Book

# Complete Normed Algebras

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## Chapters (7)

Throughout this book the symbol F will be used to denote a field that is either the real field ℝ or the complex field ℂ.
In this section A denotes an algebra over F and we consider the purely algebraic theory of irreducible left A-modules.
In this section A will denote an algebra. Definitions and results will usually be stated only for left ideals; with the obvious changes, they apply to right ideals.
Recall from § 12 that a Banach star algebra is a complex Banach algebra A with an involution * satisfying, for all x, y∈A,α∈ℂ, (i) (x+y)*=x* + y* (ii) (αx)*=α* x* (iii) x**=x (iv) (xy)* =y* x*.
Let X, Y,Z be normed linear spaces over the same field F. A mapping ø: X × Y→Z is said to be bilinear if (i) for each y∈ Y, the mapping x→ø(x,y) is linear (ii) for each x∈ X, the mapping y→ø(x,y) is linear.
A will denote a complex Banach algebra with unit. As usual, a complex polynomial in one variable is said to be monic if the coefficient of the term of highest degree is 1. We denote by Pn the set of all complex monic polynomials of degree n.