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A SPECTRAL RADIUS FORMULA FOR THE FOURIER
TRANSFORM ON COMPACT GROUPS AND APPLICATIONS
TO RANDOM WALKS
M. ANOUSSIS AND D. GATZOURAS
b be its
Abstract. Let G be a compact group, not necessarily abelian, let G
unitary dual, and for f ∈ L1 (G), let f n := f ∗ · · · ∗ f denote n-fold convolution
of f with itself and fb the Fourier transform of f . In this paper, we derive the
following spectral radius formula
1/n
kf n k1 −→ max ρ fb(R) ,
b
R∈G
where ρ fb(R) is the spectral radius of fb(R), thereby extending the wellknown Beurling–Gelfand spectral radius formula for the Fourier transform on
a (locally compact) abelian group. We also establish a partial result in this
direction for arbitrary regular Borel measures on G. As applications, we give
conceptual and rather short proofs of existing results concerning convergence
in the total variation norm of random walks on compact groups, and uniform
convergence of the corresponding densities. We also describe how these results
may be transferred to homogeneous spaces.
1. Introduction
Let G be a locally compact abelian group, let λG be Haar measure on G, and
b denote the dual group of G, i.e., the group consisting of all continuous group
let G
homomorphisms χ : G −→ S1 — the multiplicative group of complex numbers of
b let
modulus 1. For f ∈ L1 (G) and χ ∈ G,
Z
f (x)χ(x)λG (dx)
fb(χ) :=
G
be the Fourier transform of f . The Beurling–Gelfand spectral radius formula for
the Fourier transform asserts then that
1/n
kf n k
−→ maxfb(χ)
1
b
χ∈G
n
as n −→ ∞, where f is n-fold convolution of f with itself, f n := f ∗ · · · ∗ f ,
and is an instance of a more general formula, the spectral radius formula for the
Gelfand transform on a commutative Banach algebra [8, Theorem 1.30, combined
with Theorem 4.2]. In this paper we derive the analogue of this formula for arbitrary
compact groups, not necessarily abelian (Corollary 2.2), and also derive a partial
result in this direction for arbitrary (regular Borel) measures on compact groups
(Theorem 2.1). The core of these results is Theorem 3.1 below.
As applications, we present a conceptual approach to certain limit theorems for
random walks on compact groups. In particular, as applications of the spectral
Date: December 29, 2003.
1
2
M. ANOUSSIS AND D. GATZOURAS
radius formula for L1 -functions, we derive two well known results, appearing in
Kloss [13] and Shlosman [21] respectively, the first concerning convergence in the
total variation norm of µn := µ ∗ · · · ∗ µ to Haar measure, on a compact group G,
when µ is an absolutely continuous with respect to Haar measure Borel probability measure on G (under the necessary conditions on µ) (Corollary 4.2), and the
second concerning the uniform convergence of the densities f n of the µn , when µ
actually has an L1+δ -density f with respect to Haar measure (Theorem 5.2). As
an application of the corresponding result for measures, we give a short proof of
a well known result on norm-convergence of µn , for arbitrary probability measures
µ (on a compact group G) (Theorem 4.1); a weaker form of this result seems to
first have appeared in Bhattacharya [1], and the result we prove here also appears
in Mindlin [14] and Ross and Xu [17]. Finally, in the last section we also describe
briefly how these results may be transferred to homogeneous spaces acted upon by
compact groups.
Conventions and Notations. Throughout the paper, the term group will refer to a
topological group whose topology is Hausdorff. If G is a locally compact group, λ G
will denote (left) Haar measure on G, and when G is compact, λ G will be assumed
to be normalized to have total mass equal to 1. Throughout, we shall write L p (G)
for Lp (G, B(G), λG ), where B(G) is the Borel σ-algebra of G.
If G is a locally compact group, and M (G) the space of complex, regular, Borel
measures on G, then kµk will denote the total variation norm of µ. (See Rudin [18],
Chapter 6, for definitions.) It is then well known that when µ has a density f ∈
L1 (G) with respect to Haar measure, then kµk = kf k1 ; we shall occasionally denote
by M p (G), p > 1, the elements of M (G) with an Lp -density with respect to Haar
measure.
Finally, we shall write 1E for the function which is 1 on the set E and 0 elsewhere
(i.e., the characteristic function of E); in particular, 1 G will denote the function
identically equal to 1 on G.
2. Main Result
In this section we state the main result of this paper. To begin with, we first fix
some notation.
Notations. If A is a unital Banach algebra and x ∈ A, σ(x) will denote the spectrum
of x, and ρ(x) the spectral radius of x. Recall that
(2.1)
ρ(x) = lim kxn k
n−→∞
1/n
= inf kxn k
1/n
n∈N
[19, Theorem 10.13].
b of G is usually defined as the
Let G be a compact group. The unitary dual G
set of all equivalence classes of irreducible unitary representations of G; to avoid
b a fixed, complete set of mutually
abuse of language however, we shall denote by G
inequivalent, irreducible, unitary representations of G. Furthermore, recall that
b is equipped with the discrete topology.
when considered as a topological space, G
b
Finally, for R ∈ G, dR will stand for the dimension of R.
If f ∈ L1 (G), f n will denote n-fold convolution of f with itself: f n := f ∗ · · · ∗ f ,
n ∈ N. Similarly, for µ ∈ M (G), µn := µ ∗ · · · ∗ µ, n ∈ N, will denote n-fold
convolution of µ with itself.
