ADELIC ANALYSIS AND FUNCTIONAL ANALYSIS ON THE FINITE ADELE RING. Ilwoo Cho

Transcription

1 Opuscua Math. 38, no , Opuscua Mathematica ADELIC ANALYSIS AND FUNCTIONAL ANALYSIS ON THE FINITE ADELE RING Iwoo Cho Communicated by.a. Cojuhari Abstract. In this paper, we study operator theory on the -agebra M, consisting of a measurabe functions on the finite Adee ring A Q, in extended free-probabiistic sense. Even though our -agebra M is commutative, our Adeic-anaytic data and properties on M are understood as certain free-probabiistic resuts under enarged sense of noncommutative free probabiity theory we-covering commutative cases. From our free-probabiistic mode on A Q, we construct the suitabe Hibert-space representation, and study a C -agebra M generated by M under representation. In particuar, we focus on operator-theoretic properties of certain generating operators on M. Keywords: representations, C -agebras, p-adic number fieds, the Adee ring, the finite Adee ring. Mathematics Subject Cassification: 05E5, G5, R47, R56, 46L0, 46L54, 47L30, 47L55.. INTRODUCTION The main purposes of this paper are: i to construct a free-probabiity mode under extended sense of the -agebra M consisting of a measurabe functions on the finite Adee ring A Q, impying number-theoretic information from the Adeic anaysis on A Q, ii to estabish a suitabe Hibert-space representation of M, refecting our free-distributiona data from i on M, iii to construct-and-study a C -agebra M generated by M under our representation of ii, and iv to consider free distributions of the generating operators of M of iii. Our main resuts iustrate interesting connections between primes and operators via free probabiity theory. c Wydawnictwa AGH, Krakow

2 40 Iwoo Cho.. REVIEW We have considered how primes or prime numbers act on operator agebras. The reations between primes and operator agebra theory have been studied in various different approaches. For instance, we studied how primes act on certain von Neumann agebras generated by p-adic and Adeic measure spaces e.g., [2]. Meanwhie, in [], primes are regarded as inear functionas acting on arithmetic functions. In such a case, one can understand arithmetic functions as Krein-space operators under certain Krein-space representations. Aso, in [3, 4] and [7], we considered free-probabiistic structures on a Hecke agebra HG p for primes p. In [6], we considered certain free random variabes in -agebras M p of a measurabe functions on the p-adic number fieds Q p in terms of the p-adic integrations ϕ p for a primes p. Under suitabe Hibert-space representations of M p, the corresponding C - agebras M p of M p are constructed and C -probabiity on M p is studied there., where ϕ p j are In particuar, for a j Z, we define C -probabiity spaces M p, ϕ p j kind of sectionized inear functionas impying the number-theoretic data on M p, in terms of ϕ p. Moreover, from the system Mp, ϕ p } j : p, j Z, of C -probabiity spaces, we estabish-and-consider the free product C -probabiity space, M Z, ϕ Mp, ϕ p j, p, j Z caed the Adeic C -probabiity space. Independenty, in [8], by using the free-probabiistic information from a singe C -probabiity space M p, ϕ p j aso introduced as above in [6], for arbitrariy fixed p, j Z, we estabished a weighted-semicircuar eement in a certain Banach -probabiity space generated by M p, ϕ p j, and reaized that corresponding semicircuar eements are we-determined. Motivated by [6], we extended the weighted-semicircuarity, and semicircuarity of [8] in the free product Banach -probabiity space of Banach -probabiity spaces of [8], over both primes and integers, in [5]..2. MOTIVATION AND DISCUSSION Motivated by the main resuts of [5, 6] and [8], we here estabish free-probabiistic modes under extended sense started from the finite Adee ring A Q, to provide simiar framworks of [5] and [8]. Even though our structures are based on the commutativity, and hence, they are not directy foowed origina noncommutative free probabiity theory, the proceeding processes, settings, and resuts are from free probabiity theory. Thus, we use concepts and terminoogy from free probabiity theory. In this very paper, we wi consider neither weighted-semicircuarity nor semicircuarity on our free-probabiistic structures, however, ater, our main resuts woud provide suitabe toos and backgrounds for studying those semicircuar-ike aws and

3 Adeic anaysis and functiona anaysis on the finite Adee ring 4 the semicircuar aw. Readers may reaize from our main resuts that the spectra properties of operators induced by measurabe functions on A Q, and those of free reduced words induced by measurabe functions on Q p under free product over primes p, obtained in [6] are very simiar under certain additiona conditions. It guarantees that one may/can obtain weighted-semicircuarity and semicircuarity on our structures as in [5] and [8] in future..3. OVERVIEW In Section 2, we briefy introduce backgrounds and a motivation of our works. Our free-probabiistic mode on M is estabished from Adeic cacuus, and the free distributiona data on M are considered in Section 3. Then, in Section 4, we construct a suitabe Hibert-space representation of our free-probabiistic mode of M, preserving the free-distributiona data impying number-theoretic information. Under representation, the corresponding C -agebra M is constructed. In Section 5, free probabiity on the C -agebra M is studied by putting a system of inear functionas dictated by the Adeic integration. In particuar, free distributions of generating operators of M are considered by computing free moments of them. In Sections 6, we further consider reations between our free-distributiona data and Adeic-anaytic information, by computing free distributions of generating operators of the free product C -agebras. Especiay, we focus on the free distributions of certain generating opeartors, caed +-boundary operators. In Section 7, by constructing free product C -probabiity spaces which are under usua sense of noncommutative free probabiity theory from our system of C -probabiity spaces which are under extended commutativity-depending sense of Sections 5 and 6, we investigate the connections between Adeic anaysis and our free-probabiistic structures. In the ong run, we study noncommutative free probabiity theory induced by the Adeic anaysis on the finite Adee ring A Q..4. MAIN RESULTS By appying free-probabiistic settings and terminoogy, we characterize the functiona-anaytic properties of the C -agebras induced by the finite Adee ring with free-probabiistic anguage. For instance, functiona-anaytic or spectra-theoretic information of certain operators in our C -agebras are determined by the forms of free moments, or joint free moments see Sections 5, 6 and 7. Since our C -agebra M is commutative, the free-probabiistic mode for M is non-traditiona, but such a mode in Sections 5 and 6 is perfecty fit to anaize our main resuts even though the freeness on it is trivia, moreover, this non-traditiona approaches become traditiona by constructing free product C -agebras in Section 7. The constructions of our free-probabiistic modes and corresponding free- -distributiona datas of operators are the main resuts of this paper. From these, one can see the connections between number-theoretic Adeic anaysis, functiona anaysis, and operator-theoretic spectra theory via free probabiity.

