1. Introduction
Let G be a locally compact group and H a closed subgroup, this paper is concerned with the problem of extending coefficients of the regular representation of H to G. Suppose H normal in G. In 1973 [1] C. Herz proved that for
with compact support, for every
and for every U neighborhood of
in G there is
with
,
and
. In this work we want to treat the case of non normal subgroups. We succeed assuming that the subgroup H is amenable (Theorem 5). C. Fiorillo obtained [2] already this result assuming however the unimodularity of G and of H. But the AN part of the Iwasawa decomposition of
was out of reach. Even for G amenable our result is new: the case of the non-normal copy of
in the
-group was also out of reach.
Without control of norm and support of the extension, the theorem has been obtained in 1972 by McMullen [3] . With control of the norm, but not considering the supports, the statement is due Herz [1] (see also [4] ).
2. A Property of Amenable Subgroups
We denote by
the set of all complex valued continuous functions on G with compact support. We choose a positive continuous function q on G such that
, left invariant measures on G and H and a measure
on
as in Chapter 8 of [5] . The following Lemma will be used in the proof of our main result. See below the steps
and
of the proof of Lemma 2.
Lemma 1 Let
be a locally compact group,
a closed amenable subgroup,
a compact subset of
,
a neighborhood of
in
and
. Then there is
such that
,
and
![]()
Proof. Let
be a compact neighborhood of
in
with
,
and
By the Proposition 2.1 of [6] (p. 463), there is
such that
,
and such that
for every
. For every
we have
![]()
where
Consequently
![]()
3. Approximation Theorem for Convolution Operators Supported by Subgroups
We refer to [7] for
and the canonical map
of
into
(Section
p. 101). We denote by
the Banach space of all bounded operators of
.
We define a family of linear maps
of
into
where
is an arbitrary closed
subgroup of
. We precise that
is the involution of
and that for
,
and
we have
.
Definition 1. Let
be a locally compact group,
an arbitrary closed subgroup,
and
. For
we set for ![]()
.
Then
and
where
. If
then
and
is contained in
[8] .
Lemma 2. Let
be a locally compact group,
a closed amenable subgroup,
,
,
and
an open neighborhood of
in
. Then there is
with
,
,
and such that
![]()
for every
and every
.
Proof. Let
with
for every
. There is
a compact
symmetric neighborhood of
in
with
and such that
for every
. There is
open neighborhood of
in
such that
and
are both smaller than
for every
and for every
. We can choose
with
, ![]()
![]()
for every
and such that
.
Let
be a symmetric compact neighborhood of
in
contained in
with
![]()
for every
and such that
![]()
for every
and for every
(for
and
we denote by
the function defined on
by
).
We put
,
and
where
is the canonical map of
onto
.
By the preceding Lemma there is
with
and such that
![]()
is smaller than
![]()
and also smaller than
![]()
for every
. We finally put
,
and
.
1) For every
and every
we have
.
We show at first that
![]()
From
![]()
we obtain indeed
.
For every
we have
.
We have
.
But for every ![]()
![]()
and therefore
![]()
consequently
![]()
For every
we have
![]()
As above
![]()
taking in account that
we obtain
.
Proof of
Using
and
one obtains an estimate for
. We finish then the proof of 1) using
.
2) For every
and every
we have
![]()
By the Corollary 6 of section 7.2 p.112 of [7]
![]()
Consequently
![]()
But by definition of
for every
we have
![]()
3) End of the proof of Lemma 2. We are now able to define the functions
and
of the Lemma
and
. Using
and
we get
![]()
Clearly
and
. It remains to show that
. We have
![]()
But for ![]()
![]()
hence
and similarly
, we finally get
. ![]()
Theorem 3 Let
be a locally compact group,
a closed amenable subgroup,
,
a sequence of
,
a sequence of
,
and
an open neighborhood of
in
. Suppose that
the series
converges. Then there is
with
,
,
and such that
![]()
for every ![]()
Proof. We choose
with ![]()
1) There is
with
,
and such that
![]()
for every
.
There are
and
sequences of
with
![]()
and
![]()
for every
. From the convergence of
follows the existence of
such that
![]()
By Lemma 2 there is
with
,
,
![]()
and such that
![]()
for every
and every
. Consequently
![]()
2) End of the proof of Theorem 3. It suffices to put
and
to obtain
and
![]()
4. The Main Result
Definition 2 Let
be a locally compact group,
an arbitrary closed subgroup,
and
For
we put
![]()
where
and
are sequences of
such that
converges and such that
.
Then
is a linear map of
into
, for
and
one has
,
and
[8] .
Corollary 4 Let
be a locally compact group,
a closed amenable subgroup,
,
,
and
a neighborhood of
in
. Then there are
with
,
and
.
Proof. There are sequences
of
such that
converges and such that
. Let
be an open neighborhood of
in
such that
. By Theorem 3 there
is
with
,
,
and such that
![]()
for every
.
Consider an arbitrary
with
. From
![]()
and
![]()
we get
and therefore
![]()
The following theorem is the main result of the paper.
Theorem 5 Let
be a locally compact group,
a closed amenable subgroup,
,
,
and
a neighborhood of
in
. Then there is
with
and ![]()
Proof. This proof is identical with the one of Proposition
ii) p. 115 of [1] . Let
be an open neighborhood of
in
such that the closure of
in
is compact and contained in
. Using the Corollary 4 we show by induction the existence of a sequence
of
and of a sequence
of
such that
,
,
,
,
and
The function
satisfies all the requirements. ![]()