Trace-class quotient predual and normal functionals

For every $x\in M$, the functional $T\mapsto \mathrm{Tr}(Tx)$ on ${\mathcal{S}}_1(H)$ is bounded by $\mathrm{\mid }\mathrm{Tr}(Tx)\mathrm{\mid }\le \mathrm{\parallel }T{\mathrm{\parallel }}_1\mathrm{\parallel }x\mathrm{\parallel }$, using [F1]. Thus its common kernel over $x\in M$ is a closed linear subspace, namely $M_{\perp }$. If $S\in M_{\perp }$, then $\iota ([T])=\iota ([T+S])$; taking the infimum over representatives in the trace-norm estimate shows $\mathrm{\parallel }\iota ([T]){\mathrm{\parallel }}_{M^{\ast }}\le \mathrm{\parallel }[T]{\mathrm{\parallel }}_Q$. Its kernel is zero by the definition of $M_{\perp }$. The quotient $Q={\mathcal{S}}_1(H)\mathrm{/}{M}_{\perp }$ is Banach by [F1] and [F3]. The formula $\mathrm{\Psi }(x)([T])=\mathrm{Tr}(Tx)$ is independent of the representative because every element of $M_{\perp }$ annihilates $x$.

By [F3], $Q^{\ast }$ is isometrically identified with $M_{\perp }^{\perp }\subseteq {\mathcal{S}}_1(H{)}^{\ast }$. By [F2], ${\mathcal{S}}_1(H{)}^{\ast }$ is isometrically $\mathcal{B}(H)$ under the trace pairing. Under this identification, $M_{\perp }^{\perp }$ consists exactly of the bounded operators $X$ satisfying $\mathrm{Tr}(TX)=0$ for all $T\in M_{\perp }$. It remains to identify this double annihilator with $M$.

Every $x\in M$ belongs to $M_{\perp }^{\perp }$ by the definition of $M_{\perp }$. Conversely, let $X\in \mathcal{B}(H)\setminus M$. By [F6], there is a basic weak-operator neighborhood of $X$, defined by finitely many pairs $({\xi }_j,{\eta }_j)$ and tolerances, that is disjoint from $M$. The coordinate map $\theta (Y)=(⟨Y{\xi }_j,{\eta }_j⟩{)}_{j=1}^m$ has $\theta (X)\mathrm{\notin }\theta (M)$; $\theta (M)$ is a linear subspace of ${\mathbb{C}}^m$ and hence is closed. Finite-dimensional linear algebra supplies a linear functional $\mathrm{\ell }(z)={\sum }_{j=1}^m{c}_j{z}_j$ that vanishes on $\theta (M)$ but not on $\theta (X)$. Set $T={\sum }_{j=1}^m{c}_j{R}_{{\xi }_j,{\eta }_j}$, where $R_{\xi ,\eta }z=⟨z,\eta ⟩\xi$. Then $T$ is finite-rank trace class and $\mathrm{Tr}(TY)=\mathrm{\ell }(\theta (Y))$ for every $Y\in \mathcal{B}(H)$. Thus $T\in M_{\perp }$ but $\mathrm{Tr}(TX)\mathrm{\ne }0$, so $X\mathrm{\notin }{M}_{\perp }^{\perp }$. Therefore $M_{\perp }^{\perp }=M$.

Steps 1.2 and 1.3 identify the canonical map $\mathrm{\Psi }$ with the isometric duality identification of [F2]–[F3], so $\mathrm{\Psi }$ is an isometric bijection $M\to Q^{\ast }$. The Banach–Alaoglu theorem [F5] applied to $Q$ then makes the closed dual unit ball of $Q^{\ast }$ weak-star compact.

Let $(x_i)$ be a bounded increasing positive net in $M$ with supremum $x$. By [F4], $x_i\to x$ strongly; choose $C\ge 0$ with $\mathrm{\parallel }{x}_i\mathrm{\parallel }\le C$ for every $i$. For every $\xi \in H$, strong convergence gives $\mathrm{\parallel }x\xi \mathrm{\parallel }={\lim }_i\mathrm{\parallel }{x}_i\xi \mathrm{\parallel }\le C\mathrm{\parallel }\xi \mathrm{\parallel }$, so $\mathrm{\parallel }x\mathrm{\parallel }\le C$. For each $T\in {\mathcal{S}}_1(H)$, choose finite-rank $T_n\to T$ in trace norm by [F2]. Strong convergence gives $\mathrm{Tr}(T_n{x}_i)\to \mathrm{Tr}(T_{n}x)$ for each fixed $n$, since $T_n$ is a finite sum of rank-one operators. Uniformly in $i$, $\mathrm{\mid }\mathrm{Tr}((T-T_n)x_i)\mathrm{\mid }\le C\mathrm{\parallel }T-T_n{\mathrm{\parallel }}_1,\mathrm{\mid }\mathrm{Tr}((T-T_n)x)\mathrm{\mid }\le C\mathrm{\parallel }T-T_n{\mathrm{\parallel }}_1\mathrm{.}$ Given $\epsilon >0$, first fix $n$ so that the sum of these two uniform errors is less than $\epsilon \mathrm{/}2$, then choose a net index after which the finite-rank trace difference is less than $\epsilon \mathrm{/}2$. This proves $\mathrm{Tr}(T{x}_i)\to \mathrm{Tr}(Tx)$. Hence every $\varphi =[T]\in Q$ satisfies $\iota (\varphi )(x_i)\to \iota (\varphi )(x)$. If $\varphi$ is positive, then $\iota (\varphi )(x_i)$ increases and converges to $\iota (\varphi )(x)$, so $\iota (\varphi )(x)={\sup }_i\iota (\varphi )(x_i)$. In particular, for $\xi \in H$ the vector functional is represented by $[R_{\xi ,\xi }]$, since $\iota ([R_{\xi ,\xi }])(x)=\mathrm{Tr}(R_{\xi ,\xi }x)=⟨x\xi ,\xi ⟩$; it is positive and therefore normal as well.

Steps 1.1–1.3 and 2.1–2.2 prove the quotient predual, its weak-star compact dual ball, normality of positive functionals in the quotient, and normality of vector functionals. The positive-functional normality proof uses trace-norm approximation by finite-rank operators; it does not assume a positive trace-class lift for every positive functional on $M$.