Banach algebras question 2

• February 9th 2010, 08:18 AM
Arczi1984
Banach algebras question 2
Next questions about proof details.
Red shape 1: From where we have this?
Red shape 2: How we used fact that this algebra is semi-simple?

http://img413.imageshack.us/img413/7617/theorem2.png

PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
• February 9th 2010, 01:46 PM
Opalg
1. The fact that $f(Tx)\in\sigma(Tx)$ is just condition (2) from earlier in the paper (which holds because f is multiplicative).

2. Equation (10) says that $f(Txy) = f(TxTy)$ for each multiplicative linear functional f on the algebra. The semisimplicity condition says that there are enough such functions to separate elements of the algebra. So if T(xy) and T(x)T(y) have the same value at each f then they must be equal.