Banach algebra

Definition 1.

A Banach algebraMathworldPlanetmath 𝒜 is a Banach spaceMathworldPlanetmath (over ) with an multiplication law compatible with the norm which turns 𝒜 into an algebra. Compatibility with the norm means that, for all a,b𝒜, it is the case that the following product inequality holds:

Definition 2.

A Banach *-algebra is a Banach algebra 𝒜 with a map :*𝒜𝒜 which satisfies the following properties:

a** = a, (1)
(ab)* = b*a*, (2)
(a+b)* = a*+b*, (3)
(λa)* = λ¯a*λ, (4)
a* = a, (5)

where λ¯ is the complex conjugation of λ. In other words, the operator * is an involution.

Example 1

The algebra of bounded operatorsMathworldPlanetmathPlanetmath on a Banach space is a Banach algebra for the operator norm.

Title Banach algebra
Canonical name BanachAlgebra
Date of creation 2013-03-22 12:57:52
Last modified on 2013-03-22 12:57:52
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Definition
Classification msc 46H05
Synonym B-algebra
Synonym Banach *-algebra
Synonym B*-algebra
Synonym B*-algebra
Related topic ExampleOfLinearInvolution
Related topic GelfandTornheimTheorem
Related topic MultiplicativeLinearFunctional
Related topic TopologicalAlgebra