<s>
The	O
concept	O
of	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
is	O
used	O
in	O
mathematics	O
,	O
particularly	O
in	O
algebra	O
and	O
analysis	O
,	O
both	O
within	O
the	O
context	O
of	O
the	O
theory	O
of	O
group	O
representations	O
.	O
</s>
<s>
The	O
simplest	O
case	O
,	O
and	O
the	O
context	O
in	O
which	O
the	O
idea	O
was	O
first	O
noticed	O
,	O
is	O
that	O
of	O
finite	O
groups	O
(	O
see	O
primitive	B-Algorithm
permutation	I-Algorithm
group	I-Algorithm
)	O
.	O
</s>
<s>
The	O
structure	O
,	O
combinatorial	O
in	O
this	O
case	O
,	O
respected	O
by	O
translation	O
shows	O
that	O
either	O
K	O
is	O
a	O
maximal	O
subgroup	O
of	O
G	O
,	O
or	O
there	O
is	O
a	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
(	O
roughly	O
,	O
a	O
lack	O
of	O
full	O
"	O
mixing	O
"	O
)	O
.	O
</s>
<s>
In	O
order	O
to	O
generalise	O
this	O
to	O
other	O
cases	O
,	O
the	O
concept	O
is	O
re-expressed	O
:	O
first	O
in	O
terms	O
of	O
functions	O
on	O
G	O
constant	O
on	O
K-cosets	O
,	O
and	O
then	O
in	O
terms	O
of	O
projection	B-Algorithm
operators	I-Algorithm
(	O
for	O
example	O
the	O
averaging	O
over	O
K-cosets	O
of	O
elements	O
of	O
the	O
group	O
algebra	O
)	O
.	O
</s>
<s>
Then	O
(	O
U	O
,	O
X	O
)	O
is	O
a	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
for	O
G	O
.	O
</s>
<s>
If	O
the	O
action	O
of	O
G	O
on	O
the	O
elements	O
of	O
X	O
is	O
transitive	O
,	O
then	O
we	O
say	O
this	O
is	O
a	O
transitive	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
.	O
</s>
<s>
As	O
it	O
turns	O
out	O
,	O
a	O
naïve	O
approach	O
based	O
on	O
subspaces	O
of	O
H	O
will	O
not	O
work	O
;	O
for	O
example	O
the	O
translation	O
representation	O
of	O
R	O
on	O
L2(R )	O
has	O
no	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
in	O
this	O
sense	O
.	O
</s>
<s>
The	O
right	O
formulation	O
of	O
direct	O
sum	O
decomposition	O
is	O
formulated	O
in	O
terms	O
of	O
projection-valued	O
measures	O
.	O
</s>
<s>
A	O
projection-valued	O
measure	O
π	O
on	O
the	O
Borel	O
sets	O
of	O
X	O
with	O
values	O
in	O
the	O
projections	B-Algorithm
of	O
H	O
;	O
</s>
<s>
Then	O
(	O
U	O
,	O
π	O
)	O
is	O
a	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
of	O
(	O
G	O
,	O
X	O
)	O
on	O
L2μ(X )	O
.	O
</s>
<s>
This	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
is	O
sometimes	O
called	O
the	O
Koopman	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
.	O
</s>
<s>
A	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
is	O
homogeneous	O
of	O
multiplicity	O
n	O
,	O
where	O
1	O
≤	O
n	O
≤	O
ω	O
if	O
and	O
only	O
if	O
the	O
corresponding	O
projection-valued	O
measure	O
π	O
on	O
X	O
is	O
homogeneous	O
of	O
multiplicity	O
n	O
.	O
In	O
fact	O
,	O
X	O
breaks	O
up	O
into	O
a	O
countable	O
disjoint	O
family	O
 { Xn } 	O
1	O
≤	O
n	O
≤	O
ω	O
of	O
Borel	O
sets	O
such	O
that	O
π	O
is	O
homogeneous	O
of	O
multiplicity	O
n	O
on	O
Xn	O
.	O
</s>
<s>
Any	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
is	O
an	O
orthogonal	O
direct	O
sum	O
of	O
homogeneous	O
ones	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
if	O
the	O
action	O
of	O
G	O
on	O
X	O
is	O
transitive	O
,	O
then	O
any	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
on	O
X	O
is	O