4 42 Iwoo Cho 2. RELIMINARIES In this section, we briefy mention about backgrounds of our proceeding works. See [9] and [0] and cited papers therein for number-theoretic motivations. 2.. FREE ROBABILITY Readers can check fundamenta anaytic-and-combinatoria free probabiity from [2] and [4] and the cited papers therein. Free probabiity is understood as the noncommutative operator-agebraic version of cassica probabiity theory and statistics covering commutative cases. The cassica independence is repaced by the freeness, by repacing measures to inear functionas. It has various appications not ony in pure mathematics e.g., [], but aso in reated scientific topics for exampe, see [2, 3, 5] and [8]. In particuar, we wi use combinatoria approach of Speicher e.g., [2]. Especiay, in the text, without introducing detaied definitions and combinatoria backgrounds, free moments and free cumuants of operators wi be computed. Aso, we use free product of agebras in the sense of [2] and [4], without detaied introduction p-adic CALCULUS ON Q p In this section, we briefy review p-adic cacuus on the -agebras M p of measurabe functions on p-adic number fieds Q p, for p. For more about p-adic anaysis see e.g., [3]. For a fixed prime p, the p-adic number fied Q p is the maxima p-norm cosure in the set Q of a rationa numbers, where the p-norm p on Q is defined by x p a p k p p k, whenever x ap k, for some a Q, and k Z. For instance, , , 3 3 and q 3 q0, for a q \ 2, 3}. q0 Every eement z of Q p is expressed by z k N a k p k, with a k 0,,..., p }, 2. for some N N. So, from the p-adic addition and the p-adic mutipication on the eements formed by 2. under in Q p, the set Q p forms a ring agebraicay e.g., [3].

5 Adeic anaysis and functiona anaysis on the finite Adee ring 43 Moreover, one can understand this Banach ring Q p as a measure space, Q p Q p, σq p, µ p, where σq p is the σ-agebra of Q p consisting of a µ p -measurabe subsets, where µ p is a eft-and-right additive-invariant Haar measure on Q p, satisfying where Z p is the unit disk µ p Z p, Z p x Q p : x p }. of Q p, consisting of a p-adic integers x, having their forms x Moreover, if we define a k p k with a k 0,,..., p }. k0 U k p k Z p p k x Q p : x Z p }, 2.2 for a k Z, then these µ p -measurabe subsets U k of 2.2 satisfy Q p k Z U k, and µ p U k p k µ p x + U k, for a k Z, and... U 2 U U 0 Z p U U e.g., [3]. In fact, the famiy U k } k Z forms a basis of the Banach topoogy for Q p. Define now subsets k of Q p by k U k \ U k+, for a k Z. 2.4 We ca such µ p -measurabe subsets k, the k-th boundaries of U k in Q p, for a k Z. By 2.3 and 2.4, one obtains that Q p k Z k, where means the disjoint union, and for a k Z. µ p k µ p U k µ p U k+ p k, 2.5 pk+

6 44 Iwoo Cho Now, et M p be the set of a µ p -measurabe functions on Q p, i.e., M p C [χ S : S σq p }], 2.6 where C[X] mean the agebras generated by X, understood as agebras consisting of a poynomias in X, for a sets X. So, f M p if and ony if f S σq p t S χ S with t S C, where means the finite sum, and χ S are the usua characteristic functions of S σ Q p. Then it forms a -agebra over C. Indeed, the set M p of 2.6 is an agebra under the usua functiona addition, and functiona mutipication. Aso, this agebra M p has the adjoint, def t S χ S t S χ S, S σg p S σg p where t S C, having their conjugates t S in C. Let f be an eement of the -agebra M p of 2.6. Then one can define the p-adic integra of f by fdµ p t S µ p S. 2.7 Q S σq p p Note that, by 2.5, if S σq p, then there exists a subset Λ S of Z, such that Λ S j Z : S j }, 2.8 satisfying χ S dµ p Q p Q p χ S xj+ jdµ p µ p S j j Λ S j Λ S by 2.7 j Λ S µ p j j Λ S p j p j+, by 2.5, i.e., Q p χ S dµ p j Λ S p j p j+, 2.9 for a S σq p, where Λ S is in the sense of 2.8.

