# Support Projection for Characters in $\mathrm{C}^*$-Algebra with finitely many characters

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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 $$\varphi(q_C)=\varphi(fq_C)=\nu(q_C f)=0?$$

• What prevents you from taking $p_C=1$?
– Ruy
Feb 12 at 14:17
• @Ruy I really want to talk about states $\rho$ that have $\rho(p_C)=0$. I appreciate that was not explicit in the OP. Feb 12 at 14:19
• But then you need to lay out the precise conditions on $p_C$ ruling out $p_C=1$.
– Ruy
Feb 12 at 14:21
• @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. Feb 12 at 14:29

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: $$p_C=\sup\{\omega_\varphi\,:\,\varphi\in \Omega(A)\}.$$