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7330hw6

7330hw6 - Math 7330 Functional Analysis Notes/Homework 6...

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Math 7330: Functional Analysis Fall 2002 Notes/Homework 6: Banach *-Algebras A. Sengupta An involution * on a complex algebra B is a map * : B B for which (i) * ( a + b ) = * a + * b for all a, b B (ii) * ( λa ) = λ * a for all λ C and a B (iii) ( xy ) = y * x * for all x, y B (iv) ( x * ) * = x for all x B . An element a B is hermitian if a = a * . On a complex Banach algebra we also require an involution * to satisfy (v) | xy | ≤ | x || y | for all x, y B . Observe that for the identity e , we have e * = ee * and so taking * of this we get ( e * ) * = ( e * ) * e * , which says e = ee * . Thus e = e * A B*-algebra is a complex Banach algebra B on which there is an involution * for which | xx * | = | x | 2 for all x B 1. Let B be a complex Banach algebra with involution. (i) Show that if B is a B*-algebra then | x | = | x * | for all x B (ii) Suppose | y * y | = | y | 2 for all y B . Show that | y | = | y * | for all y B . 1

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(iii) Suppose | y * y | = | y | 2 for all y B . Show that | xx * | = | x | 2 for all x B . 2. Let B be a B*-algebra. (i) Show that if y B is hermitian and s is any real number then | se + iy | 2 = | s 2 e + y 2 | (ii) Show that e + iy
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