7 Adeic anaysis and functiona anaysis on the finite Adee ring 45 More precisey, one can get the foowing proposition. roposition 2. [6]. Let S σq p, and et χ S M p. Then there exist r j R, such that 0 r j in R, and Q p χ S dµ p j Λ S r j p j p j THE ADELE RING AND THE FINITE ADELE RING In this section, we introduce the Adee ring A Q, and the finite Adee ring A Q. For more information about A Q, A Q and the corresponding anaysis, see [3]. Definition 2.2. Let }, and identify Q with the Banach fied R, equipped with the usua-distance-metric topoogy. Let A Q be a set } x A Q x p p Q p for each p, where ony finitey many x q s p, are in Q q \ Z q, but a other x p s are contained in Z p of Q p 2. equipped with the addition +, and the mutipication, x p p + y p p x p + y p p, 2.2 x p p y p p x p y p p, 2.3 where Z p is the unit disk of Q p in the sense of Section 2, and where the entries x p + y p of 2.2, and the entries x p y p of 2.3 are the p-adic additioin, respectivey, the p-adic mutipication on the p-adic number fied Q p, for a p, and where x + y, and x y are the usua R-addition, respectivey, the usua R-mutipication. The Adee ring A Q is equipped with the product topoogy of the p-adic-norm topoogies for Q p s, for a p, and the usua-distance-metric topoogy of Q R, satisfying that xp p x p p, 2.4 Q p where p are the p-adic norms on Q p, for a p, and is the usua absoute vaue on R Q. From the above definition, the set A Q of 2. forms a ring agebraicay, equipped with the binary operations 2.2 and 2.3; and this ring A Q is a Banach space under its Q -norm of 2.4, by 2.. Thus, the set A Q of 2. forms a Banach ring induced by the famiy Q Q p } p Q R}.

8 46 Iwoo Cho Indeed, et X x p p A Q, and assume that there are p,..., p N, for some N N, such that x p Q p \ Z p, for a,..., N, and x q Z q, for a q \ p,..., p N }. Then, by 2. and 2.4, N N N x Q x p p x q q x p p x p p <. q \p,...,p N } From the definition 2., the Adee ring A Q is in fact the weak-direct product of Q, expressed by A Q p Q p 2.5 e.g., [0] and [3], where means the weak-direct product satisfying the conditions of 2.. So, the Adee ring A Q can be re-defined by the weak-direct product 2.5 of the famiy Q equipped with the norm 2.4. Definition 2.3. Let A Q be the Adee ring 2., or 2.5. Define a ring A Q by } A Q x p p xp Q p, for a p, and 0, x p p A Q, 2.6 set-theoreticay, equipped with the inherited operations of A Q, under subspace topoogy. Then this topoogica ring A Q of the Adee ring A Q is said to be the finite Adee ring. By 2.5 and 2.6, one can concude that A Q Q p, 2.7 p where means the weak-direct product. As in [3], one can understand the Adee ring A Q as a measure space equipped with the product measure, µ Q p µ p where µ p are the Haar measures on Q p, for a p, and µ is the usua Lebesgue measure on Q R. So, the finite Adee ring A Q of 2.6 can be regarded as a measure space equipped with the measure on the σ-agebra σa Q of A Q. µ µ p 2.8 p

9 Adeic anaysis and functiona anaysis on the finite Adee ring 47 Simiar to the -agebras M p, for p, one can define the -agebra M by M C [χ Y : Y σa Q }], 2.9 where µ is the measure 2.8 on the finite Adee ring A Q. By the definition 2.9, f M if and ony if f Y σa Q s Y χ Y, with s Y C, 2.20 where means the finite sum. Thus, one obtains the finite-adeic integration of f M by fdµ t Y µ Y, 2.2 A Y σa Q Q whenever f is in the sense of 2.20 in M. By the Adeic integration 2.2, one can naturay define a inear functiona ϕ on M by ϕf fdµ A Q Equivaenty, one can have a free probabiity space M, ϕ in the sense of [2] and [4]. Definition 2.4. Let M be the -agebra 2.9, and et ϕ be the inear functiona 2.22 on M. Then the free probabiity space M, ϕ is caed the finite-adeic -probabiity space. Remark 2.5. Remark that the term, finite-adeic probabiity space M, ϕ, does not mean it is a probabiity-measure-theoretic object. Moreover, since M is commutative, the pair M, ϕ is not a traditiona noncommutative-free-probabiity-theoretic structure, either. However, the construction of the mathematica pair M, ϕ is foowed by the definition of noncommutative free probabiity spaces in free probabiity theory we-covering commutative cases. To emphasize the construction, and to use this structure in our future research, we regard M, ϕ as a free probabiity space, and name it the finite-adeic probabiity space, even though it is not traditiona both in the measure-theoretic anaysis and in free probabiity theory. Reca that our finite Adee ring A Q is a weak-direct product of Q p } p by 2.7, i.e., A Q Q p, p and hence, Y σa Q, if and ony if there exist N N, and p,..., p N, such that Y p S p, where S p σq p, 2.23

10 48 Iwoo Cho with S p S p Q p if p p,..., p N }, Z p otherwise, for a p. Thus, if Y σa Q, then ϕ χ Y χ Y dµ χ Spdµ p A Q A Q µ S p µ p S p p p p χ Sp dµ p, Q p since µ µ p p N µ p S p µ q Z q q \ p,...,p N } by 2.23 N N µ p S p ϕ p χ Sp, 2.24 since µ q Z q, for a q. roposition 2.6. Let Y σa Q satisfy 2.23, and et χ Y be a free random variabe in our finite-adeic probabiity space M, ϕ. Then ϕ χ n Y N r Sp j j Λ Sp p j p j+, 2.25 for a n N, where r Sp j are in the sense of 2.0, for a j Λ Sp, for a,..., N. roof. Let Y be a µ-measurabe subset of A Q satisfying Then the µ-measurabe function χ Y satisfies that for a n N. So, χ n Y χ Y Y Y }} ϕ χ n Y ϕ χ Y n-times N χ Y, ϕ p χ Sp, by 2.24, for a n N. Therefore, by 2.0, we obtain the free-moment formua 2.25.

