Bounded density and finite-vector transitivity for C*-representations

If $H=\{0\}$, then $D=M=\{0\}$, the density statements are equalities, and the finite-vector prescription is the empty/zero case. The unit-vector clause is vacuous; the separate pure-state claims use nonzero GNS carriers. For the operator arguments below assume $H\mathrm{\ne }\{0\}$. Let $(e_{\lambda })$ be a positive contractive approximate unit of $D$ from [F2]. For every $d\in D$ and $\xi \in H$, $\mathrm{\parallel }(I-e_{\lambda })d\xi \mathrm{\parallel }\le \mathrm{\parallel }d-e_{\lambda }d\mathrm{\parallel }\mathrm{\parallel }\xi \mathrm{\parallel }\to 0$. Finite linear combinations of vectors $d\xi$ are dense by nondegeneracy, while $\mathrm{\parallel }I-e_{\lambda }\mathrm{\parallel }\le \mathrm{\parallel }I\mathrm{\parallel }+\mathrm{\parallel }{e}_{\lambda }\mathrm{\parallel }\le 2$; approximating an arbitrary vector by such a finite combination therefore gives $e_{\lambda }\to I$ strongly.

We first show that if $x_{\alpha }=x_{\alpha }^{\ast }\in D$ converges strongly to $x=x^{\ast }\in M$, then $h(x_{\alpha })\to h(x)$ strongly for every $h\in C_0(\mathbb{R})$ with $h(0)=0$. By [F6], $K={\mathbb{R}}^{\ast }$ is compact Hausdorff, and every $C_0(\mathbb{R})$ function extends continuously to $K$ by value $0$ at infinity. The resolvent functions $r_{\pm }(t)=(t\pm i{)}^{-1}$ also extend continuously by $0$: their positive superlevel sets are closed bounded intervals, hence compact. These functions generate a unital self-adjoint point-separating algebra: $r_{+}$ separates finite real points and is nonzero at every finite point, whereas both vanish at infinity. Stone–Weierstrass makes their *-polynomials dense in $C(K)$. If $p$ approximates the extension of $h$, replace it by $q(t)=p(t)-p(\mathrm{\infty })-(p(0)-p(\mathrm{\infty }))\mathrm{/}(1+t^2)$; then $q(0)=q(\mathrm{\infty })=0$ and $q$ still approximates $h$ arbitrarily well. For self-adjoint $y\in D$, $r_{\pm }(y)=(y\pm iI{)}^{-1}\in \tilde{D}$ and the scalar quotient of $q(y)\in \tilde{D}$ is $q(0)=0$, so $q(y)\in D$. The resolvent identity $r_{+}(x_{\alpha })-r_{+}(x)=r_{+}(x_{\alpha })(x-x_{\alpha })r_{+}(x)$ and its $r_{-}$ analogue, with all resolvent norms at most one, show strong convergence of each resolvent; finite products of uniformly bounded strongly convergent operators converge strongly. Uniform approximation by $q$ now gives $h(x_{\alpha })\to h(x)$ strongly.

An irreducible *-representation $\pi$ has scalar commutant. Indeed, if a self-adjoint $S\in \pi (A{)}^{\mathrm{\prime }}$ were nonscalar, choose disjoint neighborhoods of two points of $\sigma (S)$ and continuous nonnegative functions supported there and nonzero at those points. Their functional-calculus operators are nonzero, orthogonal, and commute with $\pi (A)$; the closure of the range of one is a nonzero proper invariant subspace, a contradiction. Taking real and imaginary parts handles every element of the commutant. A nonzero intertwiner between irreducible representations then has $V^{\ast }V$ and $V{V}^{\ast }$ scalar; normalizing $V$ gives an isometry whose range projection is a nonzero scalar projection, hence the identity, so the intertwiner is unitary. Thus inequivalent irreducible representations have zero off-diagonal intertwiners.

