Download Moments and Projections of Semistable Probability Measures on p

Moments and Projections of Semistable Probability
Measures on p-adic Vector Spaces
Makoto Maejima1 and Riddhi Shah1 2
Email addresses:
Makoto Maejima: [email protected]
Riddhi Shah: [email protected]
Address for correspondence:
Makoto Maejima, Department of Mathematics, Keio University,
3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
Tel. No.: +81 45 566 1668
Abstract
In this paper, two topics on semistable probability measures on
vector spaces are studied. One is the existence of absolute moments of operator-semistable probability measures and another is an
answer to the question whether one can get semistability of a probability measure from that of all its projections. All results obtained
here are extensions of known results for real vector spaces to p-adic
vector spaces.
p-adic
Keywords : semistable probability measure, -adic vector space, moment of
probability measure
Suggested running head:
Semistable Probability Measures on -adic Vector Spaces
p
p
Address: Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku,
Yokohama 223-8522, Japan
2 Permanent address : School of Mathematics, TIFR, Homi Bhabha Road, Mumbai,
400 005, India
1
1
1 Introduction
Let be a nite dimensional -adic vector space, where is a prime number. Let P ( ) denote the topological semigroup of probability measures
on , with weak topology and convolution ` ' as the semigroup operation
dened with respect to the additive group structure on . Let ( ) denote the group of all invertible linear operators on , namely the set of all
bi-continuous automorphisms on . In the following, R Q Z and N stand
for the set of real numbers, rational numbers, integers and natural numbers,
respectively.
A probability measure on is said to be operator-semistable if there
exist 2 ( ), 2 ]0 1 and f t gt 0 such that is embeddable
in a continuous real one-parameter semigroup f t gt 0 P ( ) as = 1
satisfying ( t) = ct x for all 0, where x denotes the Dirac measure
supported on 2 . However, if is the compact subgroup such that
, then the image of f t gt 0 on
0 = H , the normalised Haar measure of
is a continuous real one-parameter semigroup and hence it is trivial.
Thus, ( t ) = ct for all 0. We call and f tgt 0 as above ( )semistable when is a scaler automorphism, we say that (or f t gt 0 ) is
semistable.
Let k k be a -adic vector space norm on . For
that
R 2 (0 1), we1say. Note
2 P ( ) has an absolute moment of order if V k kr d ( )
that this denition is independent of norm on , since any two vector space
norms on are equivalent (see, e.g. Cassels 1], Chapter 7, Lemma 2.1).
Operator-semistable probability measures on real vector spaces have been
studied extensively. For a complete survey of results on (operator-)semistable
probability measures, see Hazod and Siebert 3] and Sato 10]. For some
related results for -adic vector spaces, see Dani and Shah 2], Shah 11],
Teloken 17] and Yasuda 19].
In this paper, we discuss the existence of absolute moments of order
of an operator-semistable probability measure on a nite dimensional
-adic vector space as an extension of the corresponding result on real
vector spaces (Theorem 1). Using this result, we investigate the relation
between semistability of a probability measure on and that of all its
one-dimensional projections under certain conditions (Theorem 2).
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2 Moments of operator-semistable probability measures on p-adic vector spaces
In this section, we discuss the existence of absolute moments of operatorsemistable probability measures on a nite dimensional -adic vector space
. There exist a lot of operator-semistable probability measures on a -adic
vector space (see Dani and Shah 2], Theorem 4.2).
