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Related to this question...

For a normal state $\varphi$ on a von Neumann algebra $\mathcal{M}$ there is a projection $p_\varphi$ with some nice properties called its support. Some properties include that for all: $f\in \mathcal{M}$: $$\varphi(f)=\varphi(fp_\varphi)=\varphi(p_\varphi f)=\varphi(p_\varphi fp_\varphi),$$ and $\varphi(p_\varphi)=1$. Also, for $q_\varphi:=1_{\mathcal{M}}-p_\varphi$, and all $f\in \mathcal{M}$ we have $$\varphi(q_\varphi)=\varphi(fq_\varphi)=\nu(q_\varphi f)=0.$$ EDIT: It is the smallest such projection with these properties.

Now suppose that $A$ is a (infinite dimensional, separable, generated by projections) $\mathrm{C}^*$-algebra with finitely many characters. Is there a projection $p_C$ such that for all characters (non-zero *-homomorphisms $\varphi:A\rightarrow \mathbb{C}$) and all $f\in A$ we have

$$\varphi(f)=\varphi(fp_C)=\varphi(p_C f)=\varphi(p_C fp_C),$$ and $\varphi(p_C)=1$. Also, for $q_C:=1_{A}-p_C$, and all $f\in A$ we have $$\varphi(q_C)=\varphi(fq_C)=\nu(q_C f)=0?$$

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  • $\begingroup$ What prevents you from taking $p_C=1$? $\endgroup$
    – Ruy
    Feb 12 at 14:17
  • $\begingroup$ @Ruy I really want to talk about states $\rho$ that have $\rho(p_C)=0$. I appreciate that was not explicit in the OP. $\endgroup$ Feb 12 at 14:19
  • $\begingroup$ But then you need to lay out the precise conditions on $p_C$ ruling out $p_C=1$. $\endgroup$
    – Ruy
    Feb 12 at 14:21
  • $\begingroup$ @Ruy the thing that is glaringly missing in the OP is that $p_C$ be the smallest such projection. I am happy with the answer I got from a correspondent. $\endgroup$ Feb 12 at 14:29
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A correspondent wrote to say that working with the enveloping von Neumann algebra $A^{**}:=\pi_{\text{GNS}}(A)''$, like any state, a character $\varphi\in S(A)$ extends to a normal state $\omega_\varphi$ on $A^{**}$, and thus has a support projection $p_\varphi\in A^{**}$ such that $\omega_\varphi(p_\varphi)=1$. Furthermore, if $\omega_\varphi(p)=1$, then $p_\varphi\leq p$, and also for all $f\in A^{**}$: $$\omega_\varphi(f)=\omega_\varphi(p_\varphi f)=\omega_\varphi( fp_\varphi)=\omega_\varphi(p_\varphi fp_\varphi).$$ If the character space $\Omega(A)$ is empty, define $p_C=0$. Otherwise, define: $$p_C=\sup\{\omega_\varphi\,:\,\varphi\in \Omega(A)\}.$$

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