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Concept
Banach algebra
Summary
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra 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 and commutative if its multiplication is commutative.
Any Banach algebra (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of . Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering 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 -adic numbers. This is part of -adic analysis.
The prototypical example of a Banach algebra is , the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. is unital if and only if is compact. The complex conjugation being an involution, is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra by definition.
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 -by- matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.
Official source
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