191 lines
13 KiB
TeX
191 lines
13 KiB
TeX
\section{Traces}
|
|
\label{section:tracial-von-neumann-algebra}
|
|
|
|
\begin{definition}[Trace]
|
|
\label{definition:trace}
|
|
Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\tau: M_+ \to [0, \infty]$, then $\tau$ is a \textbf{trace} on $M$ if:
|
|
\begin{enumerate}
|
|
\item For each $x, y \in M_+$ and $\lambda \ge 0$, $\tau(\lambda x + y) = \lambda \tau(x) + \tau(y)$.
|
|
\item For each $x \in M_+$ and $u \in U(M)$, $\tau(uxu^*) = \tau(x)$.
|
|
\end{enumerate}
|
|
\end{definition}
|
|
|
|
\begin{definition}[Tracial von Neumann Algebra]
|
|
\label{definition:tracial-von-neumann-algebra}
|
|
Let $M$ be a von Neumann algebra and $\tau: M \to \complex$ be a faithful tracial state, then $(M, \tau)$ is a \textbf{tracial von Neumann algebra} if $\tau$ is normal.
|
|
\end{definition}
|
|
|
|
\begin{definition}[Standard Representation]
|
|
\label{definition:tracial-vna-standard}
|
|
Let $(M, \tau)$ be a tracial von Neumann algebra, then its \hyperref[GNS]{definition:gns-triple} representation on $L^2(M, \tau)$ is the \textbf{standard representation} of $(M, \tau)$.
|
|
\end{definition}
|
|
|
|
\begin{proposition}
|
|
\label{proposition:cstar-trace-vna}
|
|
Let $M$ be a unital $C^*$-algebra, $\tau: M \to \complex$ be a faithful tracial state, and $(L^2(M, \tau), \pi, 1_M)$ be its \hyperref[GNS]{definition:gns-triple} triple, then:
|
|
\begin{enumerate}
|
|
\item The uniform structure of pointwise convergence\footnote{Strong operator topology.} on $\ol{B_M(0, 1)} \subset B(L^2(M, \tau))$ coincides with the uniform structure of $\ol{B_M(0, 1)} \subset L^2(M, \tau)$ induced by the $L^2(M, \tau)$ norm.
|
|
\item $\pi(M) \subset B(L^2(M, \tau))$ is a von Neumann algebra if and only if $\ol{B_M(0, 1)}$ is complete in $L^2(M, \tau)$. In which case, $\tau$ is normal.
|
|
\end{enumerate}
|
|
\end{proposition}
|
|
\begin{proof}[Proof, {{\cite[Proposition 2.6.4]{AnantharamanPopa}}}. ]
|
|
(1): For any $x, y \in \ol{B_M(0, 1)}$,
|
|
\begin{align*}
|
|
\norm{x - y}_{L^2(M, \tau)}^2 &= \tau((x - y)^*(x - y)) = \dpn{\pi(x - y)^*\pi(x - y)1_M, 1_M}{L^2(M, \tau)} \\
|
|
&= \dpn{\pi(x - y)1_M, \pi(x - y)1_M}{L^2(M, \tau)} = \norm{\pi(x - y)1_M}_{L^2(M, \tau)}^2
|
|
\end{align*}
|
|
|
|
so the uniform structure of pointwise convergence is stronger than the uniformity induced by the $L^2(M, \tau)$ norm.
|
|
|
|
On the other hand, by the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the uniform structure of pointwise convergence on $\ol{B_M(0, 1)}$ coincides with the uniform structure of pointwise convergence on any dense subset of $L^2(M, \tau)$. In particular, this includes $M \subset L^2(M, \tau)$. For each $x, y \in \ol{B_M(0, 1)}$ and $t \in M$, since $\tau$ is tracial,
|
|
\begin{align*}
|
|
\norm{(x - y)t}_{L^2(M, \tau)}^2 &= \tau(t^*(x - y)^*(x - y)t) \le \tau(tt^*(x - y)^*(x - y)) \\
|
|
&\le \norm{t}_M^2 \tau((x - y)^*(x - y)) = \norm{t}_M^2 \norm{x - y}_{L^2(M, \tau)}^2
|
|
\end{align*}
|
|
|
|
so the uniformity induced by the $L^2(M, \tau)$ norm is stronger than the uniform structure of pointwise convergence.
