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

By Corneliu Constantinescu

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Then IK x E endowed with the norm xE >~+, x > supllxyll-supllyxll yEE# yEE# if E has no unit and with the norm ]K x E > IR+, (a,m), > sup{Ic~l, IIx+c~lEll} if E has a unit 1E is a unital C*-algebra inducing on E its given norm. I f E has no unit, then the above IK x E is called the unital C*-algebra associated to E . I f E has a unit, then the unital C*-algebra associated to E will be E itself. Case1 E has no unit The map xE >~+, x, >supllxyll yEE# is obviously a seminorm. 12, it induces on E the original norm of E .

14. 13). Remark. 30 4. 1 ( 0 ) (Kaplanski) Every C*-algebra is strongly symme- tric. 40). Let E be a C*-algebra. 13). 15, for every x E Re E . 4 b =v a, E is symmetric. 9 c3 =~ cl. m ( 0 ) A commutative real C'-algebra all of whose elements me selfadjoint is a Gelfand C*-algebra. In particular, a C*-subalgebr~ of a Gelfand C*-algebra is a Gelfand C*-algebra. 2 Let E be a commutative real C*-algebra such that Re E = E . 1, /~ is symmetric. 29, for every x E E . Hence E is a Gelfand C*-subalgebra of E .

Define ~F,C" ~(G) >~ ( F ) , x', ~ ; x'lF for all unital C*-subalgebras F, G of E with F C G. for all F, G E ~ , F C G ; call T its projective limit. b) ~F,E = PF,C o ~G,E for all F , G C ~ , F C G ; call ~ " or(E) --+ T the projective limit of (~F,E)F~;~- 40 4. C*-A19ebras c) ~ is a homeomorphism. g. F is finite- dimensional), then a ( E ) is also totally disconnected. 3 c)). b) is obvious. c) 99 is continuous. By a), it is surjective. Let x', y' E a ( E ) with 99(x') = 99(y'). Then x' = y' on U F .

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