Entropy of Probability Measures on FiniteCommutative Hypergroups

奈良教育大学学術リポジトリNEAR

Entropy of Probability Measures on Finite Commutative Hypergroups

著者 FUNAKOSHI Yukari, KAWAKAMI Satoshi journal or

publication title

奈良教育大学紀要. 自然科学

volume 57

number 2

page range 17‑20

year 2008‑10‑31

URL http://hdl.handle.net/10105/712

1.Introduction

Roughly speaking, the hypergroup convolution is a probabilistic extension of the group convolution. The concept of convolution of measures on a locally com- pact group has been generalized beyond the group case in the axiomatic setting of a hypergroup, due to C.F.

Dunkl, R.I. Jewett, and R. Spector around 1975.

In this paper we establish a notion of entropy of probability measures on finite commutative hyper- groups which is compatible with usual entropy of ran- dom walks on finite symmetric regular graphs.

Wildberger⁽¹⁰⁾ studied a certain entropy of probability measure on finite hypergroups related with informa- tion theory. Our definition of entropy is different from his notion of entropy.

Let K= {c₀,c₁, …,cn} be a finite commutative hyper- group with the *-algebra A(K). We call the invariant measure μ=μK= w(ck)ckon Kthe canonicalHaar measure of K.

For a probability measureν= a_kc_kon K, we define a entropy S_μ(ν^{) of}νrelative toμby

S_μ(ν^{) =}−ν

(

)

Letν0denote the normalized Haar measure of Kwhich

is given byν0= μ.

Then we have the following results.

In Theorem 1we show 0 S_μ(ν^{) log w(K)}and we characterize the probability measureνsuch that the entropy S_μ(ν⁾attains the maximum value.

In Theorem 2 we show the following. Let H =

(H, A(H))be a generalized orbital hypergrgroup of K=

(K,A(K)) by the conditional expectation E from A(K) onto A(H)such that H= E(K). Then μH= E(μK)holds for the canonical Haar measures μKof Kand μHof H. For a probability measureνon Kwe have S_μK(ν^{) S}μH(E(ν^)).

Moreover, the equality S_μK(ν⁾ = S_μH(E(ν⁾⁾holds if and only ifν^{= E(}ν⁾ ∈A(H).

This work has been done by developing some results in bachelor's thesis⁽²⁾by the first author in 2007.

2.Preliminaries 

We recall some notions and facts on finite commu- tative hypergroups from Bloom-Heyer's Book⁽¹⁾ and Wildberger's report⁽⁹⁾. K:= (K, A) is called a finite com- mutative hypergroupif the following conditions (1)〜 (6)are satisfied.

(1) Ais a *-algebra over with the unit c₀. (2) K = {c₀, c₁, …, c_n} is a linear basis of A.

(3) K^*= K.

1 w(K)

a_k w(ck)

n

Σk=0

dν

dμ

n

Σ_k=0

n

Σk=0

Bull. Nara Univ. Educ., Vol. 57, No.2 (Nat.),2008

Entropy of Probability Measures on Finite Commutative Hypergroups

Yukari FUNAKOSHI and Satoshi KAWAKAMI

(Department of Mathematics, Nara University of Education, Nara 630-8528, Japan) (Received May 7, 2008)

Abstract

The purpose of this paper is to investigate entropy of probability measures on finite com- mutative hypergroups. In fact, we give a notion of entropy which is compatible with entropy of random walks on finite symmetric regular graphs. We study some fundamental propaties of the entropy concerning with maximality. (AMS Subject Classification : 43A62, 20N20.)

Key Words: hypergroup, entropy, random walk

(4) c_ic_j= n^k_{i j}c_k, where n^k_{i j}is a non-negative real number such that

c^*_i= c_j n⁰_{i j}> 0, c^*_i=/ c_j n⁰_{i j}= 0.

(5) n^k_{i j}=1for any i, j.

