{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
(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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}