Combining Elliott, Gong, Lin and Niu’s result and Castillejos and Evington’s result, we see that if
is a simple separable nuclear
monotracial
-algebra,
then
is
isomorphic to
,
where
is the
Razak–Jacelon algebra. In this paper, we give another proof of this. In particular, we show that if
is a simple separable
nuclear monotracial
-stable
-algebra which
is
-equivalent
to
, then
is isomorphic to
without considering tracial
approximations of
-algebras
with finite nuclear dimension. Our proof is based on Matui and Sato’s technique, Schafhauser’s
idea in his proof of the Tikuisis–White–Winter theorem and properties of Kirchberg’s central
sequence
-algebra
of . Note that
some results for
are based on Elliott, Gong, Lin and Niu’s stable uniqueness theorem. Also, we characterize
by using
properties of
.
Indeed, we show that a simple separable nuclear monotracial
-algebra
is isomorphic
to
if and
only if
satisfies the following properties:
For any
,
there exists a projection
in
such that
.
If
and
are projections in
such that
,
then
is Murray–von Neumann equivalent to
.
