3.7 Locally Compact Groups
Definition 3.7.1 (Topological Group).label A topological group is a group $G$ equipped with a topology such that the group operations
$G\ti G\to G$ with $(x,y)\mto xy$
$G\to G$ with $x\mto x\inv$
are continuous. For $A,B\suf G$ and $x\in G$ we denote
We say that $A$ is symmetric if $A=A\inv$. In particular $G$ enjoys the following properties:
- (1)
$(A\cap B)\inv=A\inv\cap B\inv$ and $xA\cap yB\neq\emp\iff x\inv y\in AB\inv$ for each $A,B\suf G$ and $x,y\in G$.
- (2)
For each open $U\suf G$ the following sets $xU,Ux,U\inv,AU,UA$ are open for any $x\in G,A\suf G$.
- (3)
For each neighborhood $U$ of $1$ is there a symmetric neighborhood $V$ of $1$ such that $VV\suf U$.
- (4)
For each compact sets $A,B\suf G$, $AB$ is also compact.
- (5)
The canonical quotient map $q:G\to G/H$ is open.
- (6)
For each subgroup $H\leq G$
- (a)
$\cl{H}$ is a subgroup
- (b)
If $H$ is open then $H=\cl{H}$
- (c)
If $H$ is closed then $G/H$ is Hausdorff
- (d)
If $H$ is normal then $G/H$ is a topological group and $\cl{H}$ is normal
- (e)
If $G$ is locally compact then $G/H$ is locally compact
- (7)
For each subsets $\emp\neq A,B\suf G$ if $B$ is neighborhood of $1$ then $AB$ is a neighborhood of $A$.
Proof.
- (1)
Observe $(A\cap B)\inv=\curl{x\inv:x\in A\cap B}=\curl{x\inv:x\in A}\cap \curl{x\inv:x\in B}=A\inv\cap B\inv$ and $xA\cap yB\neq\emp\iff \exists a\in A,b\in B, xa=yb\iff \exists a\in A,b\in B, ab\inv=x\inv y\iff x\inv y\in AB\inv$.
- (2)
The first assertion is equivalent to continuity of multiplication and inversion respectively. The second assertion follows from the first.
- (3)
By continuity pick neighborhoods $W_{1},W_{2}$ of $1$ such that $W_{1}W_{2}\suf U$ then $V\define W_{1}\cap W_{2}\cap W_{1}\inv\cap W_{2}\inv$ then $V$ is symmetric by (1) and by construction $V\suf W_{1},W_{2}$ hence $VV\suf W_{1}W_{2}\suf U$.
- (4)
If $A,B$ are compact then $AB$ is compact by continuity of multiplication.
- (5)
For each $V\suf G$ we have $q\inv(q(V))=VH$ hence if $V$ is open then $q(V)$ is open by (2).
- (6)
- (a)
Let $x,y\in \cl{H}$ then there are nets $\curl{x_\al},\curl{y_\be}$ in $H$ converging to $x,y$ respectively so by continuity of group operations $xy,x\inv\in \cl{H}$
- (b)
$G\del H=\cups{}{x\in G\del H}xH$ is open hence $H$ is closed.
- (c)
Let $q(x)\neq q(y)\in G/H$ then $xHy\inv$ doesn’t contain 1 and closed by (2). By (3) there exists a symmetric neighborhood $U$ of $1$ such that $UU\cap xHy\inv=\emp$. hence $(UxH)(UyH)\inv=UxHy\inv U$ doesn’t contain $1$ so $(UxH)\cap (UyH)=\emp$ so $q(Ux),q(Uy)$ are disjoint neighborhoods of $q(x),q(y)$ respectively.
- (d)
For each $x\in G$, conjugation by $x$, $c_{x}$ is continuous and by normality $c_{x}(\cl{H})=\cl{c_x(H)}=\cl{H}$. By normality $q(x)q(y)=xHyH=xyH=q(xy)$ for each $x,y\in G$. The result follows from (5) and Lemma 3.9.3(3).
- (e)
If $U$ is a compact neighborhood of $1$ in $G$ then for any $x\in G$, $q(Ux)$ is a compact neighborhood of $q(x)$ in $G/H$.
- (f)
Suppose $B$ is a neighborhood of $1$ then there exists $U\suf G$ open such that $1\in U\suf B$ then $AU$ is open and $A\suf AU\suf AB$.
$\square$
Lemma 3.7.2.label Let $G$ be a topological group and $G_{0}\leq G$ a clopen subgroup then $G$ is paracompact if and only if $G_{0}$ is paracompact.
