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**Extra resources for C*-Algebras Volume 3: General Theory of C*-Algebras**

**Sample text**

Hence for x,y E E , v-~(x - iy)* = ( u ( ( x - i y ) * ) * - u(x - iy) - ux + iuy. By b), then map >ExE, (x,y), > (x + iy, ux + iuy) is an isomorphism of complex (unital) algebras. f) follows from e). Remark. with m A unital real C*-algebra E is not purely real iff there is an x C E ~ x*---x, x2--1. 12 4. 9 Let E be a complex C*-algebra, A an algebraic basis of the underlying complex vector space of E , and F the real vector subspace of E generated by A . If F is an involutive real subalgebra of E , then the map ~" > E, (x,y), ~ ~+iy is a isomorphism of involutive complex algebras.

C =v a. 13 b =v a , x is selfadjoint. 4 b), o(x ~ x) c (o}, so that r ( x ~ - z ) = O. 16 b), X2--x--O. Hence x is idempotent. 16 b).

Reversing the multiplication. 50, ++ E and E are isomorphic iff there is an isomorphism of real C*-algebras E --+ ++ E which is conjugate linear. In particular if ]K = lR then E and E are isomorphic. 22 Let E be an involutive normed complex algebra and P the set of seminorms on E , for which p(xy) < p ( x ) p ( y ) , p(x) ~ < p(x* x) , p(~) ~ Ilxll for every x, y C E . Define q:E ~ IR+, x~ ~ supp(x) . pEP 20 4. C*-Algebras a) q C P . b) ~1(o) c) ~ an involut~v~ ~d~at of E . We have that q(x) = q(y) for every A 9 E/ql(0) and x, y 9 A .