Banach algebra
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 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 formula_2 (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra formula_3 so as to form a closed ideal of formula_3. Often one assumes "a priori" that the algebra under consideration is unital: for one can develop much of the theory by considering formula_3 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 formula_6, the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. formula_6 is unital if and only if "X" is compact. The complex conjugation being an involution, formula_6 is in fact a C*-algebra. More  generally, every C*-algebra is a Banach algebra by definition. Properties. Several 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 groups.) 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 elements 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. If a Banach algebra has unit 1, then 1 cannot be a commutator; i.e., formula_10  for any "x", "y" ∈ "A". This is because "xy" and "yx" have the same spectrum except possibly 0. 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: Spectral theory. Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The "spectrum" of an element  x ∈ "A", denoted by formula_11, consists of all those complex scalars "λ" such that "x" − "λ"1 is not invertible in "A". The spectrum of any element "x" is a closed subset of the closed disc in C with radius ||"x"|| and center 0, and thus is compact. Moreover, the spectrum formula_11 of an element "x" is non-empty and satisfies the spectral radius formula: Given "x" ∈ "A", the holomorphic functional calculus allows to define "ƒ"("x") ∈ "A" for any function "ƒ" holomorphic in a neighborhood of formula_14 Furthermore, the spectral mapping theorem holds: When the Banach algebra  A is the algebra L("X") of bounded linear operators on a complex Banach space "X"  (e.g., the algebra of square matrices), the notion of the spectrum in "A" coincides with the usual one in operator theory. For "ƒ" ∈ "C"("X") (with a compact Hausdorff space "X"), one sees that: The norm of a normal element "x" of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators. Let "A"  be a complex unital Banach algebra in which every non-zero element "x" is invertible (a division algebra). For every "a" ∈ "A", there is "λ  ∈ C such that a" − "λ1 is not invertible (because the spectrum of "a" is not empty) hence "a" = "λ1 : this algebra "A" is naturally isomorphic to C (the complex case of the Gelfand–Mazur theorem). Ideals and characters. Let "A"  be a unital "commutative" Banach algebra over C. Since "A" is then a commutative ring with unit, every non-invertible element of "A" belongs to some maximal ideal of "A". Since a maximal ideal formula_17 in "A" is closed, formula_18 is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that  there is a bijection between the set of all maximal ideals of "A" and the set Δ("A") of all nonzero homomorphisms from "A"  to C. The set Δ("A") is called the "structure space" or "character space" of "A", and its members "characters." A character "χ" is a linear functional on "A" that is at the same time multiplicative, "χ"("ab") = "χ"("a") "χ"("b"), and satisfies "χ"(1) = 1. Every character is automatically continuous from "A"  to C, since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm ("i.e.", operator norm) of a character is one . Equipped with the topology of pointwise convergence on "A" ("i.e.", the topology induced by the weak-* topology of "A"), the character space, Δ("A"), is a Hausdorff compact space. For any "x" ∈ "A", where formula_20 is the Gelfand representation of "x" defined as follows: formula_20 is the continuous function from Δ("A") to C given by formula_22  The spectrum of formula_23 in the formula above, is the spectrum as element of the algebra "C"(Δ("A")) of complex continuous functions on the compact space Δ("A"). Explicitly, As an algebra, a unital commutative Banach algebra is semisimple (i.e., its Jacobson radical is zero) if  and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when "A" is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between "A" and "C"(Δ("A")) . Banach *-algebras. A Banach *-algebra "A" is a Banach algebra over the field of complex numbers, together with a map * : "A" → "A" that has the following properties: In other words, a Banach *-algebra is a Banach algebra over formula_27 that is also a *-algebra. In most natural examples, one also has that the involution is isometric, that is , Some authors include this isometric property in the definition of a Banach *-algebra. A Banach *-algebra satisfying ||"x"* "x"|| = ||"x"*|| ||"x"|| is a C*-algebra.