# Thread: Banach algebras question 2

1. ## Banach algebras question 2

2. 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.