I am reading "K-theory FOR OPERATOR ALGEBRAS" Bruce Blackadar,
Proposition 23.5.1. Let B be a separable $C^{*}$-algebra. Then there is a separable commutative $C^{∗}$-algebra F, whose spectrum consists of a disjoint union of lines and planes, and a homomorphism $\phi: F \rightarrow SB \otimes K$, such that $\phi_{*}: K∗(F) \rightarrow K∗(SB)$ is surjective. I am struggling with some parts of the proof. we need to construct a commutative C^{*}-algebra $F$.
(1) Choose a set of generators for K1(B). Each generator can be represented by a unitary in $M_{n}(B)^{+}$ for some n, and hence by a ∗-homomorphism from $C_{0}(\mathbb{R})$(which is identified with the continuous functions on the unit circle which vanish at 1) into $M_{n}(B)$. Thus, if $F_{0} = C_{0}(X_{0})$, where $X_{0}$ is a disjoint union of lines, one for each generator, there is a ∗-homomorphism $\phi_{0}$ from $F_{0}$ to $B \otimes \mathbb{k}$ such that $\phi_{0}∗ : K1(F0) → K1(B)$ is surjective.
Could you please explain why if $F_{0} = C_{0}(X_{0})$, where $X_{0}$ is a disjoint union of lines, one for each generator, there is a ∗-homomorphism $\phi_{0}$ from $F_{0}$ to $B \otimes \mathbb{k}$?
Thanks a lot in advance.