For a prime number , let Q p denote the topological eld of -adic numbers with the -adic norm j jp. Namely, for any rational number 2 Q , if
= ( ) n for some integers
2 Z, where does not divide or ,
then j jp = ;n and Q p is the completion of Q with respect to this norm. Let
= dim . Then is isomorphic to Q dp . Let ( ) denote the space of all
linear operators on . Then ( ) (resp. ( )) is isomorphic to d (Q p ),
the vector space of matrices (resp. d(Q p ), the group of nonsingular
matrices) with entries in Q p , having the usual topology as a subset of
2
Q dp . Here, ( ) is a 2 -dimensional -adic vector space. Given a vector
space norm kk on , we dene a vector space norm kk on ( ) as follows:
k k = supfk ( )k : 2 k k = 1g for 2 ( ). Here, k 0 k k k k 0 k
and k ( )k k k k k for 0 2 ( ) and 2 . We dene the spectral
radius ( ) = limn!1 k nk1=n for 2 ( ) it is easy to see that the limit
exists. Note that ( ) k nk1=n for all . Clearly, ( ) is independent of
the norm dened as above, since any two vector space norms on ( ) are
equivalent (see, e.g. Cassels 1], Chapter 7, Lemma 2.1).
We now dene a vector space norm kkp on (resp. onP ( )) as follows:
d
We x a basis f 1
. For = ( 1
, let
d g on
d) =
i=1 i i 2
k kp = maxi j i jp . Using this norm on , we can dene k kp on ( ) as
follows for any 2 ( ), k kp = supfk ( )kp : k kp = 1g. Note that
if = ( ij ) 2 d (Q p ) with P
respect to the basisPmentioned above, k kp =
maxij j ij jp. Here, for = di=1 i i and = di=1 i i in , k + kp =
maxi j i + ijp = j j + j jp maxfj j jp j j jpg for some . Then k + kp maxfk kp k kpg. Therefore, for any 0, k + krp maxfk krp k krpg, and
hence 7! k krp is a continuous subadditive function on (see its denition
below).
A probability measure on is said to be full if the support of , denoted
by supp , is not contained in a proper subspace of . For 2 ( ), let
( ) = f 2 : n ( ) ! 0g. Clearly ( ) is a -invariant vector subspace
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of V . We say that is contracting on V if C ( ) = V . Note that, since C ( )
is closed in V , for any 2 P (V ), n() ! 0 if and only if supp C ( ).
A measure 2 P (V ) is said to be an idempotent if 2 = , equivalently, if
= !H , the normalised Haar measure of some compact subgroup H . Clearly,
any idempotent has an absolute moment of any order.
The following is a generalization of a result on existence of absolute moments of operator-semistable probability measures on real vector spaces to
p-adic vector spaces, (for the result on real vector spaces, see Luzak 5], 6]).
Theorem 1 Let V be a nite dimensional p-adic vector space. Let 2
P (V ) be full, non-idempotent and ( c)-semistable for some 2 GL(V ) and
some c 2 ]0:1. Then has an absolute moment of order r if and only if
s( ;1 )r c < 1.
Remark: The fullness condition is not needed for the \if" part of the above
theorem. Suppose is a non-idempotent probability measure embeddable in
a ( c)-semistable ftgt0 as = 1. Let V be the subspace generated by
supp . Since V is totally disconnected, supp t V for all t 0, and hence
V is -invariant. Let be the restriction of to V . Since is full on V , has an absolute moment of order r if and only if s( ;1 )r c < 1, by Theorem
1. Thus, the \if" part of the assertion without the fullness condition follows
since s( ;1 ) s( ;1 ).
Before proving the above theorem, let us state a result on subadditive
functions on a locally compact (Hausdor) group G. A function : G !
0 1 (resp. : G ! ]0 1 ) is said to be subadditive (resp. submultiplicative)
if (xy) (x) + (y) (resp. (xy) (x)(y)) for all x y 2 G and if there
exists a positive real number r = r() such that Ur = fx 2 G : (x) rg is a
neighbourhood of the identity e in G. Note that if a function is subadditive,
then 1 + is submultiplicative. The following result (which is perhaps well
known) follows from the same result about submultiplicative functions on G,
(see Siebert 14], Theorem 1, and Siebert 15], Theorem 5).
