Timeline for Reference request (or otherwise): Adjoint action
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| Aug 21, 2012 at 19:34 | comment | added | MTS | The existence of the modular conjugation operator requires a cyclic and separating vector for $A$ in the Hilbert space $H$. In general I don't think you'll have that. You will probably need to examine the specific representation of whatever non-associative algebra you are working with to see if you can define the $J$ in a similar way as is done for von Neumann algebras. | |
| Aug 21, 2012 at 0:31 | comment | added | SMF | Hi thank for responding. I could have been better with my definitions - although I was trying to keep my post short. What I mean by $Ad_u(\xi) = u\xi u^*$ is the following. Define right operation on elements of H as: $\xi u^* = (u^*)^0\xi$, where $b^0 = Ja^* J^*$. J is a real structure operator that acts on the hilbert space H. It is a generalization of the tomita operator. So at the level of the group (Ad u) = uJuJ^*. Exponentiating you can see that at the level of the algebra the adjoint is given as $(ad B) = B - JB^*J^*$. Focused: Is there an analogous J in the non-associative case? | |
| Aug 20, 2012 at 22:08 | history | answered | MTS | CC BY-SA 3.0 |