Download 6. Continuous homomorphisms and length functions.

6. Continuous homomorphisms and length functions.
The result of Theorem 2.24 shows that, among homomorphisms between topogroups,
those which are both open and continuous have special significance. We shall now indicate
an important special case, in which openness follows from continuity for a homomorphism
between locogroups.
We start with some preliminary considerations. First we recall a suitable version of
the Baire Category Theorem from topology.
6.1 Lemma Let X be a locally compact regular space and let hFn in∈N be a sequence
S∞
of closed subsets of X such that n=1 Fn = X . Then there exists m ∈ N such that the
interior of the set Fm in the space X is non-empty.
6.2 Corollary Every countable T1 –locogroup is discrete.
Proof. Let G be a countable T1 –locogroup. By Corollary 1.22, G is regular. Write
S
G = {gn : n ∈ N} . Then n∈N {gn } = G, and the sets {gn } are closed, since G is T1 . By
the above lemma, there exists m ∈ N such that IntG {gm } =
6 ∅ . Note that the set {gm }
−1
is open. Hence G is discrete.
is open. By Corollary 1.9, every set {gn } = gn {gm }gm
The topogroup Q shows that not all countable T1 –topogroups are discrete.
A topological space X is σ –compact, if there exists a sequence hKn in∈N of compact
S
subsets of X such that n∈N Kn = X .
We observe that a “ σ –compactly generated” topogroup is, in fact, σ –compact.
6.3 Lemma The subgroup generated by a σ –compact subset of a topogroup is σ –
compact.
Proof. Let G be a topogroup. First we note that when A and B are σ –compact subsets
of G, then the subset A−1 , A ∪ B and AB are σ –compact. For A ∪ B , σ –compactness
is obvious, and for A−1 , it follows directly from Theorem 1.8. To verify σ –compactness
S
S
of AB , write A = n∈N Kn and B = i∈N Ci , where the sets Kn and Ci are compact.
S
Then the set AB can be represented in the form AB = {Kn Ci : n ∈ N and i ∈ N} . By
Theorem 1.27, the sets Kn Ci are compact. As a consequence, the set AB is σ –compact.
Let E be a σ –compact subset of G. By the foregoing, the set E ∪E −1 is σ –compact.
Using induction, we see that (E ∪ E −1 )n is σ –compact for every n ∈ N . It follows that
S∞
the subgroup n=1 (E ∪ E −1 )n of G generated by E is σ –compact.
53
6.4 Corollary Every compactly generated topogroup is σ –compact.
By Corollaries 5.16 and 6.4, every connected locogroup is σ –compact. To exhibit
another important class of σ –compact locogroups, we recall that a topological space is
separable if the space has a countable dense subset.
6.5 Lemma Every separable locogroup is σ –compact.
Proof. Let G separable locogroup and let A be a countable subset of G such that A = G.
Let U be a compact neighborhood of the neutral element of G. By Theorem 1.19, we have
S
A ⊂ AU . As a consequence, AU = G. The set AU is σ –compact, since AU = a∈A aU
and the sets aU , a ∈ A , are compact by Theorem 1.27.
Now we prove an important Open Mapping Theorem for σ –compact T1 –locogroups.
e a T1 –locogroup and f a continuous
6.6 Theorem Let G be a σ –compact locogroup, G
e Then f is an open mapping.
homomorphism from G onto G.
e respectively.
Proof. Let e and ee be the neutral elements of G and G,
By Problem 2(b) of Exercise Set 4, openness of the mapping f follows once we show
that f is weakly open at the point e of G. Let U be a neighborhood of e in G. Since G
is locally compact, there exists a compact neighborhood W of e such that W ⊂ U . Since
S∞
G is σ –compact, there exist compact subsets K1 , K2 , . . . of G such that n=1 Kn = G.
For every g ∈ G, the set gW is a neighborhood of g by Corollary 1.11. It follows,
that for every i = 1, 2, ... , there exists a finite set An ⊂ Kn such that the sets aW ,
S∞
a ∈ An , cover the compact set Kn . Denote by A the countable set n=1 An and note
S∞
S
that G = n=1 Kn ⊂ a∈A aW . Since f is an onto mapping, it follows from the foregoing
e ⊂ S{f (aW ) : a ∈ A} . Since f is a homomorphism, we have f (aW ) = f (a)f (W )
that G
for every a ∈ A . The continuous image f (W ) of the compact set W is compact. Since the
e is T2 , the compact subset f (W ) of G
e is closed. It follows that also the
T1 –topogroup G
S
e are closed. Since A is countable and
subsets f (a)f (W ) , a ∈ A , of G
a∈A f (a)f (W ) =
e it follows from Lemma 6.1 that there exists h ∈ A such that Int f (h)f (W ) 6= ∅ . By
G,
e
G
Corollary 1.11, we have IntG
ef (W ) 6= ∅ . Since W ⊂ U , we have IntG
ef (U ) 6= ∅ . We have
shown that f is weakly open at e.
