Banach algebra - Citizendia

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For functional analysis as used in psychology see the Functional analysis (psychology article Stefan Banach ( Ukrainian: Степан Степанович Банах 1892–1945 was a Polish Mathematician who worked in interwar Poland and in In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis The algebra multiplication and the Banach space norm are required to be related by the following inequality:

$\forall x, y \in A , \|x \, y\| \ \leq \|x \| \, \| y\|$

(i. e. , the norm of the product is less than or equal to the product of the norms. ) This ensures that the multiplication operation is continuous. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

If in the above we relax Banach space to normed space the analogous structure is called a normed algebra. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to

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. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, commutativity is the ability to change the order of something without changing the end result 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 $A e$ . In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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 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. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 This is part of p-adic analysis. In Mathematics, p -adic analysis is a branch of Number theory that deals with the Mathematical analysis of functions of P-adic numbers

Examples

The set of real (or complex) numbers is a Banach algebra with norm given by the absolute value. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, a matrix norm is a natural extension of the notion of a Vector norm to matrices.
Take the Banach space Rⁿ (or Cⁿ) with norm ||x|| = max |x_i| and define multiplication componentwise: (x₁,. . . ,x_n)(y₁,. . . ,y_n) = (x₁y₁,. . . ,x_ny_n).
The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician
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. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks
Any C*-algebra is a Banach algebra. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics.
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. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of The set of all compact operators on E is a closed ideal in this algebra.
The continuous linear operators on a Hilbert space form a C*-algebra and therefore a Banach algebra. This article assumes some familiarity with Analytic geometry and the concept of a limit. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics.
If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L¹(G) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy(g) = ∫ x(h) y(h^-1g) dμ(h) for x, y in L¹(G). In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and

Properties

Let $A$ be a Banach algebra with unit $e$ . Then $xy - yx \ne e$ for any $x, y \in A$ .

Several elementary functions which 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 Mathematics, several functions or groups of functions are important enough to deserve their own names In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Complex analysis, an entire function, also called an integral function is a complex-valued function that is holomorphic everywhere on the The formula for the geometric series remain valid in general unital Banach algebras. In Mathematics, a geometric series is a series with a constant ratio between successive terms. The binomial theorem also holds for two commuting elements of a Banach algebra. In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says

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 homeomorphism) so that it forms a topological group under multiplication. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the

Unital Banach algebras provide a natural setting to study general spectral theory. The spectrum of an element x consists of all those scalars λ such that x -λ1 is not invertible. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication (In the Banach algebra of all n-by-n matrices mentioned above, the spectrum of a matrix coincides with the set of all its eigenvalues. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes ) The spectrum of any element is compact. If the base field is the field of complex numbers, then the spectrum of any element is non-empty. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

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 which is a division algebra is isomorphic to the reals, the complexes, or the quaternions. In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible Hence, the only complex Banach algebra which is a division algebra is the complexes. (This is known as Gelfand-Mazur theorem. In Operator theory, the Gelfand-Mazur theorem is a Theorem named after Israel Gelfand and Stanisław Mazur which states A complex )
Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 In Ring theory, a branch of Abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single In Topology and related branches of Mathematics, a closed set is a set whose complement is open.
Every commutative real unital noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers. In Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals.
Every commutative real unital noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
Permanently singular elements in Banach algebras are topological divisors of zero, i. In Mathematics, in a Topological algebra A, z\in A is a topological divisor of zero if there exists a neighbourhood U of zero and e. considering extensions B of Banach algebras A some elements that are singular in the given algebra A have a multiplicative inverse element in a Banach algebra extension B. Topological divisors of zero in A are permanently singular in all Banach extension B of A.

Examples

Properties

See also