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  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] .
Then we conclude this result for the Banach algebra
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).