I am working on Prop 4.8.3 of Higson & Roe's Analytic K-Homology recently. It asks me to verify that for unital C*-Algebras $\mathcal{A}$ and $\mathcal{B}$, if $u$ is a unitary of $\mathcal{A}$ and $p$ is a projection of $\mathcal{B}$, then the external product can be presented by
$\times : K_1(\mathcal{A})\times K_0(\mathcal{B})\to K_1(\mathcal{A}\otimes\mathcal{B}),\quad [u]\times[p]\mapsto[u\otimes p+1\otimes(1-p)]$
with the tensor product being the minimal tensor product.
I tried to use the isomorphism
$K_1(\mathcal{A})\times K_0(\mathcal{B})\cong K_0(C_0(\mathbb{R})\otimes\mathcal{A})\times K_0(\mathcal{B})\to K_0(C_0(\mathbb{R})\otimes\mathcal{A}\otimes\mathcal{B})\cong K_1(\mathcal{A}\otimes\mathcal{B})$
since we already know the external product between $K_0$ groups are simply $[p]\times[q]=[p\otimes q]$. But I was stuck figuring out how $[u]\in K_1(\mathcal{A})$ corresponds to the element in $K_0(C_0(\mathbb{R})\otimes\mathcal{A})$.
Any help will be sincerely grateful!