Pure-state excision and density of faithful essential vector-state orbits

Work in $\tilde{A}$, with the unique pure extension of $\varphi$ if needed, and its irreducible GNS triple $(\pi ,H,\xi )$. Put $N=\{z:\pi (z)\xi =0\}$. For $\varphi (x)=0$, let $\eta =\pi (x)\xi$, so $\eta \perp \xi$. If $\eta =0$, $x\in N$. Otherwise [F2] realizes a self-adjoint operator $h$ with $\pi (h)\xi =0$, $\pi (h)\eta =\eta$: these prescriptions are compatible with the projection onto $\mathbb{C}\eta$. Then $x-hx\in N$ and $hx=(x^{\ast }h{)}^{\ast }$ with $x^{\ast }h\in N$. Conversely Cauchy–Schwarz makes $\varphi$ vanish on $N+N^{\ast }$. Thus $\ker \varphi =N+N^{\ast }$, with no closure required.

The norm-closed left ideal $N$ gives a norm-closed $\ast$-subalgebra $C=N\cap N^{\ast }$: if $c,d\in C$, both $cd$ and $(cd{)}^{\ast }$ annihilate $\xi$. Choose a positive contractive approximate unit $e_t$ of $C$ by [F3]. For $z\in N$, the positive element $z^{\ast }z$ belongs to $C$, since $\pi (z^{\ast }z)\xi =0$; hence $\mathrm{\parallel }z(1-e_t){\mathrm{\parallel }}^2=\mathrm{\parallel }(1-e_t)z^{\ast }z(1-e_t)\mathrm{\parallel }\to 0$. Also $\pi (e_t)\xi =0$. In the unital case take $a_0=1$. In the nonunital case [F2] realizes the eigenvalue1 on $\xi$ by a positive contraction in the image $\pi (A)$; lift a self-adjoint preimage and clip it to $[0,1]$ using [F3] to obtain $a_0\in A$ with $0\le a_0\le 1$ and $\pi (a_0)\xi =\xi$. Put $u=a_0^{1\mathrm{/}2}$ and $b_t=u(1-e_t)u\in A$. Then $0\le b_t\le 1$, $\pi (u)\xi =\xi$, and $\varphi (b_t)=1$, so $\mathrm{\parallel }{b}_t\mathrm{\parallel }=1$.

For $z\in N$, $zu\in N$ since $\pi (u)\xi =\xi$. The estimate of step 1.2 therefore gives $\mathrm{\parallel }z{b}_t\mathrm{\parallel }\le \mathrm{\parallel }zu(1-e_t)\mathrm{\parallel }\mathrm{\parallel }u\mathrm{\parallel }\to 0$. For $x\in \tilde{A}$, step 1.1 writes $x-\varphi (x)1=c+d^{\ast }$ with $c,d\in N$. Consequently $\mathrm{\parallel }{b}_t(x-\varphi (x)1)b_t\mathrm{\parallel }\le \mathrm{\parallel }{b}_t\mathrm{\parallel }(\mathrm{\parallel }c{b}_t\mathrm{\parallel }+\mathrm{\parallel }d{b}_t\mathrm{\parallel })\to 0$. This is the required excision for every $x\in A$, including the nonunital construction with $b_t\in A$.

Given finitely many norm-bounded tests $x_j$ and a positive error, choose $b=b_t$ so $\mathrm{\parallel }b{x}_{j}b-\varphi (x_j)b^2\mathrm{\parallel }$ is small for every test. Put $a=b^2$, so $\mathrm{\parallel }a\mathrm{\parallel }=\varphi (a)=1$. For a state $\psi$, extend it to $\tilde{A}$ and suppose $\psi (a)>1-\delta$. Since $(1-b{)}^2\le 1-b^2$, Cauchy–Schwarz gives $\mathrm{\mid }\psi (x)-\psi (bxb)\mathrm{\mid }\le 2\mathrm{\parallel }x\mathrm{\parallel }\sqrt{\delta }$. Thus $\mathrm{\mid }\psi (x_j)-\varphi (x_j)\mathrm{\mid }\le 2\mathrm{\parallel }{x}_j\mathrm{\parallel }\sqrt{\delta }+\mathrm{\parallel }b{x}_{j}b-\varphi (x_j)b^2\mathrm{\parallel }+\mathrm{\mid }\varphi (x_j)\mathrm{\mid }\delta$. First making the excision errors small, then $\delta$ small, puts $U_{a,\delta }$ inside the prescribed neighborhood. Every such set is itself a weak-star open neighborhood of $\varphi$, proving the basis assertion on all states, not only pure states.

Let $\pi$ be the specified faithful essential irreducible representation. In a nonempty pure-state neighborhood choose a smaller $U_{a,ϵ}\cap P(A)$ from step 3.1 with $0<ϵ<1$. Faithfulness gives $\mathrm{\parallel }\pi (a)\mathrm{\parallel }=1$. The spectral range of $\pi (a)$ for $(1-ϵ,1]$ is infinite dimensional: otherwise the nonzero operator $\pi ((a-(1-ϵ){)}_{+})$ would be finite rank and belong to $\pi (A)$, contradicting essentiality. Choose a unit vector in that range orthogonal to the prescribed finite-dimensional subspace. Its expectation of $a$ is strictly greater than $1-ϵ$. Its vector state has norm1 by nondegeneracy and an approximate unit, and is pure because every nonzero vector in an irreducible carrier is cyclic. It therefore lies in the chosen neighborhood.

This proves the asserted density and all nonunital cases directly with spectral cutoffs in $A$ itself. Internal-unitary transport in [F2] identifies these vector states with the orbit of any cyclic pure vector state in the same irreducible representation. The stated choices are only the approximate unit and finitely prescribed operators/vectors; no class selector or unproved spectral multiplicity model is used.