First let $\mathcal{A}$ be any unital $\ast$-subalgebra of $\mathcal{B}(H)$, let $T\in {\mathcal{A}}^{\mathrm{\prime }\mathrm{\prime }}$, and fix a nonempty finite tuple $\xi =({\xi }_1,\dots ,{\xi }_n)$. On $H^{\oplus n}$ put $C=\overline{\{(a{\xi }_1,\dots ,a{\xi }_n):a\in \mathcal{A}\}}$ and let $P$ be its orthogonal projection by [F1,F8]. The linear subspace $C$ is invariant under every diagonal $a^{\oplus n}$ and its adjoint, so its orthogonal complement is invariant too; hence $P$ commutes with $a^{\oplus n}$. Each block $P_{ij}$ therefore commutes with every $a\in \mathcal{A}$, so $P_{ij}\in {\mathcal{A}}^{\mathrm{\prime }}$. Thus $T^{\oplus n}$ commutes with $P$. Since $I\in \mathcal{A}$, the tuple $\xi$ lies in $C$, and consequently $T^{\oplus n}\xi =P{T}^{\oplus n}\xi \in C$. By the definition of closure, one $a\in \mathcal{A}$ approximates $T$ on the whole tuple to any prescribed tolerance; the empty tuple is vacuous. Apply this argument to $\tilde{D}=D+\mathbb{C}I$, with $\tilde{D}=D$ when unital: ${\tilde{D}}^{\mathrm{\prime }}=D^{\mathrm{\prime }}$ and ${\tilde{D}}^{\mathrm{\prime }\mathrm{\prime }}=M$. Given a tuple and $\epsilon >0$, choose $r=d+\lambda I\in \tilde{D}$ with $\mathrm{\parallel }(T-r){\xi }_j\mathrm{\parallel }<\epsilon \mathrm{/}2$. Step 1.1 supplies $e_{\mu }$ with $\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\parallel }(I-e_{\mu }){\xi }_j\mathrm{\parallel }<\epsilon \mathrm{/}2$ for every $j$, so $d+\lambda e_{\mu }\in D$ approximates $T$ on the tuple within $\epsilon$. Thus $D$ is strongly dense in $D^{\mathrm{\prime }\mathrm{\prime }}=M$; the reverse closure inclusion holds because $D^{\mathrm{\prime }\mathrm{\prime }}$ is WOT closed and hence SOT closed.

If $c=c^{\ast }\in M$, step 2.1 gives a net $d_{\alpha }\in D$ with $d_{\alpha }\to c$ in WOT by [F4]. For all $\xi ,\eta \in H$, $⟨d_{\alpha }^{\ast }\xi ,\eta ⟩=\overline{⟨d_{\alpha }\eta ,\xi ⟩}\to \overline{⟨c\eta ,\xi ⟩}=⟨c^{\ast }\xi ,\eta ⟩$, so $d_{\alpha }^{\ast }\to c^{\ast }$ in WOT. Since $D$ is *-closed, $a_{\alpha }=(d_{\alpha }+d_{\alpha }^{\ast })\mathrm{/}2$ lies in $D_{\mathrm{s}\mathrm{a}}$ and converges WOT to $c$. For a finite tuple ${\xi }_1,\dots ,{\xi }_n$, coordinate testing then gives weak convergence of $(a_{\alpha }{\xi }_1,\dots ,a_{\alpha }{\xi }_n)$ to $(c{\xi }_1,\dots ,c{\xi }_n)$ in $H^{\oplus n}$. Thus the real-linear image $\{(a{\xi }_1,\dots ,a{\xi }_n):a\in D_{\mathrm{s}\mathrm{a}}\}$ is convex and the target tuple lies in its weak closure.

By [F5], the norm and weak closures of that convex image agree. Hence for every finite tuple and tolerance there is $a\in D_{\mathrm{s}\mathrm{a}}$ with $\mathrm{\parallel }(a-c){\xi }_j\mathrm{\parallel }<\epsilon$ for all $j$. This proves strong density of $D_{\mathrm{s}\mathrm{a}}$ in $M_{\mathrm{s}\mathrm{a}}$.

