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:
(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.
The prototypical example of a Banach algebra is C0(X), the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. C0(X) is unital if and only if X is compact. The complex conjugation being an involution, C0(X) is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra.
- 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 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.
- 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).
- Uniform algebra: A Banach algebra that is a subalgebra of C(X) with the supremum norm and that contains the constants and separates the points of X (which must be a compact Hausdorff space).
- Natural Banach function algebra: A uniform algebra whose all characters are evaluations at points of X.
- C*-algebra: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space.
- Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution.
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