SPECTRAL RADIUS FORMULA AND APPLICATIONS
3
If f ∈ L1 (G), fb will denote its Fourier transform:
Z
b
R x−1 f (x)λG (dx);
f (R) :=
G
\
b and that (f
∗ g)(R) = gb(R)fb(R) for
recall that fb(R) 6 kf k1 for all R ∈ G,
1
b
all R ∈ G and f, g ∈ L (G). Similarly, for µ ∈ M (G), µ
b will denote the Fourier
transform of µ:
Z
R x−1 µ(dx);
µ
b(R) :=
G
\
b and (µ
b and
again kb
µ(R)k 6 kµk for all R ∈ G,
∗ ν)(R) = νb(R)b
µ(R) for all R ∈ G
µ, ν ∈ M (G).
For a measure µ ∈ M (G), µa.c. and µs will denote its absolutely continuous and
singular parts respectively, with respect to Haar measure λ G .
Recall that kµ1 + µ2 k = kµ1 k + kµ2 k whenever µ1 , µ2 ∈ M (G) are mutually
singular. In particular, kµk = kµa.c. k + kµs k for µ ∈ M (G).
Finally, if a and b are non-negative numbers, a ∨ b denotes their maximum:
a ∨ b := max{a, b}.
The following is then the main result of this paper.
b be its unitary dual. Then, for
Theorem 2.1. Let G be a compact group and let G
any µ ∈ M (G),
1/n
1/n
lim kµn k
= sup ρ µ
b(R) ∨ inf k(µn )s k
.
n−→∞
n∈N
b
R∈G
Note 1. Recall that M 1 (G) is a closed, two-sided ideal in M (G); therefore the
quotient algebra M (G)/M 1 (G) is a (unital) Banach algebra. Notice that k(µ n )s k
is then the norm of [µ + M 1 (G)]n in M (G)/M 1 (G), since for any ν ∈ M 1 (G),
kµn + νk = k(µn )a.c. + νk + k(µn )s k
One consequence of this fact is that
inf k(µn )s k
1/n
n∈N
1/n
= ρ µ + M 1 (G) = lim k(µn )s k
.
n∈N
1
Note 2. Let µ ∈ M (G). Since M (G) is a two-sided ideal in M (G), one has that
n
µn = (µa.c. + µs ) = νn + µns with νn ∈ M 1 (G), and therefore µns = (µn )s + νn0
with νn0 = (µn )a.c. − νn ∈ M 1 (G). It follows that k(µn )s k 6 kµns k, and so one has
the estimate
1/n
inf k(µn )s k
6 ρ (µs ) 6 kµs k .
n∈N
b be its unitary dual. Then, for
Corollary 2.2. Let G be a compact group and let G
any f ∈ L1 (G),
1/n
lim kf n k1 = max ρ fb(R) .
n−→∞
b
R∈G
Remarks.
(1) Corollary 2.2 asserts an equality between spectral radii. In fact more is
true, namely that the spectrum σ(f ), in M (G), of an f ∈ L 1 (G), satisfies
[
σ fb(R) ∪ {0}
(2.2)
σ(f ) =
b
R∈G
4
M. ANOUSSIS AND D. GATZOURAS
S
unless G is finite. (When G is finite σ(f ) = R∈Gb σ fb(R) .) A proof of
this is provided
at
the end of Section 3.
(2) Let Rad M 1 (G) consist of those µ ∈ M (G) for which
1/n
inf k([ν ∗ µ]n )s k
n∈N
(2.3)
∀ν ∈ M (G);
=0
then, by Note 1 and [3, Proposition 25.1], Rad M 1 (G) is the pullback to
M (G) of the (Jacobson) radical of the quotient algebra M (G)/M 1 (G). It
is therefore a closed
two-sided
ideal in M (G), and obviously Corollary 2.2
extends to Rad M 1 (G) ; i.e.,
1/n
lim kµn k
n−→∞
= sup ρ µ
b(R)
b
R∈G
holds for all µ ∈ Rad M 1 (G) . Notice that when G is abelian, the radical
Rad M 1 (G) as defined above coincides with the radical of the ideal M 1 (G)
in M (G), as usually defined for ideals in commutative Banach algebras,
i.e., the intersection of all maximal (modular) ideals of M (G) that contain
M 1 (G) (cf. [3]).
(3) On the other hand, Theorem 2.1 shows that (2.3) does not extend to all
of M (G), and this reflects (and is actually a consequence of) the fact that
M (G) is asymmetric (cf. [4]). In fact (2.3) cannot hold on any asymmetric
subalgebra of M (G). For suppose A is asymmetric. Then there exists a
self-adjoint measure µ ∈ A, i.e., with µ∗ = µ where µ∗ (B) := µ (B −1 ),
whose spectrum σ(µ) contains a non-real complex number λ = u + iv,
and we may, without loss of generality, assume that kµk = 1 and that
v > 0. Then there exists a polynomial p, with p(0) = 0 and such that
|p(λ)| > maxx∈[−1,1] |p(x)|. (For example, consider the entire function
f (z) := ze−icz , where c is such that
|f (λ)| = |λ|ecv > 1 = max |f (x)| ,
x∈[−1,1]
and approximate it by its Taylor polynomial, uniformly on the closed unit
disc. This particular construction of p was suggested by V. Nestoridis.)
Since p is a polynomial without constant coefficient and A is an algebra,
we have that ν := p(µ) ∈ A. On the other hand, p(λ) ∈ σ(ν), whence
ρ(ν) > |p(λ)|. Since µ = µ∗ , and hence µ
b(R) is self-adjoint, we also have
b since νb(R) = p(b
that all eigenvalues of µ
b(R) are real for any R ∈ G;
µ(R)),
and therefore σ(b
ν (R)) = p(σ(b
µ(R))), we must then have that ρ νb(R) 6
b Thus the measure ν ∈ A cannot satisfy
maxx∈[−1,1] |p(x)| for all R ∈ G.