Proposition 1 Let G be a locally compact group with identity e and let P (G)
be the convolution semigroup of probability measures on G. Let ft gt0 ,
0 = e, be a continuous one-parameter semigroup in P (G) with the Levy
measure . Let U (e) denote the set of all neighbourhoods of the identity e in
G. Let be a subadditive function and let r = r() > 0 and Ur 2 U (e) be as
above. Then the following are equivalent:
4
R
(i) G dt < 1 for some t > 0.
R
(ii) sup0st G ds < 1 for all t > 0.
R
(iii) GnUr d < 1.
Moreover, if is continuous, then the above are equivalent to each of the
following statements:
(iv)
(v)
R
GnU d < 1 for all U 2 U (e).
R
GnU d
< 1 for some compact neighbourhood U 2 U (e).
Proof of Theorem 1. As in the hypothesis, since is ( c)-semistable, there
exists a ( c)-semistable one-parameter semigroup ftgt0 P (V ) with 1 =
. Now recall that C ( ) = fx 2 V : n (x) ! 0g it is a vector subspace
of V . It is shown in Dani and Shah 2] that for all t 0, t = (0)
t !H ,
0 = !H , for some compact subgroup H such that (H ) = H , f(0)
t g is a
(0)
( c)-semistable one-parameter semigroup supported on C ( ), 0 = 0 and
H \ C ( ) = f0g. Since any norm on V satises the triangular inequality, it
is easy to see that has an absolute moment of order r if and only if (0)
1
(0)
does. Also, since , and hence (0)
,
is
not
an
idempotent,
=
6
.
Hence
0
1
1
C ( ) 6= f0g. Let 1 be the restriction of to C ( ). Since 1 is contracting
on C ( ), it follows that s( 1;1) > 1. Now we are going to show that (0)
1 is
;
1
;
1
full on C ( ) and s( ) = s( 1 ).
Let VH be the subspace of V generated by H . Since (H ) = H , (VH ) =
VH . Moreover, since H is compact, if 2 is the restriction of to VH , then
it can easily be shown that for some M > 0, k 2nk < M for all n 2 Z, and
hence s( 2;1 ) = 1 = s( 2). Also, VH \ C ( ) = f0g. Since = (0)
1 !H is full
(0)
on V , V = C ( ) VH , a direct product, and 1 is full on C ( ). Now, we
have s( ;1 ) = maxfs( 1;1 ) s( 2;1)g = s( 1;1 ). In particular, s( ;1 ) > 1.
Now without loss of generality, we may assume that = (0)
1 , V = C( )
and = 1, i.e. is embeddable in a ( c)-semistable ftgt0 with 0 = 0
and is contracting on V . Let be the Levy measure of ft gt0 on V n f0g.
Then we know that is nite on V n U for any neighbourhood U of 0 in V
and () = c. Since 6= 0 and V is totally disconnected, it is easy to show
that is not a zero measure (see also Heyer 4], Theorems 6.2.10 and 6.2.3).
5
Here, V is isomorphic to Q dp , where d = dim V . Since any two norms on
V are equivalent, it is enough to consider the norm k kp dened above.
Since is contracting on V , by Lemma 3.3 in Siebert 16] there exist
distinct open compact subgroups Gn, n 2 Z, V = n Gn, \nGn = f0g, and
for all n, Gn Gn+1 and (Gn ) = Gn;1.
We rst prove the \if" part. Let r > 0 be xed such that s( ;1)r c < 1.
Since the map
R x 7!r kxkrp is subadditive, by Proposition 1, it is enough to
show that V nG0 kxkp d(x) < 1. Using the relation () = c, we have (for
0
= I ),
Z
V nG0
kxkrp d (x)
=
=
1 Z
X
Gn+1 nGn
n=0
1 Z
X
n=0 G1 nG0
1 Z
X
n=0 G1 nG0
M 1+
kxkrp d (x)
k ;n (x)krp cn d (x)
k ;n krp kxkrp cn d (x)
!