11 Adeic anaysis and functiona anaysis on the finite Adee ring 49 By 2.25, if f Y σa Q in M, where Y satisfy 2.23, then N S ϕf fdµ s Y A Y σa Q Q s Y χ Y, with s Y C, r S j j Λ S Y p Y p p j p j The formua 2.26 iustrates that the formua 2.25 provides a genera too to study finite-adeic cacuus. Notice that M p where means the weak-direct or weak-tensor product of -agebras. The isomorphism theorem 2.27 hods because of 2.7 and M p Theorem 2.7. Let M, ϕ be the finite-adeic probabiity space. Then M p M p, and ϕ p ϕ p, 2.27 where M p are in the sense of 2.6, and ϕ p are the p-adic integrations on M p, for a p. roof. The -isomorphism theorem of M in 2.27 is proven by 2.7 and The equivaence for ϕ in 2.27 is guaranteed by 2.8 and ANALYSIS OF M, ϕ In this section, we consider functiona-anaytic properties on our finite-adeic probabiity space M, ϕ. In particuar, such properties are represented by the distributiona data from eements of M, ϕ under free-probabiity-theoretic anguage, free moments. As appication, we show reations between our free moments and the Euer-totient-functiona vaues. Let M, ϕ be the finite-adeic probabiity space. From constructions, one can get that M ϕ p, 3. p M p, and ϕ p by 2.27, where ϕ p are the p-adic integrations on Q p, for a p and hence, ϕ is the Adeic integration on A Q. So, by abusing notation, one may/can re-write the reations of 3. as foows: M, ϕ M p, ϕ p. 3.2 p

12 50 Iwoo Cho Reca that, in [5, 6] and [8], we ca M p, ϕ p the p-adic probabiity spaces, for a p. Here, we concentrate on computing free distributions of eements generated by generating eements of M. Theorem 3.. Let Y,..., Y n σa Q, and χ Y M, ϕ, for,..., n, for some n N. Then there exist a unique finite subset o of, and X p σq p, for a p o, such that n ϕ χ Y p o r Xp j j Λ Xp p j p j+, 3.3 where r Xp j are in the sense of 2.0, and Λ Xp are in the sense of 2.8. roof. Let Y,..., Y n be µ-measurabe subsets of the finite Adee ring A Q, for n N. So, by 2.23, for each Y i, there exist a unique N i N, and p i,,..., p i,ni, such that Y i Sp, i with Sp i σq p, 3.4 p and S i p Sp i if p p i,,..., p i,ni }, Z p otherwise, for a p, for a i,..., n. If we et h N χ Y, and h χ Y, for,..., n, then n ϕh ϕ h i ϕ χ Yo, 3.5 where Y o n Y i i i n p i S i p p n in σa Q. For the µ-measurabe set Y o of 3.6, there exists a subset Yo of, such that for a p. Yo i S i p 3.6 n p i,,..., p i,ni } in, 3.7 i n n S Sp i p i if p Yo, i n Z p Z p otherwise, i i 3.8

13 Adeic anaysis and functiona anaysis on the finite Adee ring 5 Remark that either n Sp i Z p i or n Sp i Z p, in 3.8, case-by-case, for p Y0. However, in genera, the equaity 3.8 hods with respect to Yo of 3.7. Therefore, the formua 3.5 goes to by 2.25 ϕh ϕ χ Yo p Yo ϕ p p Yo i χ n i S i p j Λ n S p i i p Yo µ p n r j p j p j+ by 2.0, where r j are in the sense of 2.0, for a j Λ n in the sense of 2.8, for a p Yo in. Therefore, if we take o Yo of 3.7 and X p n Sp i in Q p for a p Yo, i then the formua 3.3 is we-determined. i i Si p S i p,, where Λ n i Si p The above joint free-moment formua 3.3 characterizes the free distributions of generating eements of our finite-adeic probabiity space M, ϕ. As a coroary of 2.25 and 3.3, one obtains the foowing resut. Coroary 3.2. Let Y σa Q, satisfying that Y p S p with S p σ Q p, p k S p if p p, for,..., N, 3.9 otherwise, Z p for a p, for some p,..., p N, for k,..., k N Z, for N N, where p k are the k -th boundary of basis eements U p k p k Z p of Q p, for a,..., N. Then for a n N. N ϕ χ n Y p k p k + are 3.0

14 52 Iwoo Cho roof. The proof of 3.0 is done by 2.25 and 3.3. Indeed, if Y satisfies the condition 3.9, then N N ϕ χ Y ϕ p χ p k p k p k. + Therefore, since χ n Y χ Y, for a n N, the free-moment formua 3.0 hods. By 3.3 and 3.0, we obtain the foowing coroary, too. Coroary 3.3. Let Y p S p σa Q, for,..., n, for some n N, where p t Sp k pt if p, t p,,..., p,n }, 3. otherwise, Z p where p k p are the k p -th boundaries for k p Z in Q p, for p, and where k pt,,..., k pt,n Z, for a,..., n, a p. Now, et n o p,,..., p,n } in. Then one obtains that n ϕ χ Y where p kp for a p o. p o ω p are in the sense of 3., where n if Sp, ω p 0 otherwise p kp p kp+, 3.2 roof. The proof of 3.2 is done by 3.3 and 3.0, under the condition 3.. Let Y σa Q be in the sense of 3., for,..., n, and et n X χ Y M, ϕ. 3.3 Such free random variabes X of 3.3 are caed boundary-product eements of the finite-adeic probabiity space M, ϕ. As we have seen in 3.2, if X is a boundary-product eement of M, ϕ, then there exists a subset o of such that ϕ X p o ω p p kp p kp+, for some k p Z in the sense of 3.0, for a p o, where ω p is in the sense of 3.2.