Let $g(t)=\max (-1,\min (t,1))$. Choose $R\ge 1$ and a continuous cutoff ${\chi }_R$ equal to $1$ on $[-R,R]$, zero outside $[-2R,2R]$, and between $0$ and $1$, and set $h_R=g{\chi }_R$. Then $h_R\in C_0(\mathbb{R})$, $h_R(0)=0$, and $\mathrm{\mid }g(t)-h_R(t)\mathrm{\mid }\le 2\mathrm{\mid }t\mathrm{\mid }\mathrm{/}R$. For a self-adjoint $y$ and vector $\xi$, [F7] gives $\mathrm{\parallel }(g-h_R)(y)\xi \mathrm{\parallel }\le 2\mathrm{\parallel }y\xi \mathrm{\parallel }\mathrm{/}R$. Fix a finite tuple and a self-adjoint contraction $c\in M$. By step 4.1 choose a net $y_{\alpha }\in D_{\mathrm{s}\mathrm{a}}$ with $y_{\alpha }\to c$ strongly; the finitely many $\mathrm{\parallel }{y}_{\alpha }{\xi }_j\mathrm{\parallel }$ are eventually bounded. First take $R$ large, then $\alpha$ large, and use step 1.2 with $h_R(c)=g(c)=c$ (since $\sigma (c)\subseteq [-1,1]\subseteq [-R,R]$) to obtain $g(y_{\alpha })\to c$ on the tuple. Since $g(0)=0$, [F7] puts $g(y_{\alpha })\in D_{\mathrm{s}\mathrm{a}}$, and $\mathrm{\parallel }g(y_{\alpha })\mathrm{\parallel }\le 1$. Thus the self-adjoint unit ball of $D$ is strongly dense in that of $M$.

For $T\in M$ with $\mathrm{\parallel }T\mathrm{\parallel }\le 1$, on $H\oplus H$ form the self-adjoint contraction $X=\left(\begin{array}{cc}0& T\\ T^{\ast }& 0\end{array}\right)$. Let $D_2$ be the block operators in $\mathcal{B}(H\oplus H)$ with all four entries in $D$. Block operations and adjoints preserve $D_2$, and it is norm closed because each entry is a contractive compression and $D$ is norm closed; it inherits the C*-identity from $\mathcal{B}(H\oplus H)$. The block diagonal $\mathrm{diag}(e_{\lambda },e_{\lambda })$ and step 1.1 show that $D_2$ is nondegenerate. It is strongly dense in $M_2(M):$ for any $\epsilon >0$ and finite tuple ${\zeta }^k=({\xi }_1^k,{\xi }_2^k)$, step 2.1 lets each of the four entries approximate its target block on the corresponding finite coordinate list with error less than $\epsilon \mathrm{/}4$. Each output coordinate error is then less than $\epsilon \mathrm{/}2$, so the direct-sum error is less than $\epsilon \mathrm{/}\sqrt{2}<\epsilon$. The algebra $M_2(M)$ is WOT closed because each block is recovered by a WOT-continuous coordinate compression and $M$ is WOT closed. Let $D_2^{\sim }=D_2$ when unital and $D_2+\mathbb{C}{I}_{H\oplus H}$ otherwise. By [F3], this is a unital C*-algebra between $D_2$ and $M_2(M)$, so it has the same SOT closure $M_2(M)$. Step 2.1 applied to the nondegenerate concrete C*-algebra $D_2$ shows that its SOT closure is $D_2^{\mathrm{\prime }\mathrm{\prime }}$. The preceding density and WOT closedness therefore give $D_2^{\mathrm{\prime }\mathrm{\prime }}=M_2(M)$. Apply step 5.1 to approximate $X$ strongly on vectors $(0,\xi )$ by self-adjoint contractions $Y_{\alpha }=\left(\begin{array}{cc}{a}_{\alpha }& b_{\alpha }\\ b_{\alpha }^{\ast }& d_{\alpha }\end{array}\right)$ in $D_2$. Compression to the upper-right corner gives $\mathrm{\parallel }{b}_{\alpha }\mathrm{\parallel }\le \mathrm{\parallel }{Y}_{\alpha }\mathrm{\parallel }\le 1$ and $b_{\alpha }\xi \to T\xi$ for every fixed $\xi$. Hence the unit ball of $D$ is strongly dense in the unit ball of $M$.

