Does a $\mathrm{C}^*$-algebra generated by projections contain support projections

Question

I think we have for a (normal?) state $\varphi$ on a von Neumann algebra $\mathcal{M}$ a projection $p_\varphi$ with some nice properties called its support. It arises as follows:

Define the following null space:

$$N_\varphi=\left\{g\in \mathcal{M}\,|\,\varphi(|g|^2)=0\right\}.$$

This set is a $\sigma$-weakly closed left ideal. Therefore there exists a projection $q_\varphi$ such that $N_\varphi=\mathcal{M}q_\varphi$. Some properties include the fact that $g\in N_\varphi$ if and only if $gq_\varphi=g$. Also, for all $f\in \mathcal{M}$ we have $$\varphi(q_\varphi)=\varphi(fq_\varphi)=\nu(q_\varphi f)=0.$$

Also if we define the projection $p_\varphi:=1_{\mathcal{M}}-q_\varphi$. We have $$\varphi(f)=\varphi(fp_\varphi)=\varphi(p_\varphi f)=\varphi(p_\varphi fp_\varphi),$$ and $\varphi(p_\varphi)=1$.

I understand that a $\mathrm{C}^*$-algebra generated by projections is not necessarily a von Neumann algebra... but

Question: Does a $\mathrm{C}^*$-algebra generated by projections have support projections?

Answer 1

Let $X$ be a compact space and $\phi$ be a state on $C(X)$ induced by a probability measure $\mu$ on $X$, whose support is the closed subset $S\subseteq X$.

Then $N_\phi$ is the ideal $C_0(X\setminus S)$, which is unital iff $S$ is clopen.

Thus $\phi$ has a support projection iff the support of the associated measure is clopen.

Now, if $X$ is totally disconnected but not discrete, such as the Cantor set, then $C(X)$ is generated (as a C*-algebra) by projections, but since $X$ contains some closed sets which are not clopen, any measure with support in such a set will fail to have a support projection in $C(X)$.

Answer 2

Here is a simple example. Let $A=K(H)$, the compact operators on a separable Hilbert space. It is generated by its projections, which are the finite-rank projections in $B(H)$.

Fix an orthonormal basis $\{e_n\}$, and let $\phi:A\to\mathbb C$ be the state $$ \phi(x)=\sum_n\frac1{2^n}\,\langle xe_{2n},e_{2n}\rangle. $$ This state is "normal", since it is the restriction of a normal state in $B(H)$ (the same definition above works for any $x\in B(H)$). The support projection of $\phi$ is $$ p=\sum_n \langle \cdot,e_{2n}\rangle\,e_{2n}, $$ which is not in $A$.