15 Adeic anaysis and functiona anaysis on the finite Adee ring 53 Definition 3.4. Let X be a boundary-product eement 3.3 of M, ϕ. Assume now that Y are in the sense of 3., and o is in the sense of 3.2. Assume further that, for a p o, the corresponding integers k p are nonnegative, i.e., k p 0, for a p o. 3.4 Then we say this boundary-product eement X is a +-boundary eement of M, ϕ, i.e., X is a +-boundary eement if and ony if, i X is in the sense of 3.3, and ii the corresponding finite subset o of satisfies the condition 3.4. Remark 3.5. In the rest of this section, we focus on studying +-boundary eements 3.4 of the finite-adeic probabiity space M, ϕ. As we have seen in 3.3, the free-distributiona data of χ Y, for an arbitrary µ-measurabe subsets Y of A Q, are determined by the free distributions 3.0 of boundary-product eements, or those 3.2 of of their operator products. So, it is reasonabe to restrict our interests to investigate free-distributiona information of boundary-product eements 3.3 for studying free distributions of arbitrary eements of the finite-adeic -agebra M, under the Adeic integration ϕ. However, as we checked, the free-probabiisitc information 3.2 is determined by ω p, by the chain property in 2.3, i.e., p p k k p Z p p if k p 0 in Z, if k p < 0 in Z for a p, where means the empty set. It shows that if a boundary-product eement X has a finite subset partitioned by where and o p : p k p Z p } of, + o o o + o o, p o : k p 0 in Z}, q o : k q < 0 in Z}, equppied with o, then the formua 3.2 vanishes by the roe of ω q for q o. Therefore, to avoid such vanishing cases, one had better focus on the cases where we have +-boundary eements satisfying 3.4, rather than whoe boundary-product eements in M, ϕ. As we discussed in the above remark, one can reaize that such +-boundary eements provide certain buiding bocks of the non-vanishing free distribution 3.3 under certain additiona mutipes. So, it is natura to concentrate on studying free distributions of such eements in the finite-adeic probabiity space M, ϕ. Let φ : N C be the Euer totient function defined by an arithmetic function, φn k N k n, gcdn, k }, 3.5

16 54 Iwoo Cho for a n N, where S mean the cardinaities of sets S, and gcd means the greatest common divisor. It is we-known that φn n p for a n N, 3.6 p, p n where p n means p divides n, or n is divisibe by p. For instance, φp p p p for a p, by 3.5 and 3.6. Remark that the Euer totient function φ is a mutipicative arithmetic function in the sense that φn n 2 φn φn 2, 3.7 whenever gcdn, n 2 for a n, n 2 N. If p p 2 in, then, for any n, n 2 N, and hence, by 3.6 and 3.7. gcd p n, pn2 2, φ p n pn2 2 φ pn φ pn2 2 pn pn2 2 p p 2, Theorem 3.6. Let X be a +-boundary eement 3.3 of the finite-adeic probabiity space M, ϕ satisfying 3.4. Then there exist the subset o of, and where N 0 N 0}, such that K o k p N 0 : p o } of Z, n X p kp N, p o 0 < r X in Q, pkp+ p o ϕx r X φn X. p o 3.8 roof. Let X be a +-boundary eement 3.4 in M, ϕ. Then, by 3.2, there exist the subsets o of, and K o of Z, such that ϕ X p kp p kp+, with k p K o, with k p 0 in Z.

17 Adeic anaysis and functiona anaysis on the finite Adee ring 55 where Observe that ϕx p kp p o p kp+ p o p pkp p kp+ p o p + o p kp+ p n X p o p kp φ n X, p o p kp N, p by 3.5, 3.6 and 3.7. The above theorem, especiay, the reations in 3.8, characterizes the free distributions of +-boundary eements of M, ϕ, in terms of the Euer-totient-functiona vaues. So, it iustrates the connections between our free-probabiistic structure and number-theoretic resuts. If X is a +-boundary eement 3.4 in M, ϕ, then there exist such that 0 < r X in Q, and n X N, ϕx r X φn X φn X ϕx r X, by 3.8. Observe now a converse of the above theorem. Theorem 3.7. Let n N be prime-factorized by n p kp p kp p kp N N in N, 3.9 where p,..., p N, and k p,..., k pn N, for some N N. Then there exists a +-boundary eement X of the finite-adeic probabiity space M, ϕ such that X p χ Yp M, ϕ, 3.20 having o p,..., p N },

18 56 Iwoo Cho K o k p,..., k pn } N 0, and p k Y p p if p o, otherwise Z p for a p, satisfying that φ n n p nϕx, with n p roof. Let X be a +-boundary eement 3.20 in M, ϕ. Then ϕ X p o ϕ p p kp+ p o ike in the proof of 3.8 χ p kp p kp p o where n is given as above in N p + o p p o p kp p o p kp p φn φn p o p kp p p o n p o + p φn. p o p N. 3.2 p Therefore, for any n N, with its prime-factorization 3.9, there exists a +-boundary eement X of 3.20, such that ϕx n p n φn, with n p p o p N. Therefore, one can obtain the reation 3.2, whenever n of 3.9 is fixed in N. The above two theorems provide a connection between number-theoretic resuts from the Adeic anaysis and arithmetic function theory, and our free-probabiistic

