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