homogeneous	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
the	O
action	O
of	O
G	O
on	O
X	O
is	O
ergodic	O
(	O
meaning	O
that	O
X	O
cannot	O
be	O
reduced	O
by	O
invariant	O
proper	O
Borel	O
sets	O
of	O
X	O
)	O
then	O
any	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
on	O
X	O
is	O
homogeneous	O
.	O
</s>
<s>
We	O
now	O
discuss	O
how	O
the	O
structure	O
of	O
homogeneous	O
systems	O
of	O
imprimitivity	B-Algorithm
can	O
be	O
expressed	O
in	O
a	O
form	O
which	O
generalizes	O
the	O
Koopman	O
representation	O
given	O
in	O
the	O
example	O
above	O
.	O
</s>
<s>
then	O
(	O
U	O
,	O
π	O
)	O
is	O
a	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
of	O
(	O
G	O
,	O
X	O
)	O
.	O
</s>
<s>
Conversely	O
,	O
any	O
homogeneous	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
is	O
of	O
this	O
form	O
,	O
for	O
some	O
measure	O
σ-finite	O
measure	O
μ	O
.	O
</s>
<s>
Much	O
more	O
can	O
be	O
said	O
about	O
the	O
correspondence	O
between	O
homogeneous	O
systems	O
of	O
imprimitivity	B-Algorithm
and	O
cocycles	O
.	O
</s>
<s>
A	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
(	O
U	O
,	O
π	O
)	O
of	O
(	O
G	O
,	O
X	O
)	O
on	O
a	O
separable	O
Hilbert	O
space	O
H	O
is	O
irreducible	O
if	O
and	O
only	O
if	O
the	O
only	O
closed	O
subspaces	O
invariant	O
under	O
all	O
the	O
operators	O
Ug	O
and	O
π(A )	O
for	O
g	O
and	O
element	O
of	O
G	O
and	O
A	O
a	O
Borel	O
subset	O
of	O
X	O
are	O
H	O
or	O
 { 0 } 	O
.	O
</s>
<s>
then	O
the	O
restriction	O
of	O
Φ	O
to	O
the	O
fixed	O
point	O
subgroup	O
Gx	O
is	O
a	O
Borel	O
measurable	O
unitary	O
representation	O
U	O
of	O
Gx	O
on	O
H	O
(	O
Here	O
U(H )	O
has	O
the	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
)	O
.	O
</s>
<s>
There	O
is	O
a	O
bijection	O
from	O
unitary	O
equivalence	O
classes	O
of	O
systems	O
of	O
imprimitivity	B-Algorithm
of	O
(	O
G	O
,	O
X	O
)	O
and	O
unitary	O
equivalence	O
classes	O
of	O
representation	O
of	O
Gx	O
.	O
</s>
<s>
Moreover	O
,	O
this	O
bijection	O
preserves	O
irreducibility	O
,	O
that	O
is	O
a	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
of	O
(	O
G	O
,	O
X	O
)	O
is	O
irreducible	O
if	O
and	O
only	O
if	O
the	O
corresponding	O
representation	O
of	O
Gx	O
is	O
irreducible	O
.	O
</s>
<s>
Systems	O
of	O
imprimitivity	B-Algorithm
arise	O
naturally	O
in	O
the	O
determination	O
of	O
the	O
representations	O
of	O
a	O
group	O
G	O
which	O
is	O
the	O
semi-direct	O
product	O
of	O
an	O
abelian	O
group	O
N	O
by	O
a	O
group	O
H	O
that	O
acts	O
by	O
automorphisms	O
of	O
N	O
.	O
This	O
means	O
N	O
is	O
a	O
normal	O
subgroup	O
of	O
G	O
and	O
H	O
a	O
subgroup	O
of	O
G	O
such	O
that	O
G	O
=	O
N	O
H	O
and	O
N	O
∩	O
H	O
=	O
 { e } 	O
(	O
with	O
e	O
being	O
the	O
identity	O
element	O
of	O
G	O
)	O
.	O
</s>
<s>
There	O
is	O
a	O
bijection	O
between	O
unitary	O
equivalence	O
classes	O
of	O
representations	O
of	O
G	O
and	O
unitary	O
equivalence	O
classes	O
of	O
systems	O
of	O
imprimitivity	B-Algorithm
based	O
on	O
(	O
H	O
,	O
X	O
)	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
representation	O
of	O
G	O
is	O
irreducible	O
if	O
and	O
only	O
if	O
the	O
corresponding	O
system	B-Algorithm
of	I-Algorithm
imprimitivity	I-Algorithm
is	O
irreducible	O
.	O
</s>