19 Adeic anaysis and functiona anaysis on the finite Adee ring 57 resuts on the -agebra M. In particuar, the above two theorems show that: n N if and ony if there exists X M, ϕ such that by 3.8 and 3.2. ϕx n o φn n for some n o N, 4. RERESENTATION OF M, ϕ Let M, ϕ be the finite-adeic probabiity space, M, ϕ M p, ϕ p M p, p p p ϕ p. 4. In [5, 6] and [8], we estabished and studied Hibert-space representations H p, α p of the -probabiity spaces M p, ϕ p, for p. By 4., one can construct a Hibert-space representation of M with hep of the representations, by Define a form H p, α p of M ps, for a p. [f, f 2 ] def [, ] : M M C A Q f f 2 dµ, f, f 2 M. 4.2 Then, by the definition 4.2, this form [, ] is sesqui-inear: and [t f + t 2 f 2, f 3 ] t [f, f 3 ] + t 2 [f 2, f 2 ] [f, t f 2 + t 2 f 3 ] t [f, f 2 ] + t 2 [f, f 2 ] 4.3 for a t, t 2 C and f, f 2, f 3 M. Now, observe that [f, f 2 ] f f2 dµ f 2 f dµ A Q A Q for a f, f 2 M. Let Y σa Q and t C, inducing f tχ Y M. Then [f, f] ff dµ t 2 χ Y dµ t 2 µy 0, A Q A Q A Q f 2 f dµ [f 2, f ] 4.4

20 58 Iwoo Cho where t means the omduus of t in C, and hence, For f tχ Y M, assume that [f, f] 0 t 2 µy 0 [h, h] 0, for a h M 4.5 t 2 0 or µy 0 t 0 or Y, the empty set in A Q, because µ µ p, and µ p are the Haar measures on Q p, for a p e.g., see 4. p where O means the zero eement of M. In other words, Therefore, one has f 0 χ Y O or f tχ O, [tχ Y, tχ Y ] 0 if and ony if tχ Y O in M. [f, f] 0 f O in M. 4.6 roposition 4.. The form [, ] of 4.2 on the finite-adeic -agebra M is an inner product. Equivaenty, the pair M, [, ] forms an inner product space. roof. The form [, ] of 4.2 is an inner product on M, because it satisfies 4.3, 4.4, 4.5 and 4.6. Let [, ] be the inner product 4.2 on the -agebra M. Define now the norm and the metric d on M by respectivey f [f, f], for a f M, 4.7 df, f 2 f f 2, for a f, f 2 M. 4.8 Definition 4.2. Let d be the metric 4.8 induced by the norm of 4.7 on the inner product space M, [, ]. Then maxima d-metric-topoogy cosure in M is caed the finite-adeic Hibert space, and we denote it by H. By the very definitioin of finite-adeic Hibert space H, the -agebra M is acting on H via a inear morphism α : M B H, αfh fh, for a h H, 4.9 for a f M, i.e., the agebra-action α of 4.9 assigns each eement f of M to the mutipication operator αf with its symbo f in the operator agebra B H consisting of a bounded inear operators on H. Notation 4.3. For convenience, we denote αf by α f, for a f M, where α is in the sense of 4.9. Moreover, et us denote αχ Y α χy simpy by α Y, for a Y σ A Q.

21 Adeic anaysis and functiona anaysis on the finite Adee ring 59 and By the definition 4.9, for any f, f 2 M, one has α ff 2 α f α f2 on H, 4.0 α f α f for a f M. 4. Theorem 4.4. Let H be the finite-adeic Hibert space, and et α be in the sense of 4.9. Then the pair H, α is a Hibert-space representation of the finite-adeic -agebra M. roof. It suffices to show that the inear morphism α of 4.9 is a we-determined -homomorphism from M to the operator agebra BH. Note that, by 4.0 and 4., α is indeed a -homomorphism from M to B H. By the above theorem, one can understand a eements f of M as a Hibert-space operator α f on H. Definition 4.5. Let M, ϕ be the finite-adeic probabiity space, and et H, α be the representation of M of the above theorem. Then we ca this representation, the finite-adeic representation of M. Define now the C -subagebra M of the operator agebra B H by M C M def C [α M ], 4.2 where X mean the operator-norm-topoogy cosures of subsets X of B H. We ca this C -subagebra, the finite-adeic C -agebra. 5. FUNCTIONAL-ANALYTIC ROERTIES ON M In this section, we study functiona-anaytic properties on the finite-adeic C -agebra M of 4.2 under suitabe free-probabiistic modes. Such properties are determined by the anaytic data of Section 3, impying number-theoretic information. Let M, ϕ be the finite-adeic probabiity space, and et H, α be the finite-adeic representation of M. Let M be the finite-adeic C -agebra 4.2 of M, ϕ under H, α. In this section, we wi consider free-probabiistic data on the C -agebra M by constructing a system of suitabe inear functionas on M. Define a inear functiona ϕ p,j on M by ϕ p,j T [ T χ B p j, χ B p j ], for a T M, 5. for a p, j N 0, where and N 0 def N 0}, B p j q Y q in σ A Q

22 60 Iwoo Cho with for a q, i.e., for p and j N 0. Y q p j if q p, Z q otherwise χ B p χ Z j 2 Z 3 Z 5... p... H j p-th position Remark 5.. In the definition 5., remark that we ony take j from N 0, not from Z. The reason is as foows: Let Y σa Q, with Y q S q, where S q σ Q q, where there exists a finite subset satisfying such that and where Y p,..., p N } of, for some N N, + Y Y Y + Y Y q Y : k q 0 in Z}, q Y : k q < 0 in Z}, q k S q q if q Y, otherwise Z q for a q. See 5.6 beow. Aso, et B p j be in the sense of 5., where p, and j Z. Assume first that j < 0 in Z, and p / Y in. Then Y B p j S 2 Z 2 S 3 Z 3... Z p p j... p-th position S 2 Z 2 S 3 Z , p-th position in A Q, by the chain property in 2.2, which wi gives ϕ p,j α Y 0, by 5.. Aso, see 5.5 beow. It shows that, whenever j < 0 in Z, and p / Y, one can get vanishing free-moments. Aso, suppose that j < 0 in Z, and say p p Y in, for convenience. Then Y B p j Y Bp j S 2 Z 2... p k p p j..., p -th position