1
X
n=1
f(k ;nk1p=n )r cgn where M = supfkxkrp : x 2 G1g(G1 n G0 ), which is nite. Let an =
(k ;nk1p=n)r c, P
n 2 N . Since k ;nk1p=n ! s( ;1), we get that an ! s( ;1)r c <
1, and hence ann converges. Therefore
Z
V nG0
kxkrp d (x) < 1:
Hence, has an absolute moment of order r if s( ;1 )Rr c < 1.
For the \only if" part, we need to show that V kxkrp d(x) = 1 if
s( ;1 )r c 1. If possible, suppose this integral is nite for some r > 0
satisfying s( ;1)r c 1. Now from above,
1 Z
X
Hence
n=0 G1 nG0
k
Z
;n (x)kr cn d (x) =
p
1
X
G1 nG0 n=0
Z
V nG0
kxkrp d (x) < 1:
k ;n (x)krp cn d (x) < 1:
6
Let 0 be the restriction of to G1 n G0 . Then the above implies that for
0 -almost all x,
1
X
k ;n (x)krp cn < 1 and hence k ;n (x)krp cn ! 0:
n=0
;n kp .
Let tn = k
By denition, tn 2 Q n f0g. Let n = tn ;n for all n. Then
n 2 GL(V ) and k
nkp = jtnjpk ;nkp = t;n 1tn = 1 for all n. Also, since
s( ;1 )r c 1, t;n r cn. Now we get for 0 -almost all x,
k
n (x)krp = t;n r k ;n (x)krp cnk ;n (x)krp ! 0
This implies that, for 0 -almost all x, k
n(x)kp ! 0 and hence n (x) ! 0.
Let
V 0 = fx 2 V : n (x) ! 0g:
Then V 0 is a vector subspace, hence it is closed and supp 0 V 0. Since
n = n for all n, for any x 2 V 0 ,
n( (x)) = (
n (x)) ! 0:
0 V 0 and hence (V 0 ) = V 0 as 2 GL(V ). Since =
Therefore,
P c;n n((V),) supp
V 0 n f0g. Now we show that V 0 = V . If possible,
0
n2Z
suppose V 0 is proper. Let : V ! V=V 0 be the natural projection. Then
f (t )gt0 is a continuous one-parameter semigroup with the Levy measure
() dened on (V=V 0) nf(0)g. From above, () is a zero measure. Hence,
since V=V 0 is totally disconnected, it is easy to show that (t ) = (0) , for
all t, (see also Heyer 4], Theorems 6.2.10 and 6.2.3), and hence supp1 V 0 .
But since 1 = is full on V , we get a contradiction. Hence V 0 = V . That is,
n(x) !
R 0 for all x 2 V . But k
n kp = 1 for all n, which is a contradiction.
Thus, V kxkrp d(x) must be innite if s( ;1)r c 1. This completes the
proof.
2
We now state two simple lemmas about operator-semistable probability
measures which will be used in the next section.
Lemma 1 Let V be a nite dimensional p-adic vector space. Let 2 GL(V )
and c 2 ]0 1 .
(i) If 2 P (V ) is ( c)-semistable, then for kn = c;n], n ()kn ! .
7
(ii) If
n ( )ln ! 2 P (V ) for some 2 P (V ) and fln g N
n ( ) ! 0 and ln =ln+1 ! c, then is ( c)-semistable.
such that
Proof. (i) Let 2 P (V ) be ( c)-semistable. Let ft gt0 be a continuous
one-parameter semigroup with 1 = and (t ) = ct, for all t. Then for
kn = c;n], n()kn = cnkn ! 1 = .
