Analytic theory of continued fractions III by Lisa Jacobsen

By Lisa Jacobsen

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In this regard, an ideal / of A for which /* = / , is said *-ideal. 6 Definition. A map : A —> B between two involutive algebras A, B is called -^-preserving, when {x)*, for every x G A. Thus, a *-morphism is a *-preserving morphism between two involutive algebras. In the category of involutive topological algebras (respectively topological *-algebras) a topological * -isomorphism is a *-preserving topological isomorphism. The term topological injective *-morphism applies to a * preserving topological injective morphism.

24)). Thus, one has x* = (5^) n 6 N , V x = ( z n ) n e N G C N . 1). 3) * : C(H) —> C{H) : T — » f with < T*(£), JJ >=< f, Tfa) >, for all £, r? e H. 4) * : A(Bn) -^ A(Bn) : / — > / * : f*(z) := f(z), V z e D n , 28 Chapter I. Background material n G N. 43). 4(5)). 44) each / G O(C) is of the form (ZbJneN with f\Dn E A(Bn) such that Qnm{f\nm) — / | o n , V n < m in N, where gnm, n < m, are the connecting morphisms between the Banach algebras of the inverse system (ACDn))neN (cf. 6) and the comments after it).

3 Examples. (1) Let A = C[t] and a € A with cr(<) 7^ A, for every A £ C. Then spA(a) = C. (2) Let A be the algebra of all rational polynomials in one variable with complex coefficients and r € A with r{t) ^ A, for each A £ C. Then, spA{r) = 0. 2(1)). 12) spc{x)(f) = f{X) C ftfX) = 7(X) = sp C f c W (/), V / G C6(X). where /3X denotes the Stone-Cech compactification of X. 4 Proposition (Roitman-Sternfeld). Let A be a unital algebra. The following statements are equivalent: (1) SPA{X) is bounded, for all x G A.

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