Let $\mathscr{A}$ be a W*-algebra.

A W*-algebra $\mathscr{A}$ is a C*-algebra which has a predual, i.e. there exists a Banach space $\mathscr{A}_*$ such that $\mathscr{A}$ is isometrically isomorphic to the topological dual of $\mathscr{A}_*$.

It is well known that every W*-algebra is isomorphic to a fitting von Neumann algebra in the usual sense. However, in the upper more abstract setting, is there any way to (canonically) identify the predual $\mathscr{A}_*$ with a subset of $\mathscr{A}$? This is the case for the von Neumann algebra $\mathcal{B(H)}$ where the Banach space of trace class operators can be chosen as the predual, but is a similar choice possible for general W*-algebras? If so, how?


No, this is not true in general. In fact if the von Neumann algebra, $M$, is finite von Neumann algebra then the predual, denoted $L^1(M)$ contains $M$ in a canonical way, via the map $x\rightarrow \tau(\cdot x)$ for $x\in M$. Here $\tau(\cdot)$ is a faithful trace.

  • $\begingroup$ You seem to claim that an embedding $M\hookrightarrow L^1(M)$ precludes the reverse embedding. That's not obvious to me; do you have an argument? $\endgroup$ – Martin Argerami Aug 9 '17 at 16:01

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