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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. References [1] R. N. Bhattacharya, Speed of Convergence of the n-Fold Convolution of a Probability Measure on a Compact Group, Z. Wahrsch. Verw. Gebiete 25 (1972), 1–10. [2] N. H. Bingham, Random Walk on Spheres, Z. Wahrsch. Verw. Gebiete 22 (1972), 169–192. [3] F. F. Bonsall and J. 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