(ii) Let , flng, , c be as above. Then the set T = f n ( )m : m ln n 2
N g is relatively compact (see Shah 13], Theorem 2.1 and Remark following
it). Also, since ln=ln+1 ! c, we can prove the assertion along the proof of
Theorem 4.6 of Teloken 17] using Theorem 2.3 of Teloken 17]. In Teloken
17], the fullness of is not assumed. However, it is not necessary here to
assume that is full, since, for n = n , we have n+1 n;1 = for all n. 2
Lemma 2 Let V be a nite dimensional p-adic vector space. Let 2 GL(V )
and c 2 ]0 1 . Let f
m g K , a compact subgroup of GL(V ), be such that
m commutes with for each m. Let fm g P (V ) be such that m ! 2
P (V ). Suppose each m is embeddable in a (
m c)-semistable one-parameter
semigroup f(tm) gt0 as (1m) = m , such that (0m) = 0 . Then for kn = c;n ],
n n( )kn yn ! , for some 2 K and some sequence fyng in V .
Proof. We may assume, without loss of generality, that f
m g itself converges.
Let be the limit point of it, then 2 K and commutes with . Recall
that C ( ) = fx 2 V : n(x) ! 0g, which is a vector subspace. Moreover, if
2 K commutes with , then keeps C ( ) invariant and C ( ) = C ( ) (cf.
Wang 18], Proposition 2.1). For any m 2 N , since (
m )n(m ) = (cmn ) ! 0 ,
we have that each m , and hence , is supported on C ( ). Therefore, without
loss of generality, we may assume that V = C ( ), that is, is contracting on
V.
Let n 2 N be xed. Let sn = 1 ; kncn. Here, for all m,
m = (1m) = (kmnc)n (smn ) = mn n(m )kn (smn ) :
Also, (kmnc)n = mn n (m)kn ! n n ( )kn . As m ! , the above implies that
f(smn ) gm2N is relatively compact (cf. Parthasarathy 9], Chapter III, Theorem
2.1). Let n be a limit point of it. Since sn < cn, (smn ) is a factor of (cmn ) ,
and since (cmn ) = mn n(m ) ! n n( ), n is a factor of n n( ).
8
Here, since 2 K and it commutes with , C (
) = C ( ) = V , and
hence n n ( ) ! 0 . Therefore, it follows that there exists a sequence fxng V such that n xn ! 0 (cf. Shah 13], Lemma 2.3). Now from the above
equation, we have that = n n( )kn n, for all n, and hence for yn = x;n 1 ,
n n( )kn yn ! . This completes the proof.
2
3 Semistable probability measures on p-adic
vector spaces and their projections
In this section, we compare semistability of a probability measure on a p-adic
vector space V with that of all its one-dimensional projections. Clearly, if
a probability measure on V is semistable then all its projections are also
semistable. Conversely, we are interested in nding out conditions under
which semistabilty of is implied by that of all its one-dimensional projections.
For V isomorphic toPQ dp and for x y 2 V , x = (x1 : : : xd ) and y =
(y1 : : : yd), let hx yi = di=1 xiyi. It is a continuous bi-linear map from V 2
to Q p . Any one-dimensional projection of V is of the form y 7! hx yi for
some x 2 V . For x 2 V n f0g, let Vx = Ker(y 7! hx yi) it is a subspace of
co-dimension 1 in V .
For 2 P (V ) and x 2 V , let x = (x ) denote the image of under the
map y 7! hx yi. We can see by injectivity of Fourier transform that = if and only if (x ) = (x ) for all x 2 V . Moreover, it follows from Levy's
continuity theorem that n ! in P (V ) if and only if (x n) ! (x ), for
all x 2 V .
