Let $X$ be a complex projective variety, and let $ X^{an} $ denote the topological space of complex points of $ X $ equipped with the analytic topology. Then, any algebraic vector bundle $ E \to X $ determines a topological vector bundle $ E^{an} \to X^{an} $.
How to construct a natural map $ K_0 (X) \to K_{top}^{0} (X^{an} ) $ such that : $ CH^* (X) \otimes \mathbb{Q} \simeq K_0 (X) \otimes \mathbb{Q} \to K_{top}^0 (X^{an} ) \otimes \mathbb{Q} \simeq H^{ev} ( X^{an} , \mathbb{Q} ) $ ?.
$ K_{top}^0 ( X^{an} ) $ is the Grothendieck group of topological vector bundles over $ X^{an} $.
$ K_0 (X) $ is the Grothendieck group of algebraic vector bundles over $X$.
Could you tell me furthermore why do we have : $ CH^* (X) \otimes \mathbb{Q} \simeq K_0 (X) \otimes \mathbb{Q} $ ( ring isomorphism ) and $ K_{top}^0 (X^{an} ) \otimes \mathbb{Q} \simeq H^{ev} ( X^{an} , \mathbb{Q} ) $ ?
Thanks in advance for your help.