23 Adeic anaysis and functiona anaysis on the finite Adee ring 6 and hence, ϕ p,j α Y δ kp,jϕ p,j α Y ϕ p,j α Y if k p j < 0, 0 if k p 0, by 5.. Aso, see 5.5 beow. So, for most of arbitrary choices of Y, the quantities ϕ α Y are vanishing. Therefore, to avoid-or-overcome the above two vanishing cases, we determine our system 5. of inear functionas ϕ p,j : p, j N 0 }, by taking j from N 0 in Z. Remark aso that, even though we take a system ϕ p,j : p, j Z } of inear functionas ϕ p,j in the sense of 5., one can get simiar resuts ike our main resuts of this Section containing vanishing cases. However, we need to poish ots of vanishing cases. Thus, our system of inear functionas woud be chosen by 5. for convenience. Note that a vectors h of H have their expressions, h Y σa Q t Y χ Y, with t Y C, where is a finite, or an infinite imit of finite sums under the Hibert-space topoogy induced by the metric 4.8. Note aso that every operator T of M has its expression, T Y σa Q s Y α Y, with s Y C, where is a finite, or an infinite imit of finite sums under the C -topoogy for M, and where α Y are in the sense of Notation 4.3. Therefore, the inear functionas ϕ p,j of 5. are we-defined on M, and hence, one can get the corresponding C -probabiity spaces for a p, j N 0. M p,j denote M, ϕ p,j, 5.2 Definition 5.2. Let M p,j M, ϕ p,j be a C -probabiity space 5.2, for p, j N 0. Then we ca M p,j, the p, j-finite-adeic C -probabiity space of the finite-adeic C -agebra M.

24 62 Iwoo Cho In the rest of this section, et us fix p and j N 0, and the corresponding p, j-adeic C -probabiity space M p,j of 5.2. Consider first that, et α Y α χy M p,j, for Y σa Q, satisfying Y q S q, with S q σq p, 5.3 where S q S q Z q if q Y, Z q if q / Y, where Then we have Y q : S q Z q } in. ϕ p,j α Y where B p j is in the sense of 5. in σ A Q [ ] [ χ Y χ B p, χ B p j j χ Y B p χ j B p dµ j A Q A Q χ Y B p A Q χ Y B p dµ j Bp j A Q [ ] α Y χ B p, χ B p j j j, χ B p j ] χ Y B p χ B pdµ j j χ Y B p dµ j µ Y B p j µp q Y µ q S q Z q p j µp q Y \p} µ q S q Z q Sp p j if p / Y, if p Y. 5.4 Note again that the formua 5.4 is obtained because we take j in N 0 not in Z: if j < 0, then the above formua 5.4 vanishes. By 5.4, we obtain the foowing resut. Theorem 5.3. Let α Y be a free random variabe in the p, j-adeic C -probabiity space M p,j, where Y σ A Q is in the sense of 5.3. Then ϕ αy n µ q S q Z q µ p Sp p j, 5.5 q Y p}\p} for a n N. roof. Let α Y be as above in M p,j. Then α n Y α χy n α χ n Y α χ Y α Y,

25 Adeic anaysis and functiona anaysis on the finite Adee ring 63 in M p,j, for a n N. So, ϕ p,j α n Y ϕ α Y, for a n N. The quantity ϕα Y is obtained in 5.4, re-expressed simpy by 5.5. Note that the formua 5.5, indeed, impies the two cases of 5.4 atogether. Now, et Y be in the sense of 5.3, with specific condition as foows; Y q S q, with S q σq p, where S q q k q if q Y, Z q if q / Y, 5.6 for a q, where k q Z for q Y, and Y p,..., p N } in, for some N N. If Y is in the sense of 5.6, then the corresponding free random variabe α Y the p, j-adeic C -probabiity space M p,j satisfies that ϕ αy n µ q q k q Z q µ p Sp p j q Y p}\p} of by 5.5 q Y µ q q µp k q Z q p j q Y \p} µ q qkq Z q µ p p k p p j q Y µ q q µp k q Z q p j δ j,kp p j p j+ q Y \p} µ q qkq Z q if p / Y, if p Y, if p / Y, if p Y, 5.7 for a n N, where δ means the Kronecker deta. Therefore, one obtains the foowing coroary of 5.5, with hep of 5.7. Coroary 5.4. Let Y be in the sense of 5.6 in σ A Q, and et α Y be the corresponding free random variabe of the p, j-adeic C -probabiity spae M p,j. Then ϕ p,j αy n δ j,y p j p j+ µ q q k q Z q, 5.8 q Y \p}

