We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with continuous symbol in the torus $T^2$ is the subspace spanned by $\mathcal{A}(C(T))\otimes \mathcal{K}$ and $\mathcal{K}\otimes \mathcal{A}(C(T))$. Here $\mathcal{A}(C(T))$ is the $C^*$-algebra generated by Toeplitz operators with continuous symbol in the circle $T$ and $\mathcal{K}$ is the ideal of compact operators in the Hardy space on the circle, $H^2(T)$. How can we obtain that the quotient space $\mathcal{I}/(\mathcal{K}\otimes \mathcal{K})$ is $C(T\times \{0,1\})\otimes \mathcal{K}$?
All tensor product is algebraic.
This result is written in the introdunction part of the paper of Curto and Muhly (enter link description here) and they say that it is obtained by the paper of Douglas and Howe.
By the way, what are the nice references on the tensor product of $C^*$- algebra?