(2.3).
(4) In contrast to the abelian case, formula (2.3) does hold for all central
measures on a compact simple Lie group. This follows from results of
Ragozin [15] (Corollary 3.4 and its extension to the disconnected case,
p. 228, in [15]), in conjunction with usual Gelfand theory. Note that for
such groups, the algebra of central measures Z(M (G)) is in fact symmetric
[15, p. 221].
SPECTRAL RADIUS FORMULA AND APPLICATIONS
5
3. Proof of Main Result
In this Section we prove Theorem 2.1. For µ ∈ M (G) define
kµk∗ := sup kb
µ(R)k .
b
R∈G
It is then straightforward to verify that k k∗ is a norm on M (G) (the fact that
kµk∗ = 0 implies that µ = 0 follows from the uniqueness of the Fourier transform
for regular Borel measures on compact groups), and that the completion D ∗ (G) of
M (G) with respect to this norm is a unital Banach algebra and in fact a C ∗ -algebra.
b the mapping µ 7−→ µ
Note that, for each R ∈ G,
b(R) extends uniquely from M (G)
∗
to D (G); furthermore, the mapping x 7−→ x
b on D ∗ (G) is injective. Notice also
that the group C ∗ -algebra C ∗ (G), i.e., the closure of L1 (G) with respect to k k∗ [8,
Section 7.1], is a subalgebra of D ∗ (G). We shall denote by δe the unit in M (G);
i.e., δe is the Dirac measure at the identity e of G.
The following result plays a central role in the proof of Theorem 2.1.
Theorem 3.1. Let G be a compact group and f ∈ L1 (G). Then
σD∗ (G) (f ) = σM (G) (f );
i.e., the spectrum of f as an element of the unital Banach algebra D ∗ (G) coincides
with its spectrum as an element of the unital Banach algebra M (G).
Proof. As M (G) ⊆ D ∗ (G), it is clear that
σD∗ (G) (f ) ⊆ σM (G) (f ).
We shall also show the reverse inclusion.
Since D ∗ (G) = M (G) = M 1 (G) when G is finite, we may without loss of generality assume that G is infinite. Fix f ∈ L1 (G), and λ ∈ C such that λδe − f is
invertible in D ∗ (G). We shall show that (λδe − f )−1 ∈ M (G), i.e., that λδe − f is
also invertible in M (G).
First note that we must have λ 6= 0, as no element of L1 (G) is invertible in D ∗ (G)
when G is infinite. For suppose that f ∈ L1 (G) were invertible in D ∗ (G), with
−1
b whence,
for all R ∈ G,
inverse x, say. We would then have that x
b(R) = fb(R)
−1
by the Riemann-Lebesgue lemma [11, Theorem 28.40], kb
x(R)k > fb(R) would
b and hence also G, were finite. But
be arbitrarily large, for appropriate R, unless G,
this contradicts the fact that supR∈Gb kb
x(R)k = kxk∗ < ∞.
Next let E be the algebra consisting of all finite linear combinations of matrixb and recall that E is
elements Rij of irreducible unitary representations R ∈ G,
dense in L1 (G) (in the L1 -norm) [8, Theorem 5.11]. Since |λ| > 0, there exists
b be a finite set for
g ∈ E such that kf − gk1 < |λ|; set h := f − g and let Fb ⊂ G
P
P dR
R
R
which g = R∈Fb i,j=1 cij Rij for some cij ∈ C. Since khk1 < |λ|, δe − λ−1 h is
d
invertible in M (G). Furthermore, since by the Shur orthogonality relations Q
ij (R)
b with Q 6= R [8, Theorem 5.8a], we have that
is the zero matrix whenever Q, R ∈ G
(3.1)
gb(R) = 0
b r Fb .
∀R ∈ G
Now note that
−1
−1
(3.2) δe − λ−1 h
∗ δe − λ−1 f = δe − δe − λ−1 h
∗ λ−1 g =: δe − λ−1 w,
6
M. ANOUSSIS AND D. GATZOURAS
and the right-hand side is invertible (in D ∗ (G)) because the left-hand side is. Notice
also that
h
i−1
,
(δe \
− λ−1 w)(R) = Id − λ−1 gb(R) Id − λ−1 b
h(R)
R
R
where Id denotes the d × d identity matrix, so that
(3.3)

 Id R
i−1
h
(δe \
− λ−1 w)(R) =
 IdR − λ−1 gb(R) IdR − λ−1 b
h(R)
if
if
b r Fb
R∈G
R ∈ Fb ,
by (3.1). Furthermore, the matrix (δe \
− λ−1 w)(R) is invertible for each R, since
−1
∗
δe − λ w is invertible in D (G). Set
i−1
h
− λ−1 w)(R)
(3.4)
V (R) := IdR − (δe \
b and note that V (R) = 0 for R ∈ G
b r Fb , by (3.3). Define
for R ∈ G,
X
v(x) :=
dR tr V (R)R(x) (x ∈ G),
b
R∈F
b Note also that, since Fb is finite, v is a
and note that vb(R) = V (R) for all R ∈ G.
1
continuous function, hence in L (G). Moreover,
h
−1 ib
(δ\
δe − λ−1 w
(R)
e − v)(R) = IdR − V (R) =
b by (3.3) and (3.4), and therefore
for each R ∈ G,
δe − v = δe − λ−1 w
−1
.
It now follows from (3.2) that
−1
−1
δe − λ−1 f
= (δe − v) ∗ δe − λ−1 h
∈ M (G) ∗ M (G) = M (G),
i.e., δe − λ−1 f is also invertible in M (G). This shows that
σD∗ (G) (f ) ⊇ σM (G) (f ),
and completes the proof of the theorem.
