<s>
In	O
mathematics	O
,	O
and	O
more	O
specifically	O
in	O
the	O
theory	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
,	O
the	O
noncommutative	O
tori	O
Aθ	O
,	O
also	O
known	O
as	O
irrational	O
rotation	O
algebras	O
for	O
irrational	O
values	O
of	O
θ	O
,	O
form	O
a	O
family	O
of	O
noncommutative	O
C*	B-Algorithm
-algebras	I-Algorithm
which	O
generalize	O
the	O
algebra	O
of	O
continuous	O
functions	O
on	O
the	O
2-torus	O
.	O
</s>
<s>
For	O
any	O
real	O
number	O
θ	O
,	O
the	O
noncommutative	B-Algorithm
torus	I-Algorithm
is	O
the	O
C*	O
-subalgebra	O
of	O
,	O
the	O
algebra	O
of	O
bounded	O
linear	O
operators	O
of	O
square-integrable	B-Algorithm
functions	I-Algorithm
on	O
the	O
unit	O
circle	O
,	O
generated	O
by	O
two	O
unitary	B-Algorithm
operators	I-Algorithm
where	O
is	O
the	O
parametrization	O
of	O
the	O
point	O
on	O
the	O
circle	O
in	O
.	O
</s>
<s>
Universal	O
property	O
:	O
Aθ	O
can	O
be	O
defined	O
(	O
up	O
to	O
isomorphism	O
)	O
as	O
the	O
universal	O
C*	B-Algorithm
-algebra	I-Algorithm
generated	O
by	O
two	O
unitary	B-Algorithm
elements	O
U	O
and	O
V	O
satisfying	O
the	O
relation	O
VU	O
=	O
e2π	O
i	O
θUV	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
noncommutative	O
tori	O
with	O
θ	O
irrational	O
are	O
simple	O
C*	B-Algorithm
-algebras	I-Algorithm
.	O
</s>