(6) c_ic_j= c_jc_ifor any i, j.

We often denote A by A(K) for K = (K, A). The weightof an element c_i∈Kis defined by w(c_i) := (n⁰_{i j})⁻¹ where c_j = c^*_i, and the total weight of K is given by w(K) :=Σⁿi= 0w(ci).

Let M¹(K) denote the set of probability measures on K, i.e.

M¹(K) := {ν= a_kc_k: ak 0 (k= 0, 1, …, n), a_k= 1}.

Forν⁼Σⁿk=0 a_kc_k∈A(K), supportofνis defined by supp(ν^{) := {c}k: a_k=/0, k= 0, 1, …, n}.

Letω^(K) denote the normalized Haar measure of Kwhich is given by

ω^{(K) =} c_k.

Let Abe a *-algebra with the unit c₀and Bbe a *- subalgebra of Awith the unit c₀. Then a linear mapping Efrom Aonto Bis called a conditional expectation if the following conditions are satisfied.

(1) E(c₀) = c₀.

(2) E(yxz)= yE(x)z for x∈A, y, z∈B.

(3) E(x^*x) 0.

Let H= (H, A(H))and K= (K, A(K))be finite hyper- groups such that the *-algebra A(H)is realized in the *- algebra A(K). We call Ha generalized orbital hyper- groupof Kif there exists a conditional expectation E from A(K) onto A(H)such that H= E(K). This notion is a generalization of a usual orbital hypergroup.

3.Entropy of probability measures 

Let K= {c₀, c₁, …, c_n} be a finite commutative hyper- group with the *-algebra A(K). We call the invariant measureμK= w(ck)ckon Kthe canonicalHaar meas- ure of K. This μKis often denoted by μwhen Kis obvi- ous. For a probability measureν= a_kc_kon K, we de fine a entropy S_μ(ν^)of νrelative toμby

S_μ(ν^{) =}−ν

(

)

Letν0denote the normalized Haar measure of K which is given byν0= μ. Then we have the follow- ing theorem.

Theorem 1. The entoropy S_μν( ⁾ is non-negative and S_μν( ⁾ log w(K). The entropy S_μν( ⁾attains the max- imam value log w(K) if and only ifν⁼ν0. Moreover, S_μν( ⁾

= 0if and only if ak=1for some ksuch that w(ck)= 1.

Proof.By the fact that 0 1, −a_klog 0. Then it is clear that S_μν( ⁾ 0. Suppose that S_μν( ^{) =}

0. Then −a_klog =0for all k. This implies that

= 0or 1. If =1for some kthen ak= w(ck). Since 0 a_k 1and w(ck) 1, we obtain a_k= 1and w(ck)= 1. We note that aj=0for all jsuch that j=/k. Moreover, apply- ing Jensen's inequality, it is easy to see that S_μ(ν^{) =}

log w(K)if and only if = w(K)for all k, namely ak= . This implies thatν=ν0.

[Q.E.D.]

4.Entropy and generalized orbital hypergroups 

Let H= (H, A(H))and K= (K, A(K))be finite commu- tative hypergroups such that the *-algebra A(H)is real- ized in the *-algebra A(K). We call H a generalized orbital hypergroup of K if there exists a conditional expectation Efrom A(K) onto A(H)such that H= E(K).

When an action αof a finite group Gon a hypergroup K is given, an orbital hyeprgroup H= K^αis defined by the conditional expectation Eby

E(x)= αg(x) for x∈A(K).

We note that many hypergroups are obtained as gener- alized orbital hypergroups which are not necessarily usual orbital hypergroups. Refer to our paper(4).

Theorem 2. Let H = (H,A(H)) be a generalized orbital hypergrgroup of K= (K,A(K)) by a conditional expectation Efrom A(K) onto A(H)such that H= E(K).

