Banach algebra


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Banach algebra

[′bä‚näk ′al·jə·brə]
(mathematics)
An algebra which is a Banach space satisfying the property that for every pair of vectors, the norm of the product of those vectors does not exceed the product of their norms.

Banach algebra

(mathematics)
An algebra in which the vector space is a Banach space.
References in periodicals archive ?
Clearly if A is a commutative semisimple unital complex Banach algebra with unit element e, then the maximum set M([i.sub.A](e)) has this property.
The sequence [([T.sub.n]).sub.n[greater than or equal to]1] [member of] B (H) (Banach algebra of bounded linear operators on the Hilbert space H) is called weakly statistically convergent to T [member of] B (H) if ([T.sub.n]x, y) statistically converges to (Tx, y) for any x, y [member of] H.
If E is a Banach algebra, the norm [parallel]*[[parallel].sub.L is multiplicative.
Given a semi-topological semigroup S, we shall denote by [C.sub.b](S) the Banach algebra of all bounded continuous real-valued functions on S, with the sup norm.
If X is a complex Banach algebra, condition (A) is defined by the following.
An important result by Johnson [10] is that a locally compact group G is amenable if and only if the group algebra [L.sup.1](G) is amenable as a Banach algebra; that is, the first cohomology groups [H.sup,1]([L.sup.1](G),X*) vanishes for all Banach [L.sup.1](G)-bimodules X.
In particular, the PWZ integral is used in the definition of Cameron-Storvick's Banach algebra S of functions on [C.sub.0][0, T] which is the space of generalized Fourier-Stieltjes transforms of the C-valued and finite Borel measures on [L.sup.2][0, T] [1].
Then we conclude this result for the Banach algebra [D.sup.n] (X).
A Banach space A is called a (real) Banach algebra (with unit) if there exists a multiplication A x A [right arrow] A that has the followings properties:
In particular, the Toeplitz operators for these symbols generate a commutative Banach algebra.
Let B[X] be the Banach algebra of all bounded linear operators on a Banach space X, and consider the class [[GAMMA].sub.R][X] of all operators with complemented range and the class [[GAMMA].sub.N][X] of all operators with complemented kernel.
Throughout this paper, A will denote a unital [C.sup.*]-algebra with a unit 7; namely, A is a unital Banach algebra with an involution * such that [parallel][A.sup.*] A[parallel] = [[parallel]A[parallel].sup.2] (A [member of] A).