The following proposition, which relies on Theorem 3.1, is the main ingredient
of the proof of Theorem 2.1.
b be its unitary dual, and let
Proposition 3.2. Let G be a compact group, let G
µ ∈ M (G). Then
1/n
lim kµn k
6 kµs k ∨ sup ρ µ
b(R) .
n−→∞
Proof. We shall first show that
(3.5)
b
R∈G
ρM (G) (µ) 6 ρD∗ (G) (µ) ∨ ρM (G) (µs ),
where the notation is obvious.
Fix λ ∈ C with |λ| > ρD∗ (G) (µ) ∨ ρM (G) (µs ). Then δe − λ−1 µ is invertible in
−1
D∗ (G) and δe − λ−1 µs is invertible in M (G). Set ν := δe − λ−1 µs
∗ µa.c. , and
notice that, because M 1 (G) is an ideal in M (G), ν ∈ M 1 (G). Since
−1
δe − λ−1 µs
∗ δe − λ−1 µ = δe − λ−1 ν,
SPECTRAL RADIUS FORMULA AND APPLICATIONS
7
the right-hand side is invertible in D ∗ (G), because the left side is, and
−1
−1
−1
(3.6)
δe − λ−1 µ
= δe − λ−1 ν
∗ δe − λ−1 µs
.
We know from Theorem 3.1 however, that for ν ∈ M 1 (G), δe − λ−1 ν is invertible in
−1
D∗ (G) iff it is invertible in M (G). Thus we must have that δe − λ−1 ν
∈ M (G),
−1
−1
whence, by (3.6), δe − λ µ
∈ M (G) ∗ M (G) = M (G).
This proves (3.5).
In order to establish the proposition, we now only need to show that
(3.7)
ρD∗ (G) (µ) 6 sup ρ µ
b(R) ∨ kµs k .
b
R∈G
b such
Fix > 0. By the Riemann-Lebesgue lemma, there exists a finite set Fb ⊆ G
b
b
b
that kb
µa.c. (R)k < for all R ∈ G r F . Having fixed F , choose n() ∈ N so that
n
kb
µ(R)n k 6 ρ µ
b(R) + for all R in the finite set Fb , whenever n > n(). Then
sup kb
µ(R)n k 6
b
R∈G
6
6
n
max kb
µ(R)n k ∨ sup kb
µ(R)k
b
R∈F
b F
b
R∈Gr
n
max kb
µ(R) k ∨ sup
b
R∈F
b F
b
R∈Gr
n
∨ + kµs k
sup ρ µ
b(R) + b
R∈G
n
kb
µa.c. (R)k + kb
µs (R)k
n
.
Since was arbitrary, this shows (3.7) and completes the proof of the proposition.
Proof of Theorem 2.1. Let µ ∈ M (G) and write ρ (µ) for ρM (G) (µ). Since
n
kµn k > kb
µ(R)n k > ρ µ
b(R)
b and n ∈ N, it is clear that ρ(µ) > sup b ρ µ
for all R ∈ G
b(R) . Since also
R∈G
kµn k = k(µn )a.c. k + k(µn )s k > k(µn )s k, it follows that
1/n
(3.8)
ρ(µ) > sup ρ µ
b(R) ∨ inf k(µn )s k
.
b
R∈G
n∈N
The reverse inequality follows from Proposition 3.2 applied to powers µ n of µ.
For each n ∈ N,
ρ (µn ) 6 k(µn )s k ∨ sup ρ ([b
µ(R)]n ) ,
b
R∈G
and since in any Banach algebra ρ (xm ) = [ρ(x)]
m
for all x and m, it follows that
n
b(R)
,
[ρ (µ)] = ρ (µn ) 6 k(µn )s k ∨ sup ρ µ
n
b
R∈G
which, together with (3.8), establish Theorem 2.1.
We conclude this section with a proof of the statement appearing in Remark (1)
after Corollary 2.2.
S
Proof of (2.2). It is easy to see that σD∗ (G) (f ) ⊇ R∈Gb σ fb(R) . Next recall the
group C ∗ -algebra C ∗ (G) (with the norm k k∗ ); it is well-known [11, Theorem 28.40]
that C ∗ (G) is isometrically isomorphic to the space
b := (ΛR ) b : ΛR ∈ L(CdR ), kΛR k −→ 0 as R −→ ∞ ,
C 0 (G)
R∈G
8
M. ANOUSSIS AND D. GATZOURAS
with norm supR∈Gb kΛR k, where L(CdR ) denotes the Banach space of bounded linear
b has the discrete topology.)
operators Λ : CdR −→ CdR . (Recall that G
b
Suppose now that, for some λ ∈ C r {0}, λIdR − fb(R) is invertible for all R ∈ G,
and define
i−1
h
− λ−1 IdR
(3.9)
H(R) = λIdR − fb(R)
b Since, by the Riemann-Lebesgue lemma, fb(R) −→ 0 as R −→ ∞,
for all R ∈ G.
one has that fb(R) < |λ| for all but finitely many R, and therefore the formula
∞
∞
h
h
in
in
X
X
−n b
−1
−1
−1
λ−n fb(R)
λ
f (R) − λ IdR = λ
H(R) = λ
n=1
n=0
is valid for all but finitely many R. It follows that
fb(R)
−1
kH(R)k 6 |λ|
|λ| − fb(R)
for all but finitely many R, and therefore, by the preceding paragraph, H(R)
defines an element h in C ∗ (G). Then x := λ−1 δe + h is in D ∗ (G). Since also
i−1
h
,
x
b(R) = λ−1 IdR + H(R) = λIdR − fb(R)
b
R∈G
−1
by (3.9), we must have that x = (λδe − f ) . Thus λδe − f is invertible in D ∗ (G).
