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.
References in periodicals archive ?
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.