C*-algebras Volume 2: Banach Algebras and Compact Operators by Corneliu Constantinescu

By Corneliu Constantinescu

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Example text

Take x E E . Then (x, O) -- (x, O)(a, b) - (xa, xb) , (x, O) = (a, b)(x, O) = (ax, bx) , so that x=xa=ax. Hence a is a unit of E . The final assertions are trivial. 6), E and L(E) the complexification of E and L(E), respectively, and L(E) the complex unital algebra of linear maps k ~ E. f unital algebras. Thus the unital algebras o L o L(E) and (E) may be identified. 7.

I Algebras ~ a2 x s + 4 - as 4 2a _ as (xy + yx) + 2a s 4 4 - a 21 - -4- --- - a5 1 Oe2 - ~ 1 + 4 - - O~2 ~s b~ - 4 - as -4- --' - ~a ab = 4 4a 4 4 ~ a 2 x y + 4 - a 2 ( a x y - 1) = 4a 4 4 ~ a ~xy + 4 L a~ xy - + 4 a2 - 4 - ' --- - -a' ~ x a2 - a2- 4 = --'----~ 4 1 = -1, 4a 1 - a2 - 4 " - '-- - 'a~ x y~ - a~ 4a c~s 4 - a~ 4 4 ~ a 2 2a sx2y a a2- 4 -4- '-- - -a~ 1 = 4 - a ~ 2a 4 4 - a 2y + ~ y x y 2a 2a 4 - a 2y 4a a ~(ay + x) = 4 4 - a2 a u y + 4 ---Z---~x = 4 - a ~ x = x , ba = x a 2 -- - x a x -- a2b - bx - x a x - = 4 4 - a 2y + 4- 4a - 4 " - - ' - ~a x - -4- --' - 'a~ y + 4 --1, -xb , -b , -- -x2a ~ a.

Assume that x r R. Then there is a maximal proper left ideal F of E with x r F . Since {yx + a I Y E E, a E F} is a left ideal of E containing F W {x}, it contains 1. 1 Algebras ~9 a= 1-yx. By b), there is a z E E with 1 -" z ( 1 - yx). But then l=zaEF, I which is a contradiction. 22 The radical of a unital algebra is an ideal. Let E be a unital algebra. R its radical, and take (x, y) E E x R . Assume y x ~ R . Then there is a maximal proper left ideal F of E such that y x ~ F . Then { z y x + a [ ( z , a ) E E x F } i s a l e f t ideal of E containing F O { y x } .

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