S
This shows that σD∗ (G) (f ) ⊆ R∈Gb σ fb(R) ∪ {0}. It follows that σD∗ (G) (f ) =
S
b
b σ f (R) ∪ {0} unless G is finite (cf. proof of Theorem 3.1). By Theorem
R∈G
3.1, this completes the proof for G infinite. The case of finite groups is dealt with
easily.
4. An Application: Convergence in the Total Variation Distance of
Random Walks on Compact Groups
Let G be a compact group and µ a regular Borel probability measure on G, and
assume, without loss of generality, that µ is not concentrated on a proper closed
subgroup of G. It is then well known that µn := µ ∗ · · · ∗ µ converges weak∗ iff
µ is not concentrated on a coset of a proper, closed, normal subgroup of G, and
when this is the case, µn −→ λG (cf. [23]; see also [12] and [13]). In the sequel, we
shall refer to this statement as the weak∗ limit theorem for random walks on the
compact group G. Furthermore, we shall use the following, more or less standard
terminology, suited to the statement of the above limit theorem.
Terminology. If G is a compact group and µ a regular Borel probability measure
on G, we shall say that the pair (G, µ) is adapted if µ is not supported by a proper
closed subgroup of G. When (G, µ) is adapted, we shall say that µ is strictly
aperiodic if it is not concentrated on a coset of a proper, closed, normal subgroup
of G.
Now assume for the moment that G is a compact abelian group, whence one
has the usual Gelfand theory available, and fix a regular Borel probability µ on G,
such that the pair (G, µ) is adapted and µ is strictly aperiodic. Let S(M (G)) be
the spectrum of M (G) (cf. [8], Section 1.2), and recall that S(M (G)) is a compact
b of the group
subset of M (G)∗ , the Banach-dual of M (G), that each character χ ∈ G
SPECTRAL RADIUS FORMULA AND APPLICATIONS
9
b is part of the
G induces an element of S(M (G)) via the Fourier transform (i.e., G
spectrum of M (G)), and that kψk = 1 for each ψ ∈ S(M (G)) [8, Proposition 1.10].
Furthermore, for ψ ∈ S(M (G)), write νb(ψ) := ψ(ν) for the Gelfand transform of
ν ∈ M (G). Then,
kµn − λG k
1/n
−→
sup
ψ∈S(M (G))r{1G }
|b
µ(ψ)| =: a
[8, Theorem 1.13], and we claim that, unless all powers µ, µ 2 , . . . of µ are singular
with respect to Haar measure λG , the quantity a on the right is < 1, and hence
kµn − λG k tends to 0 as an ; that is, in this case, the convergence in the limit
theorem for random walks holds in (the total variation) norm.
Indeed, assume first that µa.c. 6= 0. Observe that
\
a=
sup
(µ − λG )(ψ) ,
ψ∈S(M (G))
and since S(M (G)) is compact, the supremum on the right is attained for some
b then
ψ ∈ S(M (G)). Fix such a ψ. If ψ ∈ S(M (G)) r G,
µs (ψ)| 6 kµs k < 1,
− λG )(ψ) = |b
a = (µ\
b implies that νb(ψ) = 0 for all absolutely continuous
because ψ ∈ S(M (G)) r G
1
measures ν ∈ M (G) [10, p. xx], and because µa.c. 6= 0 implies
that kµs k < 1. If,
b then either ψ = 1G , whence a = (µ\
on the other hand, ψ ∈ G,
− λG ) (1G ) = 0, or
ψ 6= 1G , whence
µ(ψ)| ,
− λG )(ψ) = |b
a = (µ\
which is again < 1, this time by adaptedness and strict aperiodicity of µ (see Lemma
4.3 below). Thus, in any case, a < 1.
Finally, if µa.c. = 0 but the absolutely continuous part of some µn is non-trivial,
then the discussion of the preceding paragraph applies to µ n , and since
n
ρ (µn − λG ) = ρ ((µ − λG )n ) = [ρ(µ − λG )] = an ,
we again get that a < 1.
The spectral radius formula of Theorem 2.1 enables one to follow this approach
in the non-abelian case as well. In particular, using Theorem 2.1 we prove the
following.
Theorem 4.1. Suppose G is a compact group, and µ a regular Borel probability
measure on G. Then
1/n
1/n
−→
sup
ρ µ
b(R) ∨ inf k(µn )s k
kµn − λG k
.
b
R∈Gr{1
G}
n∈N
In particular, kµn − λG k −→ 0 iff
(1) (G, µ) is adapted and µ is strictly aperiodic, and
(2) µ, µ2 , . . . are not all singular with respect to Haar measure λG .
Moreover, the convergence kµn − λG k −→ 0 takes place exponentially fast when it
holds.
10
M. ANOUSSIS AND D. GATZOURAS
Corollary 4.2. Suppose G is a compact group, and µ a regular Borel probability
measure on G. Suppose further that µ is absolutely continuous with respect to Haar
measure λG on G. Then
1/n
lim kµn − λG k
= max ρ µ
b(R) .
n−→∞
b
R∈Gr{1
G}
If in particular (G, µ) is adapted and µ strictly aperiodic, then kµn − λG k −→ 0
exponentially fast.
Remarks.
(1) Corollary 4.2 already appears in [13] [13, Theorem 13 and §5.2] (see also
[12]). (As also pointed out by Bhattacharya [1] however, the assertions
made in [13] regarding absolutely continuous measures ((5.9) and §5.2.3)
are inaccurate, as they lack the necessary hypotheses of adaptedness and
strict aperiodicity.) However, Corollary 4.2 also gives the rate of decay of
kµn − λG k; in particular, the estimate
1/n
lim kµn − λG k
= max ρ µ
b(R) =: a
n−→∞
b
R∈Gr{1
G}
appears to be different than the one in [13], and this estimate is sharp, in
the sense that, for each > 0, one has that
an 6 kµn − λG k 6 (a + )n ,
the left inequality for all n ∈ N (cf. (2.1)), the right for sufficiently large n.