ThenμH= E(μK) holds for the canonical Haar measures

μKof KandμHof H. For a probability measureνon K we have SμK(ν^{) S}μH(E(ν)). Moreover, the equality S_μK(ν⁾

= S_μH(E(ν⁾⁾holds if and only if ν^{= E(}ν⁾∈A(H).

Proof. Let Kand Hbe given by K= {c₀,c₁,…,c_n} and H= {d₀,d₁,…,d_m}, where c₀is the unit of Kand d₀ is the unit of H, and c₀= d₀. For each dj∈H, set

Σ_g∈G

1

|^G|

w(ck) w(K)

w(ck) ak

a_k w(ck)

a_k w(ck) a_k

w(ck)

ak

w(ck) ak

w(ck) 1

w(K)

a_k w(ck)

n

Σ_k=0

dν

dμ

n

Σk=0 n

Σk=0

w(c_k) w(K)

n

Σk=0

n

Σ_k=0

n

Σ_k=0

n

Σ_k=0

n

Σk=0

Yukari Funakoshi・Satoshi Kawakami 18

K(j)= {c∈K: E(c)= d_j}

= {c₁(j), c₂(j), …, c_n

j(j)}

We note that

K= K(j) and n_j = n

Moreover, it is easy to see that each dj∈His written as d_j= a_i(j)c_i(j) where a_i(j)= 1.

By this fact, we see that djμK=μKfor each dj∈H. Hence d_jE(μK)= E(djμK)= E(μK). This implies that the measure E(μK) is H-invariant so that E(μK)is a Haar measure of H.

Therefore E(μK)is written by E(μK)= cμHfor some con- stant c> 0. Since μKand μHis represented as

μK= w(c_k)c_k, μH= w(d_j)d_j,

and E(c₀)= d₀, we see that the constant cmust be 1so that μH= E(μK)holds. The canonical Haar measure μKof Kis given by

μK= w(ci(j))ci(j), where

K(j)= {c₁(j),c₂(j), …,c_n

j(j)} and K= K(j).

Since E(ci(j))= dj,

E(μK)=

(

)

By the fact thatμH= E(μK), we see that w(d_j)= w(c_i(j)).

For a probability measureν^{= a}kc_k= a_i(j)c_i(j) of K, E(ν) is given by

E(ν^{) =}

(

)

where bj= a_i(j). Then we get the following equalities.

S_μK(ν^{) =}− a_i(j)log ,

S_μH(E(ν⁾⁾=− b_jlog . We may assume that ai(j)> 0. Hence we see that

− log = log

log = log = −log ,

by Jensens' inequality. Hence we see that

− a_i(j)log −b_jlog .

Therefore we obtain that S_μK(ν^{) S}μH(E(ν)). Moreover, it is also obtained that the equality holds if and only if

= for all i=1, 2, …, nj. This implies that a_i(j)ci(j)= bjd_j, namely, ν^{= E(}ν⁾∈A(H).

[Q.E.D.]

Remark. When His an orbital hypergroup of K by an actionαof a group Gon K, the conditionν^{= E(}ν⁾

∈A(H) is equivalent to say thatνisα-invariant.

Therefore we note that the equality S_μK(ν^)=SμH(E(ν⁾⁾

holds if and only ifνisα-invariant.

Example. Let K= {c₀, c₁, c₂} be the cyclic group 3

of order three such that c³₁= c₀, c²₁= c₂, c^*₁= c₂, and c₂^*= c₁. Let H= {d₀, d₁} be the hypergroup of order two aris- ing from random walk on edges of a regular triangle where d²₁= d₀+ d₁, d^*₁= d₁, and w(d₁) = 2. Then we note that the hypergroup His realized in A(K)by the relation d₀= c₀and d₁= c₁+ c₂. We can interpret that this hypergroup His an orbital hypergroup by an action αof the group G= {e, g} (g²= e)of order two on Ksuch thatαg(c₁)= c₂andαg(c₂)= c₁. We can also interpret that this hypergroup His a generalized orbital hypergroup by the conditional expectation E from A(K) onto A(H) such that E(c₀)= d₀and E(c₁)= E(c₂)= c₁+ c₂= d₁. In this case the Haar measures μKof K and μHof Hare given by

μK= c₀+ c₁+ c₂, μH= d₀+ 2d₁. We note that

E(μK)= E(c₀)+ E(c₁)+ E(c₂)

= d₀+ d₁+ d₁

= d₀+ 2d₁

= μH.

