For a von Neumann algebra $\mathcal{A} \subseteq \mathcal{B(H)}$ where $\mathcal{B(H)}$ is the space of all bounded linear operators on the Hilbert space $\mathcal{H}$, there is a Banach space $ \mathcal{A}_∗$, called the predual of $\mathcal{A}$, such that the Banach dual of $\mathcal{A}_*$ coincides with $\mathcal{A}$ with norm topology, whereas the weak-$∗$ topology coincides with the ultra-weak topology of $\mathcal{A}$. In other words, predual of a von Neumann algebra is the Banach space of all ultraweakly continuous linear functionals on $\mathcal{A}$ In fact, Sakai showed that a von Neumann algebra can be characterized in the class of $C^∗$-algebras by this property of having predual.

Since there is an isometrically isomorphism, $\mathcal{B_1(H)^*} \cong \mathcal{B(H)}$ under the map $A \longrightarrow tr(.A)$. Predual of $\mathcal{B(H)}$ is $\mathcal{B_1(H)}$, the space of trace-class operators. That is ultra-weak topology on $\mathcal{B(H)}$ is the topology such that all the trace class operators are continuous as seminorms. For a von Neumann algebra $\mathcal{A} \subseteq \mathcal{B(H)}$ ultra-weak topology on $\mathcal{A}$ can be seen as subspace- topology induced from $\mathcal{B(H)}$. But I am rather interested in explicit description of ultra-weak topology in terms of semi-norms, that is the Banach space of all ultraweakly continuous linear functionals on $\mathcal{A}$. If there is some isomorphism between predual $\mathcal{A}_*$ and Banach space containing the space of trace-class operators $\mathcal{B_1(H)}$?

A Course in Functional Analysisby Conway. $\endgroup$ – Nik Weaver May 14 '14 at 23:42Measure Theory and Functional Analysistoo ... $\endgroup$ – Nik Weaver May 14 '14 at 23:42