This estimate, valid when µ has an L1 -density, is also to be compared to a
well known estimate of Diaconis and Shahshahani [6], of use only when µ n
P
2
2
µ(R)n k2 (here k k2 denotes
has an L2 -density: kµn − λG k 6 R6=1G dR kb
Hilberti–Schmidt norm). This latter inequality has been used, for central
measures whence it takes a simpler form, by Diaconis and Shahshahani (see
[5], [6], [7]), and by Rosenthal [16], to establish certain cut-off phenomena,
as they call them.
(2) A weak form of Theorem 4.1 seems to first have appeared in Bhattacharya
[1, Theorems 2 and 3]. In the general form presented here, it also appears
in Mindlin [14, Theorem 1], and in Ross and Xu [17, Theorem 4.1] (except
again for the precise rate of decay to 0).
(3) Examples where µn does not converge to Haar measure in norm although
µn −→ λG weak∗ , abound. For an elementary such example, take G to
be the multiplicative group S1 —the unit circle—and let µ be any Borel
probability measure concentrated on the rationals (i.e., the roots of unity),
having positive mass at every rational. Notice however, that in contrast
to this abelian example one also has the case described in the following
remark.
(4) Utilizing results of Ragozin [15], and arguing as in the discussion in the
beginning of this section pertaining to abelian groups, one can actually say
more for central measures on compact, simple Lie groups. If G is such a
group which is also connected and µ is central, then µ n −→ λG weak∗ iff
µn −→ λG in norm iff µc 6= 0, where µc stands for the continuous (i.e.,
SPECTRAL RADIUS FORMULA AND APPLICATIONS
11
atomless) part of µ. Furthermore,
1/n
kµn − λG k
−→
sup
b
R∈Gr{1
G}
d−1
R |µ (χR )| ,
where χR (x) := tr R(x) , x ∈ G, stands for the character of the repreR
b and µ (χR ) :=
χ dµ. In particular, kµn − λG k −→ 0
sentation R ∈ G,
G R
exponentially fast when µc 6= 0. These facts remain valid for disconnected
(compact, simple, Lie) groups, except that the condition µ c 6= 0 may no
longer suffice for the convergence µn −→ λG to hold. (In relation to this
see also the next remark.)
(5) When G is a connected compact group, and µ an absolutely continuous,
with respect to Haar measure, regular Borel probability measure on G,
then, automatically, (G, µ) is adapted and µ strictly aperiodic. This follows
from the fact that, if G is (locally) compact and connected, then the only
closed subgroup of G having positive λG -measure is G itself. In fact when
G is connected, the condition that (µn )a.c. 6= 0 for some n ∈ N, already
suffices for adaptedness and strict aperiodicity of µ.
We now proceed to the proof of Theorem 4.1. We shall use the following lemma,
which is also implicit in the proof of Theorem 3.3.5 in [23]. For the convenience of
the reader, we present a short argument here as well.
Lemma 4.3. Let G be a compact group, and µ a regular Borel probability
measure
on G not supported by a proper closed subgroup of G. If ρ µ
b(R) = 1 for some
b with R 6= 1G ,
non-trivial, irreducible, unitary representation R of G, i.e., R ∈ G
then there exists a closed, normal, proper subgroup H of G with µ(gH) = 1 for
some g ∈ G.
b since µ
Proof. Suppose that ρ µ
b(R) = 1 for some R ∈ G;
b(R) is finite-dimensional,
it must then have an eigenvalue ξ ∈ C of modulus 1. Let u ∈ C dR be a corresponding
eigenvector of norm 1. We shall show that then R(x) has the eigenvalue ξ with the
same eigenvector u, i.e., that R(x)u = ξu, for µ-a.e. x ∈ G. Indeed, one has that
Z
Z
2
R x−1 u, ξu µ(dx),
R x−1 u, µ
b(R)u µ(dx) =
1 = kb
µ(R)uk =
G
G
and since
the integrand
is of modulus 6 1 and µ is a probability measure, it follows
that R x−1 u, ξu =1 for µ-a.e. x. It then follows from the Cauchy–Schwartz
inequality that R x−1 u = ξu for µ-a.e. x. Thus µ is concentrated on the set
C := x ∈ G : R(x)u = ξu , which is easily seen to be both a left and right coset of
the subgroup H := {x ∈ G : R(x)u = u}. Note also that H is a proper subgroup,
since R 6= 1G and R is irreducible. Finally, {x ∈ G : xH = Hx} is a closed subgroup
of G, containing the coset C; since G is assumed to be the smallest closed subgroup
of G supporting µ, we must have that xH = Hx for all x ∈ G, i.e., H is normal in
G. (This last argument is taken from Lemma 3.3.3 of [23].)
Proof of Theorem 4.1. When µn ⊥ λG for all n ∈ N, then kµn − λG k = 2 for all
n ∈ N, whence µn cannot converge to λG in norm.
