Let $\mathcal{T}$ be the Toeplitz algebra. I.e. the $C^*$ algebra generated by the shift operator $S\in B(l^2(\Bbb N))$.
In page 6, line 8 of a proof we have a unitary element $u \in \mathcal{T} \otimes \mathcal{T}$, and it is claimed that $u$ is homotopic to the identity by a path of unitaries.
The claim seems to be quite general. Is there any reference/ similar result, which I could read to understand more about this?
The unitaries you consider there are self-adjoint. In that particular case, you can write down an easy formula for the path. Let $u$ be a self-adjoint unitary in a C*-algebra. Define $h := (1-u)/2$. Then $e^{\pi i h} = u$ and the path $t \mapsto e^{\pi i th}$ connects $1$ to $u$.