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By Corneliu Constantinescu

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Extra resources for C*-algebras Volume 2: Banach Algebras and Compact Operators

Sample text

Take j E INn, and v E s ~) such that ~JUX --" e j . Define n w" IKn } lKn Z i=1 Uj : - - t / ~ W . 10, every unital algebra homomorphism u" E(IK ") --+ E is injective. The final assertion follows from the fact that an element of s n) is invertible iff the corresponding determinant does not vanish. 8 Let E be a normed space and take u, v E s U ' O "-- V U with , P E ]K[tl, and Q e lK[s, tl. Then P(u)' = P(u') , An element of s Q(u, v)' = Q(u', v') . is idempotent iff it is a projection in E . 5.

Thus /3--1, O~2 <4 xy + yx = al . Take a, b, c E lR with ~. Banach Algebras 44 a l + bx + cy = O. Then a x - bl + c x y = 0 and a x - bl + c y x = O. Then 2ax-2bl+cc~l=O, -4a 2=(2b-ca) 2ax=(2b-ca)l, 2>_0, a=0, b=c=0. Hence 1, x, y are linearly independent. 20 Let E II be a f i n i t e - d i m e n s i o n a l real unital algebra, which is a division algebra, and take z , y E E linearly i n d e p e n d e n t with X2 = y2 ___ _ I . T h e n there is an injective unital algebra h o m o m o r p h i s m u : IH -4 E such that x=us, ue u ( ~ ) .

21 a =~ b, again, there is a z E E with 1 = z ( 1 - (1 - y ) ) = z y . 3, y is invertible and y-l=l-x. Hence 1 - x is invertible and R E ~. Take F E ~ and x E F . Take y E E . Then y x E F and so 1 - y x is invertible. 21 b =~ a, x E R . Hence F C R and R is the greatest element of ~. 24 I ( 4 ) Let R be the radical of the algebra E and take x E R . Then o(~) = { 0 } . 19, we may assume E to be unital. 9, x is not invertible. Thus 0 E a ( x ) . Take a E IK\{0}. 23, 1 - ~x 1 is invertible. 25 ( 7 ) I/E (01.

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