Next let µ be a regular Borel probability measure on G such that (G, µ) is adapted
and µ is strictly aperiodic, and assume first that µ has a non-trivial absolutely
12
M. ANOUSSIS AND D. GATZOURAS
n
continuous part. It is straightforward to verify that µ n − λG = (µ − λG ) , so that
1/n
1/n
kµn − λG k
−→ sup ρ (µ\
− λG )(R) ∨ inf k(µn )s k
=
n∈N
b
R∈G
sup
b
R∈Gr{1
G}
1/n
ρ µ
b(R) ∨ inf k(µn )s k
n∈N
,
by Theorem 2.1. Since µa.c. 6= 0, we must have that kµs k < 1. Since (G, µ) is
b
adapted and µ is strictly aperiodic, we also have that ρ µ
b(R) < 1 for all R ∈ G
with R 6= 1G , by Lemma 4.3, and
it then follows from the Riemann-Lebesgue
b is a finite set for which
lemma that supR∈Gr{1
ρ
µ
b
(R)
< 1; for if Fb ⊆ G
b
G}
b r Fb, then
kb
µa.c. (R)k < 12 (1 − kµs k) for all R ∈ G
kb
µ(R)k 6 kb
µa.c. (R)k + kb
µs (R)k 6 12 1 − kµs k + kµs k = 12 1 + kµs k
b r Fb , whence
for all R ∈ G
sup
ρ µ
b(R) 6
b
R∈Gr{1
G}
6
Thus
sup
b
R∈Gr{1
G}
max
br{1G }
R∈F
max
br{1G }
R∈F
ρ µ
b(R) ∨ sup kb
µ(R)k
ρ µ
b(R) ∨
b F
b
R∈Gr
1
2
+
1
2
kµs k < 1.
1/n
ρ µ
b(R) ∨ inf k(µn )s k
< 1,
n∈N
and so kµn − λG k −→ 0, exponentially fast.
It remains to consider the case where µa.c. = 0 but the absolutely continuous part
of some µn is non-trivial. But in this case, we know from the preceding paragraph
that ρ (µn − λG ) < 1, and since
n
n
[ρ (µ − λG )] = ρ (µ − λG ) = ρ (µn − λG ) ,
we again have that kµn − λG k −→ 0 exponentially fast.
(Notice that in the last paragraph we used the fact that when (G, µ) is adapted
and µ is strictly aperiodic, then this is also the case for any power µ k of µ. This can
be seen as follows: if µk were concentrated on a proper closed subgroup
H of G or
on a coset of a proper, closed, normal subgroup, then the sequence µnk : n ∈ N
would either converge weak∗ to Haar measure λH on H or not converge at all,
by the weak∗ limit theorem; however, this contradicts the fact that {µ n } itself
converges weak∗ to Haar measure on G, since (G, µ) is adapted and µ is strictly
aperiodic.)
5. Another Application: Convergence of Densities
Given Corollary 4.2, it is trivial to derive the following local limit theorem for
compact groups.
Corollary 5.1. Suppose that G is a compact group, and µ a regular Borel probability measure on G such that (G, µ) is adapted and µ is strictly aperiodic. If µ has
a density f with respect to Haar measure λG , then
f n = f ∗ · · · ∗ f −→ 1G
(λG -a.e.).
SPECTRAL RADIUS FORMULA AND APPLICATIONS
13
Proof. Given Corollary 4.2, the proof is a trivial application of the Borel–Cantelli
lemma. By the Markov inequality, for any given > 0,
λG {x ∈ G : |f n (x) − 1| > } 6 −1 kf n − 1G k1 = −1 kµn − λG k 6 −1 cn
for all sufficiently large n, where, by Theorem 4.2, we may choose c < 1.Thus
∞
X
λG {x ∈ G : |f n (x) − 1| > } < ∞,
n=1
and the Borel-Cantelli lemma now implies the result.
1+δ
In fact however, the following stronger result obtains for measures with an L density. A similar result also appears in [21] as Corollary 3, with an entirely different
(and rather more elaborate) proof, however.
Theorem 5.2. Let G be a compact group and µ a regular Borel probability measure
on G. If µ has an Lp -density f with respect to Haar measure λG , for some p > 1,
then
1/n
sup |f n (x) − 1|
−→ max ρ µ
b(R) .
b
R∈Gr{1
G}
x∈G
If, in particular, (G, µ) is adapted and µ is strictly aperiodic, then f n −→ 1G
uniformly, exponentially fast.
Remark. Theorem 5.2 is optimal, in the sense that there exists f ∈ L 1 (S1 ) for
which f n is unbounded for all n ∈ N (where S1 denotes the multiplicative group of
complex numbers of modulus 1). An example of such an f is given in [20], p. 140;
see also [17], p. 417.
The proof of Theorem 5.2 relies on Corollary 2.2 and the following.
Lemma 5.3. Suppose G is a unimodular, locally compact group, and p > 1. Let
k = d1/(p − 1)e, and set
p
(n ∈ N).
pn :=
p − (n ∧ k)(p − 1)
If g ∈ L1 (G) ∩ Lp (G), then g n ∈ L1 (G) ∩ Lpn (G) for all n ∈ N, and furthermore,
g n ∈ C0 (G) for all n > k and
sup |g n (x)| 6 kgk g k g n−k−1 ,
x∈G
p
q
1
where q = p/(p − 1) is the exponent conjugate to p.
Note. For a positive number a, dae denotes the smallest integer > a. For a, b > 0,
a ∧ b stands for the minimum min{a, b}.
Proof of Lemma 5.3. This follows from the Riesz-Thorin interpolation theorem [9,
Theorem 6.27], in combination with Propositions 2.39a and 2.40 of [8] (cf. [8], p. 54),
0
00
and the general fact that Lr ∩ Lr ⊆ Lr for 0 < r 0 6 r 6 r 00 . Notice that the
definition of k ensures that pn > q for all n > k.