|
|
|
|
(2): Using (1), it is sufficient to show that $\ol{B_M(0, 1)} \subset B(L^2(M, \tau))$ is complete if and only if it is strong-operator closed. By \autoref{proposition:complete-closed}, $\ol{B_M(0, 1)}$ is complete if and only if it is closed in $L^2(M, \tau)^{L^2(M, \tau)}$ with respect to the product topology. Finally, the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} implies that $\ol{B_M(0, 1)}$ is closed in $L^2(M, \tau)^{L^2(M, \tau)}$ if and only if it is strong-operator closed in $B(L^2(M, \tau))$.
|
|
\end{proof}
|
|
|
|
\begin{proposition}
|
|
\label{proposition:tracial-separable-condition}
|
|
Let $(M, \tau)$ be a tracial von Neumann algebra, then the following are equivalent:
|
|
\begin{enumerate}
|
|
\item $M$ is separable.
|
|
\item $\ol{B_M(0, 1)}$ is strong-operator separable.
|
|
\item $\ol{B_M(0, 1)}$ is separable in $L^2(M, \tau)$.
|
|
\item $L^2(M, \tau)$ is separable.
|
|
\end{enumerate}
|
|
\end{proposition}
|
|
\begin{proof}
|
|
(1) $\Rightarrow$ (2): Let $H$ be a separable Hilbert space with $M \subset B(H)$ and $D \subset H$ be countable and dense. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the strong operator topology on $\ol{B_M(0, 1)}$ coincides with the topology of pointwise convergence on $D$. By \autoref{theorem:uniform-metrisable} and \autoref{proposition:separable-product}, $H^D$ is metrisable and separable. Therefore $\ol{B_M(0, 1)} \subset H^D$ is also metrisable and separable.
|
|
|
|
(2) $\Rightarrow$ (3): By \autoref{proposition:cstar-trace-vna}.
|
|
|
|
(3) $\Rightarrow$ (4): By \autoref{lemma:tracial-von-neumann-gns}, the standard representation $(L^2(M, \tau), \pi, 1_M)$ is faithful and normal. As $M$ is dense in $L^2(M, \tau)$, $\ol{B_M(0, 1)}$ being separable in $L^2(M, \tau)$ implies that $L^2(M, \tau)$ is separable.
|
|
\end{proof}
|
|
|
|
\begin{proposition}
|
|
\label{proposition:tracial-von-neumann-factor}
|
|
Let $(M, \tau)$ be a tracial von Neumann factor, then exactly one of the following holds:
|
|
\begin{enumerate}
|
|
\item There exists $n \in \natp$ such that $M \iso M_n(\complex)$.
|
|
\item $M$ is of type $\vnII_1$.
|
|
\end{enumerate}
|
|
\end{proposition}
|
|
\begin{proof}
|
|
(1): Suppose that $M$ has a minimal projection. By \autoref{theorem:type1-bh}, there exists a complex Hilbert space $H$ such that $M \iso B(H)$. Let $\seqi{e} \subset H$ be an orthonormal basis, then $e_i \otimes e_i \in \text{Proj}(M)$ is minimal for all $i \in I$, and all such rank one projections are equivalent. As such, for any $j \in I$,
|
|
\[
|
|
\tau(1_M) = \sum_{i \in I}\tau(e_i \otimes e_i) = |I| \cdot \tau(e_j \otimes e_j)
|
|
\]
|
|
|
|
so $I$ must be finite, and $M \iso M_{|I|}(\complex)$.
|
|
|
|
(2): Suppose that $M$ has no minimal projections. By \autoref{lemma:abelian-minimal-factor}, $M$ has no abelian projections. As $\tau$ is a faithful tracial state, $1_M$ is finite. Therefore $M$ is a $\vnII_1$ factor.
|
|
\end{proof}
|
|
|
|
\begin{lemma}
|
|
\label{lemma:diffuse-factor-subsplit}
|
|
Let $M$ be a diffuse factor, $p \in \text{Proj}(M)$ be a non-zero projection, then there exist non-zero projections $p_1, p_2 \in \text{Proj}(M)$ with $p_1 \sim p_2$ and $p_1 + p_2 \le p$.
|
|
\end{lemma}
|
|
\begin{proof}[Proof, {{\cite[Lemma 4.1.1]{AnantharamanPopa}}}. ]
|
|
Since $p$ is not minimal, there exists a projection $q \le p$ with $0 \ne q \ne p$. As $M$ is a factor, $Z(q)Z(p - q) = 1_M$. By \autoref{proposition:central-support-mvn}, there exist non-zero projections $p_1, p_2 \in \text{Proj}(M)$ with $p_1 \sim p_2$, $p_1 \le q$, and $p_2 \le p - q$.
