# Banach algebra

(redirected from Banach *-algebra)

## 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.
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Then, equipped with the adjoint (7.2.4), this Banach algebra [??] forms a Banach *-algebra.
Let [??] be a Banach *-algebra (7.2.3) in B([??]), for p [member of] P.
Construct now the tensor product Banach *-algebra [mathematical expression not reproducible] by
We concentrate on such generating operators [mathematical expression not reproducible] of the Banach *-algebra [mathematical expression not reproducible] of (7.2.5).
Fix p [member of] P, and let [mathematical expression not reproducible] be the tensor product Banach *-algebra (7.2.5), and let [E.sub.p] : [mathematical expression not reproducible] be the linear transformation (7.2.6).
If we understand [mathematical expression not reproducible] as a Banach *-algebra, we call it the radial-Adelic (Banach *-)algebra.
Define now a tensor product Banach *-algebra [mathematical expression not reproducible] by
Since [[??].sub.p] is a von Neumann algebra and [mathematical expression not reproducible] is a Banach *-algebra, the tensor product [mathematical expression not reproducible] of (9.6) is a well-defined Banach *-algebra under product topology.
Act [mathematical expression not reproducible] of (10.2) on the Banach *-algebra [mathematical expression not reproducible] of (9.6), as the [mathematical expression not reproducible]-bimodule action, denoted by #, from left-and-right, i.e.,
Note that every element [mathematical expression not reproducible] of the system [[??].sub.p,J] of (11.5) are well-defined elements of the tensor product Banach *-algebra [mathematical expression not reproducible].
Let ([A.sub.k], [[phi].sub.k]) be arbitrary topological *-probability spaces (e.g., [C.sup.*]-probability spaces, or [W.sup.*]-probability spaces, or Banach *-probability spaces, etc.), consisting of topological *-algebras [A.sub.k] (e.g., [C.sup.*]-algebras, or [W.sup.*]-algebras, or Banach *-algebras, etc.), and corresponding (bounded or unbounded) linear functionals [[phi].sub.k], for k [member of] [DELTA], where [DELTA] is an arbitrary countable (finite or infinite) index set.
At several places in the development of the theory of topological *-algebras (especially, of the theory of Banach *-algebras and locally m-convex *-algebras) a self-adjoint square root for a self-adjoint element with a positive spectrum is needed.

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