Assume now $M=\mathcal{B}(H)$. The span of the prescribed finite tuple is finite-dimensional, hence closed by [F8]; let $p$ be its orthogonal projection and let $r=r^{\ast }\in \mathcal{B}(H)$ be the target self-adjoint operator. If $p=0$, take $a=0$. Otherwise, the self-adjoint operator $b=prp+(I-p)rp+pr(I-p)$ agrees with $r$ on $pH$ and satisfies $\mathrm{\parallel }b\mathrm{\parallel }\le 3\mathrm{\parallel }rp\mathrm{\parallel }$: each of its three terms has norm at most $\mathrm{\parallel }rp\mathrm{\parallel }$, and the last two are adjoints. If $\mathrm{\parallel }rp\mathrm{\parallel }=0$ choose $a=0$. Otherwise put $L=3\mathrm{\parallel }rp\mathrm{\parallel }>0$. By [F8], choose an orthonormal basis $u_1,\dots ,u_m$ of $pH$, where $m\ge 1$. Apply the self-adjoint unit-ball conclusion of step 5.1 to $b\mathrm{/}L$ on this basis, choosing a self-adjoint contraction $d\in D$ with $\mathrm{\parallel }(d-b\mathrm{/}L)u_k\mathrm{\parallel }<1\mathrm{/}(2L\sqrt{m})$ for each $k$. Set $a_0=0$, $a_1=Ld$ and define $r_0=r$, $r_1:=r-a_1$. Then $a_1\in D_{\mathrm{s}\mathrm{a}}$ and $\mathrm{\parallel }{a}_1\mathrm{\parallel }\le 3\mathrm{\parallel }rp\mathrm{\parallel }$. For any unit $v={\sum }_{k=1}^m{\alpha }_k{u}_k\in pH$, Cauchy–Schwarz gives ${\sum }_k\mathrm{\mid }{\alpha }_k\mathrm{\mid }\le \sqrt{m}$, so $\mathrm{\parallel }(b-a_1)v\mathrm{\parallel }<L\sqrt{m}\mathrm{/}(2L\sqrt{m})=1\mathrm{/}2$; hence $\mathrm{\parallel }{r}_{1}p\mathrm{\parallel }=\mathrm{\parallel }(b-a_1)p\mathrm{\parallel }<1\mathrm{/}2$. Recursively, each residual remains self-adjoint; for $n\ge 1$, if $r_{n}p=0$, set $a_{n+1}=0$ and $r_{n+1}=r_n$. Otherwise put $b_n=p{r}_{n}p+(I-p)r_{n}p+p{r}_n(I-p)$ and $L_n=3\mathrm{\parallel }{r}_{n}p\mathrm{\parallel }$, so $b_n=b_n^{\ast }$, $b_{n}p=r_{n}p$, and $\mathrm{\parallel }{b}_n\mathrm{\parallel }\le L_n$. Use the self-adjoint unit-ball conclusion of step 5.1 on $b_n\mathrm{/}{L}_n$ and the same basis, choosing a self-adjoint contraction $d_n\in D$ with $\mathrm{\parallel }(d_n-b_n\mathrm{/}{L}_n)u_k\mathrm{\parallel }<2^{-n-1}\mathrm{/}(L_n\sqrt{m})$; set $a_{n+1}=L_n{d}_n$ and $r_{n+1}:=r_n-a_{n+1}$. The same coordinate estimate gives $\mathrm{\parallel }{r}_{n+1}p\mathrm{\parallel }=\mathrm{\parallel }(b_n-a_{n+1})p\mathrm{\parallel }<2^{-n-1}$ and $\mathrm{\parallel }{a}_{n+1}\mathrm{\parallel }\le 3\mathrm{\parallel }{r}_{n}p\mathrm{\parallel }$. Thus $\mathrm{\parallel }{r}_{n}p\mathrm{\parallel }<2^{-n}$ for every $n\ge 1$, and ${\sum }_{n\ge 1}\mathrm{\parallel }{a}_{n+1}\mathrm{\parallel }<\mathrm{\infty }$. By [F1] choose this sequence recursively. The norm-convergent sum $a={\sum }_{n\ge 1}{a}_n\in D_{\mathrm{s}\mathrm{a}}$ satisfies $r_{n}p=(r-{\sum }_{j=1}^n{a}_j)p\to 0$, hence $ap=rp$ and realizes the exact prescription.

