A graded C*-algebra A is inner-graded if there exists a self-adjoint unitary $\varepsilon$ in the multiplier algebra M(A) of A which implements the grading automorphism $\alpha$ on A: $\alpha(a)=\varepsilon a \varepsilon$.

My question are:

- If A is inner-graded, is M(A) inner-graded?
- How can we prove that $A\widehat{\otimes}B\subseteq M(A)\widehat{\otimes}M(B)\subseteq M(A\widehat{\otimes}B)$, where the tensor is minimal graded tensor product. What about if we replace minimal graded tensor product by maximal graded tensor product?