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Let A be a C*-algebra. We denote by A,*, the quotient of the C*-algebra of bounded continuous functions (1, oo) —>• ^4 by the ideal of the functions which vanish at oo . We show that the the canonical map A -» A A^ be the "standard embedding" given by i(a) = (consta) (mod CA). Let us take, using the Bartle-Graves Theorem, a section p : Aoo —► CbA for the map nA : CbA —► Aoo with the following properties: (i) p(Xx) = Xp(x) for all x G A, X G C ; (ii) p is continuous. If we take $ : Aoo —► A to be the map defined, for each t G (1, oo), by $(x) = p(x)(t), then using the terminology of (CH) we get an asymptotic homomorphism from Aoo to A. It is easy to show that (i7/ o i : A —► A gives an asymptotic homomorphism which is homotopic to Id^ . So t : A —► Aoo has a left inverse in E( A&, A). As noted in (CH), every asymptotic homomorphism y/t : B —► A is equivalent to one of the form $ o y/ with y/ : B —► A^ a *-homomorphism.

On the 𝐾-groups of certain 𝐶*-algebras used in 𝐸-theory