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Banach Algebra

A Banach algebra is an algebra B over a field F endowed with a norm ||·|| such that B is a Banach space under the norm ||·|| and

||xy||<=||x||||y||.

F is frequently taken to be the complex numbers in order to ensure that the operator spectrum fully characterizes an operator (i.e., the spectral theorems for normal or compact normal operators do not, in general, hold in the operator spectrum over the real numbers).

If B is commutative and has a unit, then x in B is invertible iff x^^(phi)!=0 for all phi, where x|->x^^ is the Gelfand transform.

See also

C-*-Algebra, Banach Space, Gelfand Transform

Portions of this entry contributed by Mohammad Sal Moslehian

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References

Helemskii, A. Ya. Banach and Locally Convex Algebras. Oxford, England: Oxford University Press, 1993.Katznelson, Y. An Introduction to Harmonic Analysis. New York: Dover, 1976.Rudin, W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, 1987.

Referenced on Wolfram|Alpha

Banach Algebra

Cite this as:

Moslehian, Mohammad Sal and Weisstein, Eric W. "Banach Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BanachAlgebra.html

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