7330hw3 - Math 7330 Functional Analysis Notes/Homework 3...

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Math 7330: Functional Analysis Fall 2002 Notes/Homework 3: Banach Algebras A. Sengupta A complex algebra is a complex vector space B on which there is a bilinear multipli- cation map B × B B : ( x,y ) 7→ xy which is associative. Bilinearity of multiplication means the distributive law x ( y + z ) = xy + xz, ( y + z ) x = yz + zx for all x,y,z B , and ( λa ) b = λ ( ab ) = a ( λb ) for all a,b B and λ C . In particular, a complex algebra is automatically a ring. An element e B is a multiplicative identity (or unit element ) if xe = ex = x for all x B . If e 0 is also a multiplicative identity then e = ee 0 = e 0 Thus the multiplicative identity, if it exists, is unique. Suppose B is a complex algebra with unit e . An element x B is invertible if there exists an element y B , called an inverse of x , such that yx = xy = e If y 0 is another element for which both xy 0 and y 0 x equal e then y = ey = ( y 0 x ) y = y 0 ( xy ) = y 0 e = y 0 Thus if x is invertible then it has a unique inverse, which is denoted x - 1 . The set of all invertible elements in B will be denoted G ( B ). It is clearly a group. Assume, moreover, that there is a norm on the complex algebra
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7330hw3 - Math 7330 Functional Analysis Notes/Homework 3...

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