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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:

  1. If A is inner-graded, is M(A) inner-graded?
  2. 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?
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2  
I tidied the layout a bit. For Q1, surely the same formula, $x \mapsto \epsilon x \epsilon$, gives a grading of $M(A)$?? – Matthew Daws May 11 '11 at 19:08
    
thanks. M(A) is inner-graded which implies the statement in 2. – m07kl May 12 '11 at 19:03

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