I've been working computing generator for several $C^*$-algebras involved in my Master's thesis, however I've got stucked with the generators of $K_1(C(\mathbb{T})\otimes\mathbb{K})$, which is in my opinion weird, since the stabilization of $C(\mathbb{T})$ is a fairly known algebra. This difficulty led me to the next question:
It is a well known fact that for every $C*$-algebra $A$ it holds true that $$K_0(A)\cong K_0(A\otimes\mathbb{K})\quad\text{and}\quad K_1(A)\cong K_1(A\otimes \mathbb{K}).$$ The first isomorphism is actually induced by the map $a\mapsto a\otimes e_{11}$, where $e_{11}$ is a rank 1 projection in $\mathbb{K}$. This means that if we know what are some generators for $K_0(A)$, it would be possible to obtain the generator of $K_0(A\otimes \mathbb{K})$ using this result. It is however a bit trickier in the case of $K_1$, since the second isomorphism follows from a non constructive isomorphism given by the continuity over direct limits of the $K_1$-functor (at least this is the proof that I know).
Does anyone know how to get some generators for the group $K_1(A\otimes\mathbb{K})$ knowing the ones for $K_1(A)$ or could help me a little computing some generators for $K_1(C(\mathbb{T})\otimes\mathbb{K})$?
Fix a (minimal) projection $q$ in $\mathbb K$, and let $p=1_A\otimes q$. Then the inclusion $p(A\otimes\mathbb K)p\hookrightarrow A\otimes\mathbb K$ induces an isomorphism on $K$-theory. By virtue of $q$ being mininmal, we have $p(A\otimes\mathbb K)p\cong A$. To simplify matters, let's assume $A$ is unital. Then for $u\in M_n(A)$ a unitary, the image of $[u]\in K_1(A)$ under these isomorphisms is the class of $(1+(u-1)\otimes q)\otimes1_n$ in $M_n((A\otimes\mathbb K)^+)$.
For the case $A=C(\mathbb T)$, the element $z\in C(\mathbb T)$ given by $z(e^{i\theta})=e^{i\theta}$ generates $K_1(C(\mathbb T))\cong\mathbb Z$ (this can be seen in several ways), and thus $1+(z-1)\otimes q$ generates $K_1(A\otimes\mathbb K)$.