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Encyclopedia > Banach algebra

In 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. The algebra multiplication and the Banach space norm are required to be related by the following inequality: Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Stefan Banach Stefan Banach (March 30, 1892 in KrakÃ³w, Poland â€“ August 31, 1945 in Lviv, Ukraine), was a Polish mathematician, one of the moving spirits of the LwÃ³w School of Mathematics in pre-war Poland. ... In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

(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; that is, a mapping which preserves the topological structure. ...

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 set. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

Banach algebras can also be defined over fields of p-adic numbers. This is part of p-adic analysis. This article may be too technical for most readers to understand. ... P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers. ...

• 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 Rn (or Cn) with norm ||x|| = max |xi| and define multiplication componentwise: (x1,...,xn)(y1,...,yn) = (x1y1,...,xnyn).
• 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 is unital if and only if the original space is compact. Also, since every continuous function on a compact space is automatically bounded, we do not need to assume the boundedness of the functions in this case.
• Any C*-algebra 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 closed ideal in this algebra. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... For the square matrix section, see square matrix. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... C*-algebras are an important area of research in functional analysis. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...

• The continuous linear operators on a Hilbert space form a C-star-algebra and therefore a Banach algebra.
• 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 mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... C*-algebras are an important area of research in functional analysis. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... This article is about the mathematical concept of convolution. ...

## Properties

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. The formula for the geometric series and the binomial theorem also remain valid in general unitary Banach algebras. In mathematics, several functions are important enough to deserve their own name. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, so that it forms a topological group under multiplication. In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...

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 the Banach algebra of all n-by-n matrices mentioned above, the spectrum of a matrix coincides with the set of all its eigenvalues.) 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. Scalar is a concept that has meaning in mathematics, physics, and computing. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ...

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.
• 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.
• Every commutative real unital noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
• Every commutative real unital noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
• Permanently singular elements in Banach algebras are topological divisiors of zero, i.e. considering extensions B of Banach algebras A some elements that are singular in the given algebra A have an 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. Results from FactBites:

 Station Information - Banach algebra (550 words) The algebra of bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the supremum norm) is a Banach algebra. The algebra of continuous real- or complex-valued functions on some compact space (again with pointwise operations and supremum norm) is a Banach algebra. The set of invertible elements in any unitary Banach algebra is an open set, and the inversion operation on this set is continuous, so that it forms a topological group under multiplication.
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