Get C*-Algebras Volume 3: General Theory of C*-Algebras PDF

By Corneliu Constantinescu

ISBN-10: 0444507515

ISBN-13: 9780444507518

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Additional info for C*-Algebras Volume 3: General Theory of C*-Algebras

Example text

Let y' C a(F) such that F ~(y') - ~. By a), E . 16 C*-algebra E . - ~. 1 The General Theory a) 39 There is a unique map ~" a ( E ) ~ a(F), such that E F X=XO~O for every x c F . b) x' C a(E) :=> ~(x') = x ' l F . c) ~ is continuous and surjective. d) ~ is bijective iff E = F . e) Every character of F can be extended to a character of E . 4), then ~ - ~. 15 a) ,b) ,c) ,d) ,e) . e) follows from b) and c). f) By E F ~(~') = ~(~(~')) F for every x' e or(E). Since we have identified or(x) with a(F) via ~ we see E ~.

7 c), x'(1) = y ' ( 1 ) = IlY'II = Ilx']l, So assume lo x' C E+. 1. Then (1 - 10) 2 = 1 - 10- l o + 10 = 1 - I11- Ioli = I. 10, 33 ~4. C*-Algebras Take 99 E]0, 2] and x E E such that I1~11 = s i n ~ . Then 99)21o x(1 1. 7 c). 7 c) again, ! x' E E + . 9 I If T is a locally compact space and # is a bounded Radon measure on T , then # is positive iff , ( T ) : Ilall. 7 c). If T is not compact, let T* be its Alexandroff compactification. 14). 10 x 6 E. I ( 0 ) Let E be a Gelfand unital C*-algebra and take Then the following are equivalent: a) x is not invertible.

21. 4). Given x E E , define 5"E >E , y~ ) xyx*. Then ~ = 5" E ~ for every x E U, the map U ~, x: ;5 is a continuous group homomorphism, and EC= {x E E ] ~ = identity map} = {x E U I ~ = identity map}. Take x E U. Then 5 is linear, bijective, 51 = xlx* = 1, ~y* = xy*x* = ( ~ j ) * = (~y)*, 22 ,~. C * - A l g e b r a s for every y E E , and ~(yz) = ~ z ~ * = ~y~*xz~* = (~y)(~z), for all y , z C E . Hence ~ is an involutive unital algebra homomorphism and = 5*. 21, 5 is an isometry of C*-algebras.

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C*-Algebras Volume 3: General Theory of C*-Algebras by Corneliu Constantinescu


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