$\displaystyle \gamma : \left[0, 1\right]\rightarrow \mathbb{T},t\mapsto e^{i2\pi t}$

Let $\displaystyle S$ be the closed ideal consisting of functions $\displaystyle \{f\in C_{0}(\left[0,1\right],\mathbb{T}):f(1)=0\}$

Where $\displaystyle C_{0}(\left[0,1\right],\mathbb{T})$ denotes the continuous functions vanishing at infinity

from $\displaystyle \left[0,1\right]$ to the unit circle, $\displaystyle \mathbb{T}$

Show that the map $\displaystyle \gamma_A:A\otimes_{*}S\rightarrow S(A)$

where $\displaystyle S(A)=\{f\in C_{0}(\left[0,1\right],\mathbb{T}):f(1)=0=f(0)\}$

is an injective *-homomorphism

Linearity is easy, so is multiplicativity, but how will I show that the adjoint is preserved and that the map is injective?