By Hahn-Banach it holds in any Banach space that the norm of a positive element attained by a functional of norm $1$. But why is it true in a $C^*$-algebra that norm of any positive element is attained by a positive, linear functional of norm $1$?
Asked
Active
Viewed 133 times
3
-
1Is that true? If you take a negative element (with positive norm), then any positive functional must map it to a negative number, i.e. not its norm. – Theo Bendit Apr 14 '18 at 14:48
-
Right, I meant positive elements. – Gaute Schwartz Apr 14 '18 at 15:06
1 Answers
4
Let $A$ be a $C^*$-algebra, with $a\in A$ positive, and let $B$ be the $C^*$-subalgebra of $\tilde A$ generated by $1$ and $a$. Then there is a character $\tau_0$ on $B$ such that $\tau_0(a)=\|a\|$. By the Hahn-Banach theorem, $\tau_0$ extends to a linear functional $\tau_1$ on $\tilde A$ of norm $1$, and since $\tau_1(1)=\tau_0(1)=1$, it is positive. The restriction $\tau$ of $\tau_1$ to $A$ is a positive linear functional of norm $1$, and $\tau(a)=\|a\|$.
Aweygan
- 23,232