In the following, we consider semistable probability measures on V . Here,
we identify a 2 Q p with the map x 7! ax. Recall that a probability measure
on V is semistable if it is (a c)-semistable for some a 2 Q p n f0g, jajp < 1,
and some c 2 ]0 1 , that is, is embeddable in a continuous one-parameter
semigroup ftgt0 P (V ) as = 1 and a(t ) = ct, for all t 0. This
automatically implies that C (a) = V and 0 = 0 . Now for a probability measure on V , we denote ;() = fc 2 ]0 1 : is embeddable in ftgt0 P (V ) such that a(t ) = ct t 0 for some a 2 Q p n f0gg. Clearly, ;()
is always nonempty as 1 2 ;() and if is (a c)-semistable for some a 2
Q p nf0g, then c c;1 2 ;(). Let Zp denote the ring of p-adic integers, namely,
9
Zp = fx 2 Q p : jxjp 1g. Let Zp(1) = fx 2 Zp : jxjp = 1g. It is the maximal
compact subgroup of the multiplicative group Q p n f0g. For 2 P (V ), let
Inv() = f 2 GL(V ) : () = g. It is a closed subgroup of GL(V ).
The following theorem is a generalization (to p-adic vector spaces) of
Theorem 1 in Maejima and Samorodnitsky 8] which is for real vector spaces.
For semistable probability measures on a real vector space R d , see Maejima
7] and references cited therein.
Theorem 2 Let V be a nite dimensional p-adic vector space and let 2
P (V ). Suppose that for all one-dimensional projections of V , () is
semistable and ; = \ ;( ()) 6= f1g. Then there exist unbounded sequences
fan g and fdng in N such that pan ()dn ! in particular is embeddable.
Moreover, if Inv() \ Zp(1) = fz 2 Zp(1) : z () = g is an open subgroup of
Zp(1), then is semistable.
Remark: 1. Given any subgroup K of Zp(1), there exist a lot of semistable
probability measures on V which are K -invariant. In the proof of Theorem
4.2 in Dani and Shah 2], for any contracting 2 GL(V ), (we may choose
= aI , a 2 Q p , 0 < jajp < 1), and any c 2 ]0 1 , one can construct a Levy
measure such that () = c and is K -invariant for this one has to take
a K -invariant subgroup H0 and a K -invariant measure 0 on H0 n H1, which
will imply that the corresponding one-parameter semigroup ftgt0 is such
that it is ( c)-semistable and each t is K -invariant (see also Yasuda 19]).
2. Any closed innite subgroup of Zp(1) is open, so the additional condition in the above theorem leaves out only those probability measures which
have nite invariance subgroups in Zp(1).
Proof of Theorem 2. Step 1. We may assume that 6= 0 . Let W be the
subspace generated by supp . If dim W = 1, then the statement is trivial. So
we assume that dim W 2. Since scaler automorphisms keep any subspace
invariant, for all one-dimensional projections of W , () is semistable and
\ ;( ()) 6= f1g. In view of the above arguments, we may assume that
V = W . That is, it is enough to prove the theorem for a full probability
measure on V .
From the hypothesis, x = (x ) is semistable for all x 2 V and ; =
\x ;(x ) 6= f1g. Choose c 2 ; such that c 2 ]0 1 . Then for each x 2 V nf0g,
x is (ax c)-semistable for some ax 2 Q p n f0g, jaxjp < 1.
10
Step 2. Suppose for any x 2 V n f0g, x is also (dx c)-semistable for
some dx 2 Q p n f0g. We show that jaxjp = jdxjp. This can be shown by
using Theorem 1, but we give a direct proof here. Let ax = bxpm(x) , and
dx = sxpk(x) , where bx sx 2 Zp(1) and m(x) k(x) 2 N n f1g. Let kn = c;n],
n 2 N . Then by Lemma 1 (i), anx(x)kn ! x and dnx(x)kn ! x. If
possible, suppose m(x) 6= k(x). Without loss of generality, we may assume
that m(x) > k(x). Then for i(x) = m(x) ; k(x) 2 N ,
x = lim
an( )kn = lim
(a d;1)ndnx(x)kn = lim
bn s;npi(x)ndnx(x)kn :
n x x
n x x
n x x
Since fbnxs;x ng Zp(1), which is compact, pi(x)n ! 0 in M (V ) and since
dnx(x)kn ! x, we have x = 0 , and hence supp Vx, which is proper as
x 6= 0. This is a contradiction as is full on V . Therefore m(x) = k(x) and
hence jaxjp = p;m(x) = jdxjp.
