Banach algebra

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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. 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.

If in the above we relax Banach space to normed space the analogous structure is called a normed algebra.

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 Ae so as to form a closed ideal of Ae. Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering Ae 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. This is part of p-adic analysis.

[edit] Examples

  • 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.
  • 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.
  • The continuous linear operators on a Hilbert space form a C*-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).

[edit] 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. The formula for the geometric series remain 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 homeomorphism) so that it forms a topological group under multiplication.

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.

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. Hence, the only complex Banach algebra which is a division algebra is the complexes. (This is known as Gelfand-Mazur theorem.)
  • 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 divisors of zero, i.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.

[edit] See also