x_n y_n\right).$
* The quaternions 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 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 linear operators on a Banach space $E$ (with functional composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on $E$ is a Banach algebra and closed ideal. It is without identity if $\dim E = \infty.$
* If $G$ is a locally compact Hausdorff topological group and $\mu$ is its Haar measure, then the Banach space $L^1(G)$ of all $\mu$-integrable functions on $G$ becomes a Banach algebra under the convolution $x y(g) = \int x(h) y\left(h^ g\right) d \mu(h)$ for $x, y \in L^1(G).$
* Uniform algebra: A Banach algebra that is a subalgebra of the complex algebra $C(X)$ with the supremum norm and that contains the