Step 3. Let V (1) = fx 2 V : kxkp = 1g. Then V (1) is compact. We
now dene a map m : V (1) 7! N as follows: If x is (ax c)-semistable, then
m(x) = ; logp(jaxjp), i.e. ax = bxpm(x) , where jbxjp = 1. From the above
arguments, the map m is well-dened.
For each x, we x ax and an (ax c)-semistable one-parameter semigroup
f(x )t gt0 with (x )1 = (x ).
We now show that the image of m, F = fm(x) : x 2 V (1)g, is a nite subset in N . If possible, suppose that for some sequence fxl g V (1), m(xl ) !
1. Then axl ! 0 in Q p . This implies that axl (xl ) = (xl )c ! 0 . But
(xl ) is a factor of (xl )c]n0 for some xed n0 2 N with n0 c > 1. From
above, (xl )c]n0 ! 0. But since xl 2 V (1), fxl g is relatively compact,
and for any limit point x of it, x 2 V (1) V n f0g and x = (x ) is a
factor of 0. Therefore, x = g for some g 2 Q p and hence x = 0 as x is
semistable. Now supp Vx, a proper subspace, hence is not full, which
is a contradiction. This implies that F is nite.
Step 4. We next show that the map m from V (1) to F N is continuous.
Let fxl g V (1), xl ! x in V (1). Since F is nite, we may assume that
m(xl ) = i0 for all l. We have to show that m(x) = i0 . Here, (xl ) is
(axl c)-semistable. Then by Lemma 2, there exist b 2 Zp(1) and a sequence
fyng Q p such that bn pi0 n (x )kn yn ! (x ). Since (x ) is (ax c)semistable, we have that anx(x )kn ! (x ). Let ax = bxpm(x) as above. If
11
possible, suppose m(x) < i0. Then
lim
(x ) yn;1 = lim
bnpi0 n(x )kn = lim
(bb;x 1 )np(i0 ;m(x))n anx(x )kn = 0 n
n
n
since f(bb;x 1 )ng Zp(1), which is compact, p(i0 ;m(x))n ! 0 in M (V ) and
since anx(x )kn ! (x ). This implies that fyn;1g is relatively compact and
(x ) = y for some limit point y of fyng in Q p . Hence (x ) = 0 as it is
semistable, Therefore, we have supp Vx, a proper subspace, and this is a
contradiction. Therefore, m(x) i0.
If possible suppose m(x) > i0 . Then for zn = (bxb;1 )np(m(x);i0 )n(yn;1),
(x ) = lim
an(x )kn = lim
(b b;1 )np(m(x);i0 )n bnpi0 n(x )kn yn ] zn :
n x
n x
Therefore, arguing as earlier, we get that (x ) zn;1 ! 0. Now we get a
contradiction as above. Hence m(x) = i0. That is, the map m from V (1) to
F is continuous.
Step 5. Now we show that F consists of only one natural number. If
possible, suppose there exist g1 g2 2 V (1) such that m(g1) 6= m(g2). We may
also assume that m(g1 ) > m(g2). Here, pm(gi )bgi (gi ) = (gi )c, for i = 1 2.
