The modular automorphism group of a von Neumann algebra 
with a faithful normal state 
is the one-parameter group
 |
where 
is the modular operator in the standard Hilbert-space representation. It is constructed
from the polar decomposition of the closure
of 
.
The fixed algebra 
is the centralizer of the state. For a faithful tracial
state, the modular automorphism group is trivial. For 
and 
with positive invertible 
of trace 1, it is 
. The continuous
core retains this modular action by forming a crossed
product.