Proof. Since $G_{0}$ is a subgroup, $G=\dcups{}{i\in I}g_{i} G_{0}$ for some coset representatives $\curl{g_i}_{i}\in I$. Since left translation is a homeomorphism by Definition 3.7.3, $g_{i}G_{0}$ is homeomorphic to $G_{0}$ for each $i\in I$.
$(\implies)$ Since $G_{0}$ is a closed subspace of $G$, any open cover $\cali{U}$ of $G_{0}$ gives an open cover $\cali{V}=\cali{U}\cup \curl{G\del G_0}$ of $G$ thus if $G$ is paracompact then $G_{0}$ is paracompact.
$(\impliedby)$ Suppose $G_{0}$ is paracompact and $\cali{U}$ be an open cover of $G$. For each $i\in I$ the family
is an open cover of $g_{i}G_{0}$. Since paracompactness is a topological property, there exists a locally finite open refinement $\cali{V}_{i}$ of $\cali{U}_{i}$ for each $i\in I$. Then $\cali{V}=\cups{}{i\in I }\cali{V}_{i}$ is an open cover of $G$ that refines $\cali{U}$. Let $x\in G$ then $\exists! i\in I$ such that $x\in g_{i}G_{0}$. Since $g_{i}G_{0}$ is open, $\exists W\in\cali{N}(x)$ such that $W\suf g_{i}G_{0}$. Since $\cali{V}_{i}$ is locally finite $W$ can be taken such that $W$ intersect only finitely many sets of $\cali{V}_{i}$ and hence of $\cali{V}$.$\square$
Definition 3.7.3 (Translates and Inversions).label Let $G$ be a topological group, $f$ a (complex-valued) function on $G$ and $y\in G$ then left, right translates of $f$ through $h$ and inversion
are homeomorphisms and the maps $y\mto L_{y},R_{y}$ are group homomorphisms. Similarly for each subset $E\suf G$, translates and inversion acts by
We say $f$ is left (resp. right) uniformly continuous if $\norm{L_yf-f}{\sup}\to 0$ (resp. $\norm{R_yf-f}{\sup}$). If $f\in C_{c}(G;\com)$ then $f$ is left and right uniformly continuous.
Proof. Since $G$ is a group, translates and inversion are bijective and by Definition 3.7.1(2) they are homeomorphisms. For homomorphisms
and similarly for right translate.
Let $x\in K$, $\exists U_{x}\in\cali{N}(1)$ such that $\abs{f(xy)-f(x)}<\frac{\e}{2}$ for $y\in U_{x}$ hence there exists symmetric neighborhood $V_{x}$ of 1 such that $V_{x}V_{x}\suf U_{x}$. By compactness of $K$ there exists $x_{1},...,x_{n}\in K$ such that $K\suf \cups{n }{i=1}x_{j}V_{x_j}$ so pick $j$ depending on $x$ such that $x_{j}\inv x\in V_{x_j}$. Let $y\in V=\caps{n }{i=1}V_{x_j}$ then $xy=x_{j}(x_{j}\inv x)y\in x_{j}U_{x_j}$ and
Since $V$ is symmetric a similar argument works for $xy\in K$.$\square$
Remark 3.7.4 (Reminder).label The definition above for left translates comes from the fact that the action of $G$ on itself is covariant but contravariant on $\text{Fun}(G,\com)$
Lemma 3.7.5.label Let $G$ be a topological group then if $G$ is
- (1)
$T_{1}$ then $G$ is Hausdorff
- (2)
not $T_{1}$ then $G/\cl{\curl{1}}$ is a Hausdorff topological group
hence WLOG every topological group is Hausdorff unless stated otherwise.
Proof.
- (1)
By Definition 3.7.1(6c) with $H=\curl{1}$
- (2)
By Definition 3.7.1(6a,6d) $\cl{\curl{1}}$ is a normal subgroup hence $G/\cl{\curl{1}}$ is a Hausdorff topological group by Definition 3.7.1(6c,6d)
$\square$
Definition 3.7.6 (Locally Compact Group).label A locally compact group is a topological group $G$ whose topology is locally compact and Hausdorff. Moreover, $G$ enjoys the following properties:
- (1)
There exists a clopen $\s$-compact subgroup $G_{0}\leq G$.
- (2)
$G$ is paracompact.
- (3)
$G$ is (T4).
Proof.
- (1)
Let $U$ be a symmetric compact neighborhood of $1$ then $G_{0}\define \cups{\infty}{n=1}\prods{n}{i=1}U$ is the group generated by $U$. $G_{0}$ is open, closed and $\s$-compact by Definition 3.7.1(7,6b,4) respectively.
- (2)
By Corollary 3.13.11 and Lemma 3.7.2
- (3)
$\square$
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