Let $\varphi ,\psi$ be pure states with inequivalent GNS representations and cyclic unit vectors ${\xi }_{\varphi },{\xi }_{\psi }$. The direct-sum image $\mathrm{\Delta }(A)=\{{\pi }_{\varphi }(a)\oplus {\pi }_{\psi }(a):a\in A\}$ is a concrete C*-algebra by [F9]. Choose a positive contractive approximate unit $(e_{\lambda })$ of $A$ by [F2]. Each GNS representation is nondegenerate [F9]; contractivity [F7] makes ${\pi }_{\varphi }(e_{\lambda })$ and ${\pi }_{\psi }(e_{\lambda })$ approximate units of their image algebras, so step 1.1 gives strong convergence to the identities on their respective carriers. Hence $\mathrm{\Delta }(e_{\lambda })\to I$ strongly and $\mathrm{\Delta }(A)$ is nondegenerate. By [F9] the two pure GNS representations are irreducible. A block operator in its commutant has diagonal blocks in the two scalar commutants and off-diagonal blocks intertwining the two representations; step 1.3 makes the latter zero. Thus its commutant is $\mathbb{C}I\oplus \mathbb{C}I$, so its bicommutant is $\mathcal{B}(H_{\varphi })\oplus \mathcal{B}(H_{\psi })$ and contains $I\oplus (-I)$. Apply step 5.1 to approximate this self-adjoint contraction by self-adjoint contractions $d\in \mathrm{\Delta }(A)$ on $({\xi }_{\varphi },{\xi }_{\psi })$; their expectations approach $1$ and $-1$. Write $d=\mathrm{\Delta }(a)$; then the coset $a+J$, where $J=\ker \mathrm{\Delta }$, has quotient norm at most one. Since $d=d^{\ast }$, $a^{\ast }-a\in J$. By [F9] choose a representative $b$ of this coset with $\mathrm{\parallel }b\mathrm{\parallel }\le 1+\epsilon$; replacing it by $(b+b^{\ast })\mathrm{/}2$ keeps it in the same coset and does not increase its norm. Both states vanish on $J$, so their difference on this self-adjoint representative is the same as on $a$ and approaches $2$. Rescaling it to the unit ball and letting the approximation error and $\epsilon$ tend to zero gives $\mathrm{\parallel }\varphi -\psi \mathrm{\parallel }\ge 2$; the reverse bound follows because both states have norm one.

For the interval clause, include the stated orthonormal eigenvectors in the finite tuple of step 6.2, so its $a$ agrees with $r$ on all of them. Apply to this $a$ the continuous map that clips each real number to the nearest point of $J$ (an infinite endpoint imposes no clipping on that side). This map fixes every point of $J$ and sends $0$ to $0$. Thus $f(0)=0$, $f(a)\in D_{\mathrm{s}\mathrm{a}}$, its spectrum is contained in $J$, and $f(a){\xi }_j=f({\lambda }_j){\xi }_j={\lambda }_j{\xi }_j$ on each selected eigenvector.

