A predual of a Banach space 
is a Banach space 
together with an isometric identification of 
with the continuous dual 
.
The identification defines the weak-star topology
on 
by pointwise
convergence on 
. A predual need not exist for
every Banach space, and a Banach
space can admit different preduals. A von Neumann
algebra 
has its canonical predual 
of normal linear functionals. For 
, trace pairing identifies
this predual with matrices carrying the trace norm.
Separable predual is the countability assumption used in the continuous-core and
bicentralizer statements.
Predual
See also
Bicentralizer, Continuous Core, Dual Normed Space, von Neumann Algebra, Weak-Star TopologyExplore with Wolfram|Alpha
References
Anantharaman, C. and Popa, S. An Introduction to
Factors. https://www.math.ucla.edu/~popa/Books/IIun.pdf.
Cite this as:
Weisstein, Eric W. "Predual." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Predual.html