An irrational rotation algebra 
, for an irrational real number 
, is the universal unital C*-algebra
generated by unitaries 
with
 |
Its canonical tracial state takes value zero on 
unless 
.
In the associated tracial representation, its closure as a von
Neumann algebra 
is the hyperfinite type 
factor. The C*-algebra 
and its closure as a von Neumann algebra are
different objects. For a continuous function 
on the unit circle, the operator 
in 
is an example relevant to the invariant
subspace problem with the projection restricted to 
.