|
|
\end{proof}
|
|
|
|
\begin{lemma}
|
|
\label{lemma:diffuse-factor-split}
|
|
Let $M$ be a diffuse factor, $p \in \text{Proj}(M)$ be a non-zero projection, then there exist $p_1, p_2 \in \text{Proj}(M)$ with $p_1 \sim p_2$ and $p_1 + p_2 = p$.
|
|
\end{lemma}
|
|
\begin{proof}[Proof, {{\cite[Lemma 4.1.1]{AnantharamanPopa}}}. ]
|
|
Let
|
|
\[
|
|
\mathcal{F} = \bracsn{(q, r) \in \text{Proj}(M)^2|q \sim r, q \perp r, q + r \le p}
|
|
\]
|
|
|
|
and $\mathcal{M} \subset \mathcal{F}$ be a maximal family such that $\bracsn{q, r|(q,r) \in \mathcal{M}}$ are all mutually orthogonal. Let $q = \sum_{(q, r) \in \mathcal{M}}q$ and $r = \sum_{(q, r) \in \mathcal{M}}r$, then by \autoref{lemma:mvn-equivalent-direct-sum}, $q \sim r$, $q \perp r$, and $q + r \le p$.
|
|
|
|
If $q + r \ne p$, then by \autoref{lemma:diffuse-factor-subsplit} applied to $p - (q + r)$, there exist non-zero projections $q', r' \in \text{Proj}(M)$ with $q' \sim r'$, $q' \perp r'$, and $q' + r' \le p - (q + r)$, which contradicts the maximality of $\mathcal{M}$.
|
|
\end{proof}
|
|
|
|
\begin{corollary}
|
|
\label{corollary:diffuse-factor-dyadic}
|
|
Let $M$ be a diffuse factor, then for each $n \in \natp$, there exist mutually orthogonal equivalent projections $\bracsn{p_j}_1^{2^n} \subset \text{Proj}(M)$ such that $\sum_{j = 1}^{2^n}p_j = 1$. If $\tau: M \to \complex$ is a tracial state, then $\tau(p_j) = 2^{-n}$ for all $1 \le j \le 2^n$.
|
|
\end{corollary}
|
|
\begin{proof}
|
|
By \autoref{lemma:diffuse-factor-split}.
|
|
\end{proof}
|
|
|
|
\begin{proposition}
|
|
\label{proposition:tracial-factor-simple}
|
|
Let $M$ be a factor, then $M$ admits at most one tracial state.
|
|
\end{proposition}
|
|
\begin{proof}[Proof, {{\cite[Proposition 4.1.3]{AnantharamanPopa}}}. ]
|
|
Using \autoref{proposition:tracial-von-neumann-factor}, assume without loss of generality that $M$ is diffuse.
|
|
|
|
Let $p \in \text{Proj}(M)$ with $0 \le p < 1_M$. Let $n \in \natp$, then by \autoref{corollary:diffuse-factor-dyadic}, there exist mutually orthogonal equivalent projections $\bracsn{p_j}_1^{2^n} \subset \text{Proj}(M)$ such that $\sum_{j = 1}^{2^n}p_j = 1$. By the \hyperref[corollary:vna-factor-comparability]{theorem:vna-comparability}, there exists a unique $0 \le k \le 2^n - 1$ such that
|
|
\[
|
|
\sum_{j = 1}^k p_j \preceq p \prec \sum_{j = 1}^{k+1}p_j
|
|
\]
|
|
|
|
As such, for any tracial state $\tau: M \to \complex$,
|
|
\[
|
|
\frac{k}{2^n} \le \tau(p) \le \frac{k+1}{2^n}
|
|
\]
|
|
|
|
so $\tau(p)$ is independent of the choice of $\tau$. By \autoref{theorem:vn-projection-norm-dense}, $\tau$ is unique.
|
|
\end{proof}
|
|
|
|
\begin{proposition}
|
|
\label{proposition:tracial-factor-unique-trace}
|
|
Let $(M, \tau)$ be a tracial von Neumann algebra, then $M$ is a factor if and only if $\tau$ is the unique normal faithful tracial state on $M$.