Assume $M=\mathcal{B}(H)$. For unit vectors $\xi ,\eta \in H$, choose $z$ as in [F11], so $⟨\xi ,z\eta ⟩$ is real. If $\xi$ and $z\eta$ are collinear, the identity unitary carries one to the other up to phase. Otherwise $\xi +z\eta$ and $\xi -z\eta$ are nonzero orthogonal vectors. The self-adjoint operator $r=\pi P_{\mathbb{C}(\xi -z\eta )}$ has eigenvalues $0$ and $\pi$ on their respective spans. Step 6.2 realizes these values by some $a\in D_{\mathrm{s}\mathrm{a}}$; step 7.1 clips it to $[0,\pi ]$ without changing them. By [F12], its exponential $u=e^{ia}\in U(D^{\sim })$ fixes $\xi +z\eta$ and negates $\xi -z\eta$, so $u\xi =z\eta$. For $D=\pi (A)$, closedness of the image in [F9] gives a self-adjoint preimage $h\in A$ of $a$ by self-adjointizing any preimage; the unital extension $A^{\sim }\to \mathcal{B}(H)$, $b+\lambda 1\mapsto \pi (b)+\lambda I$, is a $\ast$-homomorphism, so its continuous functional calculus sends $e^{ih}$ to $u$.

For orthogonal unit vectors in one irreducible carrier, [F9] and step 1.3 give $D\mathrm{\prime }=\mathbb{C}I$ and hence $D\mathrm{\prime }\mathrm{\prime }=\mathcal{B}(H)$. Apply step 6.2 to the self-adjoint operator with eigenvalues $1$ and $-1$ on those vectors, then step 7.1 with $J=[-1,1]$. The resulting self-adjoint contraction $d\in D$ has vector-state values $1$ and $-1$. The vector functionals on $A$ are states: contractivity gives norm at most one, and a positive contractive approximate unit converges strongly to $I$ by step 1.1, so their norms are at least one. If $D=\pi (A)$, lift $d$ to a self-adjoint representative in $A$ of norm at most $1+\epsilon$ using the quotient norm as in step 6.3; rescaling and letting $\epsilon \downarrow 0$ proves that the two states have norm distance $2$.

If pure states $\varphi ,\psi$ have norm distance less than $2$, step 6.3 shows their GNS representations cannot be inequivalent. By [F9] the GNS representation ${\pi }_{\varphi }$ is irreducible, so step 1.3 gives ${\pi }_{\varphi }(A{)}^{\mathrm{\prime }}=\mathbb{C}I$ and ${\pi }_{\varphi }(A{)}^{\mathrm{\prime }\mathrm{\prime }}=\mathcal{B}(H_{\varphi })$. Let $U$ be a unitary intertwiner and put $\eta =U^{\ast }{\xi }_{\psi }$ in the carrier of ${\pi }_{\varphi }$; then $\psi$ is the vector state of $\eta$. The construction of step 8.1 gives a self-adjoint $h\in {\pi }_{\varphi }(A)$ whose exponential sends ${\xi }_{\varphi }$ to a phase multiple of $\eta$. By [F9], ${\pi }_{\varphi }(A)$ is closed; choose a preimage $b\in A$ of $h$ and replace it by its self-adjoint part $b_{\mathrm{s}\mathrm{a}}$. The representation extends to the minimal unitization by ${\pi }_{\varphi }^{\sim }(a+\lambda 1)={\pi }_{\varphi }(a)+\lambda I$ (with the unital case unchanged); this is a unital $\ast$-homomorphism, so [F7] gives ${\pi }_{\varphi }^{\sim }(e^{i{b}_{\mathrm{s}\mathrm{a}}})=e^{ih}$. Thus $v=e^{i{b}_{\mathrm{s}\mathrm{a}}}\in U(A^{\sim })$ implements the same vector transport. The phase cancels in a vector state, giving $\psi =\varphi \circ \mathrm{Ad}(v^{\ast })$.