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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.