Several questions about state space

Question

Here are several questions about the state space of a C*-algebra $A$:

1. Let $A$ be a unital and separable C*-algebra, can we find a faithful state $\phi \in S(A)$. ( The $S(A)$ denotes the state space of $A$)

2. It is not hard to verify that: if C*-algebra $A$ is unital, then $S(A)$ is weak-* closed convex set. What about the case that $A$ is non-unital?

3. Let $A$ be unital C*-algebra and $h\in A$ be any self-adjoint element. Let $C^{*}(h,~1_{A})$ be the unital C*-algebra generated by $h$ and $1_{A}$. Can we find a pure state $\psi\in P(C)$ such that $|\psi(h)|=||h||$ ?

Answer 1

1. Deduce that there is countable set $\{q_n\}$ dense in the unit ball. Let $r_n=q_n^*q_n$. Given any $a\in A^+_1$, there exists $b\in A$ with $b^*b=a$. Given $\varepsilon>0$ there exists $q_n$ with $\|q_n-b\|<\varepsilon/2$. Then $$ \|r_n-a\|=\|q_n^*q_n-b^*b\|\leq\|q_n^*(q_n-b)\|+\|(q_n^*-b^*)b\|\leq2\|q_n-b\|<\varepsilon. $$ So the sequence $\{r_n\}$ is dense in $A^+_1$.

For each $n$, let $\varphi_n\in S(A)$ with $\varphi_n(r_n)=\|r_n\|$.Then define the state $\phi$ by $$ \phi(a)=\sum_n\frac{\varphi_n(a)}{2^n}. $$ If $a\in A^+_1$ and $a\ne0$, then there exists $r_n$ with $\|a-r_n\|<\|a\|/4$. Then $\|r_n\|\geq3\|a\|/4$, and $$ \varphi_n(a)=|\varphi_n(a-r_n)+\varphi_n(r_n)|=|\varphi_n(a-r_n)+\|r_n\|\,| \geq\|r_n\|-|\varphi_n(a-r_n)|\\ \geq3\|a\|/4-|\varphi_n(a-r_n)|>3\|a\|/4-\|a\|/4=\|a\|/2. $$ Then $\phi(a)>0$, and so $\phi$ is faithful.

2. When $A$ is not unital and $\tilde A$ is the unitization of $A$, $S(\tilde A)\cup\{\hat 1\}$, where $\hat 1$ is the state $\hat 1(a,\lambda)=\lambda$. While convex, $A(A)$ is not weak$^*$-closed.

3. Yes, that can be done in any C$^*$-algebra.