Proof of Theorem 5.2. It is easily
R checked that
R g ∗ h − 1G = (g − 1G ) ∗ (h − 1G )
whenever g, h ∈ L1 (G) satisfy G gdλG = G hdλG = 1. Therefore, f n − 1G =
n
(f − 1G ) , since f is a probability density. Theorem 5.2 now follows by applying
Corollary 2.2 to g = f − 1G , and using Lemma 5.3 and the inequality
n
sup |f n (x) − 1| = sup |(f − 1G )n (x)| > k(f − 1G ) k1 = kf n − 1G k1 ,
x∈G
x∈G
14
M. ANOUSSIS AND D. GATZOURAS
valid by compactness of G. Thus
sup |f n (x) − 1|
x∈G
1/n
−→
max
b
R∈Gr{1
G}
ρ µ
b(R) ,
and the last quantity is < 1 when (G, µ) is adapted and µ strictly aperiodic, by the
Riemann-Lebesgue lemma and Lemma 4.3.
6. Homogeneous Spaces
In what follows, if X and Y are measurable spaces, T : X −→ Y a measurable
map, and µ a measure on X, then µ ◦ T −1 denotes the measure µ ◦ T −1 (A) :=
µ T −1 (A) on Y .
In this section, we assume that K is a compact homogeneous (Hausdorff) space,
acted upon transitively by the compact group G, on the left. One may then consider
random walks on K induced by the action of G on K: If µ is a regular Borel
probability measure on G, and z0 a fixed base-point in K (the starting point of the
random walk), define π : G −→ K by π(x) := xz0 , and consider the random walk
ν n := µn ◦ π −1 , where, as before, µn = µ ∗ · · · ∗ µ. The results of the previous
sections apply then to µn , and may be transferred to K via the mapping π. In
particular, notice that if mK is the unique G-invariant probability measure on K
[8, Theorem 2.49], then mK = λG ◦ π −1 , and therefore kν n − mK k 6 kµn − λG k.
Strict inequality may occur here, however, if µ happens to be H-invariant on the
right (i.e., µ ∗ λH = µ), where H is the isotropy subgroup of G fixing the basepoint z0 , then one actually has that kν n − mK k = kµn − λG k. Notice also that
µn −→ λG weak∗ implies ν n −→ mK weak∗ , and that one also has the reverse
implication when µ is H-invariant on the right. (The implications regarding Hright-invariant measures follow from the fact that such measures are completely
determined by how they integrate the functions of the form ϕ ◦ π, ϕ ∈ C(K).)
Finally, note that ν n ⊥ mK always implies that µn ⊥ λG , and that again, one also
has the reverse implication when µ is H-invariant on the right.
When ν (= µ ◦ π −1 ) is invariant under the action of H, one can always consider
the measure µ̄ instead of µ, where µ̄ is the unique bi-invariant under H measure on G
projecting to ν on K. Indeed, if µ̄ := µ ∗ λH , then µ̄ is the unique H-right-invariant
measure on G projecting to ν on K. Since λH ∗ µ̄ is also H-right-invariant, and since
it also projects to ν, by the H-invariance of ν, it follows from the uniqueness of
such right-invariant measures that λH ∗ µ̄ = µ̄, whence µ̄ is also left-invariant, and
hence bi-invariant under the action of H. Furthermore, ν n as defined above, i.e.,
ν n = µn ◦ π −1 , also satisfies ν n = µ̄n ◦ π −1 , since, by the left-invariance λH ∗ µ̄ = µ̄
of µ̄,
µ̄n ◦ π −1 = (µn ∗ λH ) ◦ π −1 = µn ◦ π −1 = ν n .
Therefore kν n − mK k = kµ̄n − λG k −→ 0, unless µ̄n ⊥ λG for all n, provided
of course µ̄ is adapted and strictly aperiodic. In particular, kν n − mK k −→ 0 iff
ν n −→ mK weak∗ and not all of ν, ν 2 , . . . are singular with respect to mK .
Reversing the point of view, in case a measure ν is given on K, which is invariant under the action of the stabilizer H of a point z 0 in K, one may take this
z0 as the base-point defining the map π, and µ to be the unique measure on G
projecting to ν via π and which is H-invariant from the right. This measure is
then also H-invariant from the left, and hence bi-invariant. (In this case, the measure ν n = µn ◦ π −1 corresponds to convolution ν ∗ · · · ∗ ν as usually defined for
invariant measures on homogeneous spaces.) One may then restrict attention to
SPECTRAL RADIUS FORMULA AND APPLICATIONS
15
the sub-algebra M (H\G/H) of measures on G bi-invariant under the action of H.
If furthermore (G, H) is a Gelfand pair, one can actually treat questions pertaining
to norm-convergence of ν n via the usual Gelfand theory in the way indicated by the
methods of the present paper (see beginning of Section 4), as then the sub-algebra
M (H\G/H) of interest is commutative. Theorem 4.1 however, allows one to also
study such questions for random walks on homogeneous spaces for which (G, H) is
not a Gelfand pair, in the way indicated above.
By way of illustration, we mention that “the drunkard’s walk on the sphere”,
considered by Su in [24], and the random walks considered by Bingham in [2] in
the discrete-parameter case (cf. [2]), may be described in the above manner. (In
fact the random walks in [24] are special cases of those in [2].) Thus in particular,
Proposition 2b of [2] may be obtained directly from what we called the weak ∗ limit
theorem for random walks on G = SO(k + 1) in this paper (cf. Section 4). And
one may in fact obtain convergence in norm of these random walks to the uniform
distribution on the k-sphere, from Theorem 4.1. This norm-convergence of these
random walks seems to be new.
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[21]
Department of Mathematics, University of the Aegean, 832 00 Karlovasi – Samos,
Greece
E-mail address: [email protected]
Agricultural University of Athens, Mathematics, Iera Odos 75, 118 55 Athens, Greece
E-mail address: [email protected]