54
Example 2.22 shows that the Open Mapping Theorem does not hold without the
requirement that G is σ –compact.
We mention here only one consequence of the Open Mapping Theorem.
6.7 Theorem Let A be a σ -compact closed subgroup of a topogroup G and let H be
a closed normal subgroup of G. If the subgroup AH of G is locally compact, then the
topogroups AH/H and A/(A ∩ H) are topologically isomorphic.
Proof. The natural homomorphism ϕH : AH → AH/H is continuous, and thus the
restriction f = ϕH |A is a continuous homomorphism. Moreover f is an onto mapping
A → AH/H . The topogroup A is σ -compact, and A is locally compact, as a closed
subtopogroup of the locogroup AH . On the other hand, since H is closed, the quotient
topogroup AH/H of the locogroup AH is a T1 -locogroup. By Theorem 6.6, the continuous
homomorphism f : A → AH/H is an open mapping. By Theorem 2.24, we have AH/H ∼
=
A/Kerf . Since Ker f = A ∩ H , we have AH/H ∼
= A/(A ∩ H) .
In Theorem 4.11 we showed that a topogroup has a compatible length function if the
neutral element of the topogroup has a countable neighborhood base. We shall use the
following topological result to show that, in a locogroup, the existence of a compatible
length function follows from a weaker condition.
6.8 Lemma A point x of a locally compact regular space X has a countable neighborhood
base if the set {x} is the intersection of countably many neighborhoods of x .
T∞
Proof. Let O1 , O2 , ... be neighborhoods of x such that n=1 On = {x} . Since X is locally
compact, the point x has a compact neighborhood U . Using regularity of X , we see that
there exists a sequence hVn i∞
n=1 of neighborhoods of x such that Vn+1 ⊂ Vn ⊂ On ∩ U
for every n > 0 . We show that sequence hVn i∞
n=1 is a neighborhood base of x . Let O
be an open neighborhood of x . Since X is regular, we have {x} ⊂ O . Since we have
T∞
T∞
{x}
and,
for
every
n
>
0
,
V
⊂
O
,
we
see
that
O
=
n
n
n
n=1 Vn = {x} . From the
n=1
foregoing it follows that the open sets X \ Vn , n = 1, 2, ... , cover the compact set U \ O .
Sm
Therefore there exists m > 0 such that U \ O ⊂ n=1 (X \ Vn ) . For every n > 0 , we have
Sm
X \ Vn ⊂ X \ Vn+1 , and it follows that n=1 (X \ Vn ) = X \ Vm . By the foregoing, we
have U \ O ⊂ X \ Vm that is Vm ∩ U ⊂ O . Since Vm ⊂ U , it follows that Vm ⊂ O . We
have shown that the sequence hVn i∞
n=1 is a neighborhood base of x .
55
Lemma 6.8 and Theorem 4.11 give us the following result.
6.9 Theorem A locogroup has a compatible length function if, and only if, the set {e}
is the intersection of countably many open sets.
Our last theorem shows that many locogroups can be “approximated” by locogroups
with compatible length functions.
6.10 Theorem Let G be a locogroup, which is either σ –compact or locally invariant, and
let U be a neighborhood of the neutral element of G. Then there exists a compact normal
subgroup K of G such that K ⊂ U and the quotient topogroup G/K has a compatible
length function.
Proof. Let e be the neutral element of G. We note first that there exists a sequence
S∞
hAn i∞
n=1 of subsets of G such that
n=1 An = G and, for every n > 0 , we have An ⊂ An+1
and the following condition is satisfied:
(*) for every V ∈ ηe there exists W ∈ ηe such that for every x ∈ An we have xW x−1 ⊂ V .
If G is locally invariant, then we can choose An = G for every n > 0 . On the other
S∞
Sn
hand, if G = n=1 Kn , where the sets Kn are compact, then we can choose An = k=1 Kk
for every n > 0 . The sets An are then compact and condition (*) holds by Theorem 1.26.
