C* -Algebras Volume 1: Banach Spaces - download pdf or read online

By C. Constantinescu

ISBN-10: 0080528376

ISBN-13: 9780080528373

ISBN-10: 0444507493

ISBN-13: 9780444507495


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Additional info for C* -Algebras Volume 1: Banach Spaces

Sample text

For example, the Euclidean norm will be the appropriate one in the case of Hilbert spaces while the supremum norm will be needed in the case of C*-algebras. 3 ( 0 ) Let E be a normed space and take (~,~ c IK. Then the map ExE >E, (x,y), >c~x+~y is uniformly continuous. We have II(~x~ + ,~yl) -(o~x2 + ~y~)ll = IIc~(Xl z2) + ,/~(yl - - - y~)ll <_ I~l IIx~ - ~11 + 19111y, - y~ I _ < (1~1 ~- + iZl~)~ (ll~ - x~ll ~ +ly, = (Ic~l ~ + I~l~) ~ II(x~, y~)- - y~tl~) ~ - (x~, y~)l for all Xl,X2, yl, y2 E E , which proves the assertion.

5, F is complete with respect to the induced norm and so it is a closed subset of E. 1 35 Normed Spaces ( 0 ) Let A be a nonempty subset of the normed space E . Given x E E , we set dA(x) := inf ] i x - YII yEA dA(X) is called the distance o f x f r o m A . 2 ( 0 ) Let F be a proper closed vector subspace of the I1~11- 1 normed space E . Then there is an x E E with dR(x) > -2 Take y E E \ F . and 1 Since F is closed, dE(y) > 0. There is a z E F with lly- zll < 2dR(y). We set 1 x:=~(y-z). IlY- zll Then llxll = 1.

By Step 1, ~/OQxe ~E EJ <_ a + a - - 2a. By Step 2, II~~FI-~II~,r~,~ 9 ~,-~~-4o,~1 imo~,x~ll m 12 1. 1 ( 0 ) Given sets E and T , we denote by E T the set of all maps of T into E . If E is a vector space (additive group), then E T becomes a vector space (additive group) when the operations are defined pointwise. If E is an additive group, we define E (r) := {x E ETI{x # 0} is finite }. Given a subset A of T , define eA'=eT'T ~IK t, ~ ~ 1 iftEA ( 0 if t E T \ A ' eTA is called the characteristic Junction of A on T .

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C* -Algebras Volume 1: Banach Spaces by C. Constantinescu

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