Let fng Q p nf0g be such that jnjp ! 0. Let hn = ng1 + g2 n 2 N . Here,
hn ! g2 and kg2kp = 1, we get that, for all large n, khnkp = 1. Without loss
of generality, we may assume that khnkp = 1 for all n. We know that since is full on V , for any x 2 V , (x ) is full on the image space fhx yi : y 2 V g
moreover, since (x ) is semistable, it is not an idempotent. Now for i = 1 2,
let ri = ; log c=m(gi) log p. By Theorem 1,
Z
Qp
jxjsp d(gi )(x) =
Z
V
jhgi y ijsp d(y ) < 1 if and only if s < ri :
Now for a xed s 2 r1 r2 and a xed n, using the subadditivity of the
function x 7! jxjsp, we get that
Z
Qp
jxjsp d(hn )(x)
=
Z
V
jhhn y ijsp d(y )
jn jsp
= jnjsp
= 1
Z
ZV
jhg1 y ijsp d(y ) ;
Qp
Z
jxjsp d(g1 )(x) ;
12
ZV
jhg2 y ijsp d(y )
Qp
jxjsp d(g2 )(x)
as the rst integral above with respect to the measure (g1 ) is innite and
the second one with respect to (g2 ) is nite for s 2 r1 r2 . Since hn ! g2
and hn 2 V (1), by continuity of the map m, we get that, for all large n,
m(hn) = m(g2 ), i.e. (hn ) is (bn pm(g2 ) c)-semistable and jbnjp = 1. Hence,
from Theorem 1 and above equations, we get that s ; log c=m(g2) log p =
r2. But s 2 r1 r2 , which is a contradiction. Therefore, our assumption
that m(g1) 6= m(g2) is wrong. Hence there exists a unique u0 2 N such that
F = fu0g and x is (ax c)-semistable, where ax = bxpu0 , bx 2 Zp(1), for all
x 2 V (1). That is, bxpu0 (x )t = (x )ct, for all x 2 V (1) and all t > 0.
Step 6. Let m1 = p ; 1 and let mn = mpn;;11 for n 2. Since bx 2 Zp(1),
we know that bmx n ! 1 in Zp(1). Let an = u0mn and let dn = c;mn ]. Clearly,
an ! 1, dn ! 1 and cmn dn ! 1. Then for all x 2 V (1),
lim (x pan ()dn ) = nlim
pan (x dn )
!1
= nlim
bmn pu0 mn (x dn )
!1 x
= nlim
(x )cmn dn
!1
= (x ):
n!1
Since for x 2 V n f0g (x ) = kxk;p 1 (x0 ) x0 2 V (1), the above equation
also holds for all x 2 V n f0g. Hence pan ()dn ! . By Theorem 1.5 of Shah
12], is embeddable.
Step 7. We suppose K = Inv() \Zp(1) is open in Zp(1). Then Zp(1)=K is
a nite group and K Inv(x) for all x. Let q 2 N be the order of Zp(1)=K .
Then bqx 2 K for all x 2 V (1). This implies that pu0q (x ) = (x )cq . That
is, for j0 = u0q and r = cq , we get that pj0 (x ) = (x )r , x 2 V (1). Now
for each n, let ln = r;n]. Since 0 < r < 1, rn ! 0, ln ! 1 and lnrn ! 1.
Then we have that for x 2 V (1),
lim (x pj0n()ln ) = nlim
pj0n(x )ln = nlim
(x )lnrn = (x ):
!1
!1
n!1
Again, since for x 2 V n f0g (x ) = kxk;p 1 (x0 ) x0 2 V (1), the above
equation also holds for all x 2 V nf0g. Hence pj0n()ln ! . Since pj0n() !
0 , pj0(n+1) =pj0n = pj0 is contracting on V and ln=ln+1 ! r, by Lemma 1 (ii),
is (pj0 r)-semistable, i.e. is semistable. This completes the proof. 2
13
Acknowledgements
This work was done while the second author was visiting Keio University on
a long-term Invitation Fellowship from the Japan Society for the Promotion
of Science (JSPS). She is grateful to the JSPS for the fellowship and Keio
University for the hospitality during her stay. She would like to thank W.
Hazod for a useful discussion on subadditive functions. The rst author was
partially supported by the 21st Century COE program of the Ministry of
Education, Culture, Sport, Science and Technology in Japan. The authors
would also like to thank the referee for helpful suggestions.
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