Let O be a compact neighborhood of e. Let U0 = U ∩ O , and use induction to
define a sequence hUn i∞
n=1 of open neighborhoods of e so that, for every n > 0 , we have
T∞
Un2 ⊂ Un−1 , Un−1 ⊂ Un−1 and xUn x−1 ⊂ Un−1 for every x ∈ An . Let H = n=0 Un . By
Theorem 2.6, the set H is a closed subgroup of G. We see easily that, for all x ∈ G and
n ≥ 0 , there exists m ≥ 0 such that xUm x−1 ⊂ Un : we can choose m = max(n + 1, k) ,
where k satisfies the condition x ∈ Ak . From the foregoing it follows by Theorem 2.6 that
the subgroup H is normal. Since H is closed, H ⊂ U0 ⊂ O and O is compact, also H is
compact. Moreover H ⊂ U0 ⊂ U .
We show that the quotient topogroup G/H has a compatible length function. Since
H is closed, G/H is T1 by Theorem 2.18, and therefore the closure of the subset {H}
of G/H is {H} . By Theorem 5.2, G/H is a locogroup. The existence of a compatible
length function follows by the foregoing and Theorem 6.9, once we show that the set
{H} is the intersection of countably many open subset of G/H . By Theorem 2.15, the
natural homomorphism ϕH is an open mapping. As a consequence, the set Vn = ϕH (Un )
56
T∞
Vn = {H} . For every n > 0 ,
T∞
we have e ∈ Un and therefore H = ϕH (e) ∈ Vn . It follows that {H} ⊂ n=1 Vn . To
T∞
T∞
show that n=1 Vn ⊂ {H} , let J be an element of the set n=1 Vn . For every n > 0 ,
is open in G/H for every n > 0 . We show that
n=1
−1
since J ∈ Vn , we have ϕ−1
H (J) ∩ Un 6= ∅ . Let g be an element of the set ϕH (J) .
Then ϕ−1
H (J) = gH , and thus, for every n > 0 , we have gH ∩ Un 6= ∅ and hence
gH ∩ Un 6= ∅ . Since the set gH is compact and Un ⊂ Un−1 for every n > 0 , it follows
T∞
from the foregoing that gH ∩ n=1 Un 6= ∅ . By Corollary 1.21 we have, for every n > 0 ,
that Un ⊂ Un2 . It follows further, since Un2 ⊂ Un−1 , that Un ⊂ Un−1 . From this it follows
T∞
T∞
that n=1 Un ⊂ n=0 Un = H . By the foregoing, we have gH ∩ H 6= ∅ , in other words,
g ∈ H . Since g ∈ ϕ−1
H (J) , it follows from the foregoing that J = ϕH (g) = H . We have
T∞
shown that n=1 Vn ⊂ {H} .
57
III. The invariant integral of a compact group.
The Peter–Weylin Theorem.
7. Existence and uniqueness of the integral.
The deeper results in the theory of locally compact groups are based on the existence
of a so-called left– (or right–) invariant integral, or Haar integral, in every locally compact
group. In the following, we prove the existence of the invariant integral for every compact
topogroup.
Throughout Chapter 7, the symbol G refers to a compact topogroup and e to the
neutral element of G.
We denote the collection {f : f is a continuous function G → C} by K(G) and the
subfamily {f ∈ K(G) : f (G) ⊂ [0, ∞)} of K(G) by K+ (G) . A constant function, with
value either 0 or 1 , is simply denoted by 0 or 1.
For every f ∈ K(G) and for all g, h ∈ G, we denote by g fh the function G → C
defined by the formula
g fh (x)
= f (gxh) . Moreover, we let
gf
=g fe and fh =e fh .
By Theorem 1.8, the mapping x 7→ gxh is a homeomorphism from G onto G. As a
consequence, the following result obtains.
7.1 Lemma For all f ∈ K(G) and g, h ∈ G, we have g fh ∈ K(G) and g fh (G) = f (G) .
The set K(G) is made into a linear space by defining, for all c ∈ C and f, l ∈ K(G) ,
the functions cf and f +l by the formulas (cf )(x) = c·(f (x)) and (f +l)(x) = f (x)+l(x) ;
clearly, we have cf ∈ K(G) and f + l ∈ K(G) . We see easily, that with these definitions
of scalar multiplication and addition, K(G) becomes a complex linear space.
7.2 Definition An invariant integral of G is a a linear mapping M : K(G) → C which
satisfies the following three conditions:
(i) M (g fh ) = M (f ) for all f ∈ K(G) and g, h ∈ G.
(ii) M (f ) > 0 when f ∈ K+ (G) and f 6= 0 .
(iii) M (1) = 1 .
We now start to prove the existence of the invariant integral. First we give some
auxiliary results.
The first result shows that continuous functions G → C are “uniformly continuous”.
58