|
|
\end{proposition}
|
|
\begin{proof}[Proof, {{\cite[Proposition 4.1.4]{AnantharamanPopa}}}. ]
|
|
If $M$ is a factor, then $\tau$ is unique by \autoref{proposition:tracial-factor-simple}. Suppose that $M$ is not a factor, then $M$ admits a non-trivial central projection $z \in Z(M)$. In which case, for any $\alpha \in (0, 1)$,
|
|
\[
|
|
\tilde \tau(x) = \frac{\alpha}{\tau(z)} \tau(xz) + \frac{1 - \alpha}{1 - \tau(z)}\tau(x(1 - z))
|
|
\]
|
|
|
|
is also a normal faithful tracial state. In particular,
|
|
\[
|
|
\tilde \tau(z) = \frac{\alpha}{\tau(z)} \tau(z) + \frac{1 - \alpha}{1 - \tau(z)}\tau(z(1 - z)) = \alpha
|
|
\]
|
|
|
|
so if $\alpha \ne \tau(z)$, then $\tilde \tau \ne \tau$.
|
|
\end{proof}
|
|
|
|
\begin{proposition}
|
|
\label{proposition:tracial-factor-calkin}
|
|
Let $(M, \tau)$ be a tracial factor, then $M$ admits no non-trivial two-sided ideal.
|
|
\end{proposition}
|
|
\begin{proof}[Proof, {{\cite[Proposition 4.1.5]{AnantharamanPopa}}}. ]
|
|
Let $I \subset M$ be a non-zero two-sided ideal. By \autoref{proposition:vna-two-sided-ideal}, $I$ admits a non-zero projection $e \in I$. Let $\mathcal{P} \subset \text{Proj}(M)$ be a maximal family of mutually orthogonal projections equivalent to $e$. Since $\tau$ is faithful, $\tau(p) > 0$ for all $p \in \mathcal{P}$, so $\mathcal{P}$ is finite.
|
|
|
|
Let $q = 1_M - \sum_{p \in \mathcal{P}}p$, then by the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $q \preceq e$ or $e \prec q$. If $e \prec q$, then there exists a subprojection $p \le q$ with $e \sim p$, which contradicts the maximality of $\mathcal{P}$. Thus $q \preceq e$.
|
|
|
|
Let $\mathcal{Q} = \mathcal{P} \sqcup \bracsn{q}$, then $p \preceq e$ for all $p \in \mathcal{Q}$ and $1_M = \sum_{p \in \mathcal{Q}}p$. For each $p \in \mathcal{Q}$, let $v_p \in M$ be a partial isometry such that $v_pv_p^* = p$ and $v_p^*v_p \le e$. As such, $p = v_pev_p^* \in I$ for all $p \in \mathcal{Q}$. Therefore $1_M = \sum_{p \in \mathcal{Q}}p \in I$, and $I = M$.
|
|
\end{proof}
|
|
|
|
\begin{proposition}
|
|
\label{proposition:ii1-trace-projection-equivalence-classes}
|
|
Let $M$ be a $\vnII_1$ factor with tracial normal state $\tau: M \to \complex$, then the mapping $p \mapsto \tau(p)$ induces a bijection from the equivalence classes of $\text{Proj}(M)$ to $[0, 1]$.
|
|
\end{proposition}
|
|
\begin{proof}[Proof, {{\cite[Proposition 4.1.6]{AnantharamanPopa}}}. ]
|
|
(Injective): Let $p, q \in \text{Proj}(M)$ with $\tau(p) = \tau(q)$. By the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $p \prec q$, $p \sim q$, or $q \prec p$. Since $\tau$ is faithful, $p \prec q$ implies that $\tau(p) < \tau(q)$, and $q \prec p$ implies that $\tau(q) < \tau(p)$. Therefore the only possibility is $p \sim q$.
|
|
|
|
(Surjective): By \autoref{lemma:diffuse-factor-split}, there exists a family $\bracsn{p_r|r \in \mathbb D \cap [0, 1]} \subset \text{Proj}(M)$ such that:
|
|
\begin{enumerate}[label=(\roman*)]
|
|
\item For each dyadic rational number $r \in \mathbb D \cap [0, 1]$, $\tau(p_r) = r$.
|
|
\item For each $r, s \in \mathbb D \cap [0, 1]$ with $r \le s$, $p_r \le p_s$.
|
|
\end{enumerate}
|
|
|
|
For each $t \in [0, 1]$, let $p_t = \sup\bracsn{p_r|r \in \mathbb D \cap [0, 1], r \le t}$, then since $\tau$ is normal,
|
|
\begin{align*}
|
|
\tau(p_t) &= \tau(\sup\bracsn{p_r|r \in \mathbb D \cap [0, 1], r \le t}) \\
|
|
&= \sup\bracsn{\tau(p_r)|r \in \mathbb D \cap [0, 1], r \le t} = t
|
|
\end{align*}
|
|
|
|
|
|
\end{proof}
|
|
|