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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

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

This note was uploaded on 01/27/2010 for the course MATH 7330 taught by Professor Davidson,m during the Spring '08 term at LSU.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online