For ν=a₀c₀+a₁c₁+a₂c₂∈M¹(K), E(ν)=a₀d₀+ (a₁+ a₂)d₁, we have

S_μK(ν⁾= −a₀ log a₀−a₁log a₁−a₂log a₂, S_μH(E(ν⁾⁾= −a₀ log a₀−(a₁+ a₂)log . Theorem asserts that the equality S_μK(ν⁾ =S_μH(E(ν⁾⁾

holds if and only if a₁ = a₂ which is equivalent to say thatνis anα-invariant measure.

a1 + a₂ 2 1 2 1 2 1

2 1 2 1 2 1 2

n_j

Σ_i=1

w(dj) bj

w(ci(j)) ai(j)

b_j w(dj) a_i(j)

w(ci(j))

n_j

Σi=1

b_j w(dj) w(ci(j))

b_j

nj

Σi=1

w(ci(j)) a_i(j) a_i(j)

b_j

nj

Σi=1

w(ci(j)) a_i(j) a_i(j)

b_j

n_j

Σ_i=1

a_i(j) w(ci(j)) a_i(j)

b_j

n_j

Σ_i=1

b_j w(d_j)

m

Σ_j=0

a_i(j) w(ci(j))

n_j

Σ_i=1

m

Σ_j=0

n_j

Σ_i=1

m

Σj=0 nj

Σ_i=1

m

Σj=0

nj

Σ_i=1

m

Σj=0 n

Σ_k=0

nj

Σi=1 nj

Σ_i=1

m

Σj=0

m

j=0 n_j

Σ_i=1

m

Σ_j=0

m

Σj=0 n

Σ_k=0

nj

Σ_i=1

nj

Σ_i=1

m

Σj=0 m

j=0

References

(1) Bloom, W.R. and Heyer, H. : Harmonic Analysis of Probability Measures on Hypergroups, 1995, Walter de Gruyter, de Gruyter Studies in Mathematics 20.

(2) Funakoshi, Y. : Entropy of probability measures on finite commutative hypergroups, Bachelor's thesis(Japanese), 2007.

(3) Funakoshi, Y. and Kawakami, S. : Entropy of probability measures on compact commutative hypergroups, in prepa- ration.

(4) Heyer, H., Jimbo, T., Kawakami, S., and Kawasaki, K. : Finite commutative hypergroups associated with actions of finite abelian groups, Bull. Nara Univ. Educ., Vol. 54 (2005), No.2., pp.23-29.

(5) Heyer, H., Katayama, Y., Kawakami, S., and Kawasaki, K. :

Extensions of finite commutative hypergroups, Scientiae Mathematicae Japonicae, 65, No. 3 (2007), pp.373-385.

(6) Kawakami, S. : Extensions of commutative hypergroups, to appear in Infinite Dimesional Harmonic Analysis IV, World Scientific, 2008.

(7) Kawakami, S. and Nakano, F. : Entropy of states on finite commutative hypergroups, in preparation.

(8) Kawakami, S. and Tai, M. : Entropy of probability measures on motion hypergroups, in preparation.

(9) Wildberger, N.J. : Finite commutative hypergroups and applications from group theory to conformal field theory, Applications of Hypergroups and Related Measure Algebras, Amer. Math. Soc., Providence, 1994, pp.413-434.

(10)Wildberger, N.J. : Duality and entropy for finite abelian hypergroups, preprint, Univ. of NSW (1989).

Yukari Funakoshi・Satoshi Kawakami 20