26 64 Iwoo Cho for a n N, where δ j,y δ j,kp if p Y, otherwise, where Y is in the sense of 5.6. roof. The free-moment formua 5.8 hods by 5.5 and 5.7. If we simpify the expression 5.7, then the formua 5.8 is obtained. Note that the operator α Y of the above coroary is nothing but an operator induced by a boundary-product eement χ Y of the finite-adeic probabiity space M, ϕ, and hence, they provide buiding bocks of free distributions of a operators in M from 5.8. So, as in Section 3, we focus on studying free-distributiona data of these operators α Y for investigating free distributions of a operators of M. Definition 5.5. Let α Y be the operator of the finite-adeic C -agebra M, generated by the µ-measurabe subset Y of 5.6. Then we ca such an operator α Y a boundary-product operator of M. As we discussed above, in the rest of this paper, we focus on studying free-distributiona data of certain operators of M, generated by boundary-product operators α Y s in M p,j, for a p, and j N 0, where Y are in the sense of 5.6 in σa Q. Note that, if Y is in the sense of 5.6 and if k p N 0 in Z, for a p Y, 5.9 then it is regarded as Y q Y B q k q in σa Q, 5.0 where B q k q are the µ-measurabe subsets of A Q in the sense of 5., for q Y and for now k q Z, where Y is the subset 5.6 of. Note that the above set-equaity 5.0 hods ony if the condition 5.9 of Y is satisfied. Therefore, the corresponding boundary-product operator α Y is understood as α Y α q Y B q kq α B q kq q Y 5.

27 Adeic anaysis and functiona anaysis on the finite Adee ring 65 in M p,j, under 5.0. So, one can get that ϕ p,j α Y ϕ p,j p Y α B q kq by 5.0 [ [ χ p Y α B q kq q Y p}b q kq χ B p j, χ B p j ] ], χ B p j by identifying k p j in N 0 δ j,y q Y p} µ q q k q δ j,y q Y p} q kq q kq+, 5.2 where δ j,y is in the sense of 5.8. Therefore, from a different approach from 5.9, we obtain the foowing specia case of 5.8. Coroary 5.6. Let Y be in the sense of 5.6 with additiona condition 5.9 in σa Q, and et α Y be the corresponding boundary-product operator in the p, j-adeic C -probabiity space M p,j, for p, and j N 0. Then α Y in M, and ϕ p,j αy n δ j,y q Y α B q kq q Y p} q kq q kq+, 5.3 with identification: k p j in Z, for a n N, where δ j,y is in the sense of 5.8. roof. The operator-identity in 5.3 is shown by 5., and the free-moment formua in 5.3 is proven by 5.8 and 5.2, since αy n α Y in M, for a n N, for a p, j N 0. Y Now, et Y and et Y be in the sense of 5.6 not necessariy with 5.9. Then is partitioned by Y + Y Y in, 5.4 where and + Y Y q Y : k q 0 in Z}, q Y : k q < 0 in Z}.

28 66 Iwoo Cho Then the formua 5.8 can be refined as foows with hep of 5.3. Theorem 5.7. Let Y be in the sense of 5.6, inducing a finite subset Y + Y Y of, as in 5.4. If α Y M p,j, for p, j N 0, then ϕ p,j αy n δj,y q Y if q kq q kq + Y, 0 if Y, 5.5 for a n N, where means the empty set. roof. Suppose α Y and et be given as above in M p,j. Assume first that Y in Y, Y q,..., q t }, for some t N in N, i.e., k qs < 0 in Z, for a s,..., t. Then Y B p j S 2 Z 2... q k q Z q q -th position S 2 Z ,... qt k qt Z qt q t-th position... by the chain property of Q qs in 2.2, for s,..., t, in A Q. Thus, ϕ p,j α n Y ϕ α Y 0, by 5.8, for a n N. Assume now that Y, equivaenty, suppose Y + Y in. Then, by 5.3, one obtains that ϕ p,j αy n δ j,y q kq q kq+ q + Y p} for a n N, where δ j,y δ j,kp if p + Y Y, otherwise, by 5.3. Therefore, the refined free-moment formua 5.5 of 5.8 hods. The above free-distributiona data 5.5 shows that, a µ-measurabe subsets Y inducing non-empty subset Y generate non-zero operators α Y in our p, j-adeic C -probabiity spaces M p,j, having vanishing free distributions, for a p and j N 0.

29 Adeic anaysis and functiona anaysis on the finite Adee ring 67 Assumption and Notation 5.8 in short, AN 5.8 from beow. Let Y be in the sense of 5.6 satisfying 5.4, and et α Y be the corresponding boundary-product operator in the finite-adeic C -agebra M. In the rest of this paper, we automaticay assume Y, equivaenty, Y + Y, i.e., from beow, to avoid the vanishing cases in 5.5. Y q : k q 0 in Z} + Y, To avoid confusion, we wi say such boundary-product operators α Y are +-boundary-product operators of M. The notation is reasonabe because such +-boundary operators α Y are induced by +-boundary eements χ Y of M. From beow, a boundary-product operators woud be +-boundary operators of AN 5.8 in M. Again, notice that a boundary-product operators α Y, which are not +-boundary operators in M p,j, have vanishing free distributions, for a p, and j N 0, by 5.5. So, we focus on studying free-distributiona data of +-boundary operators in p, j-adeic C -probabiity spaces M p,j, for a p, j N 0. Theorem 5.9. Let α Y be a +-boundary operator of AN 5.8 in the finite-adeic C -agebra M. Let us understand α Y as a free random variabe in the p, j-adeic C -probabiity space M p,j, for p, and j N 0. Then there exist n Y q kq, with identification: k p j in N 0, q + Y p} in N, such that where where δ j,y ϕ p,j α n Y δ j,y n Y n p,j φn p,j, for a n N, 5.6 n p,j q + Y p} q in N, is in the sense of 5.8, and φ is the Euer totient function. roof. Reca that, for a fixed p, j N 0, if Y is a µ-measurabe set of A Q, satisfying both 5.6 and 5.9, then the corresponding operator α Y forms a +-boundary operator of AN 5.8 in M p,j, and it satisfies ϕ p,j α Y δ j,y q kq q kq+, 5.7 q Y p} with identification: k p j in Z, since Y + Y, where δ j,kp if p Y + Y δ j,y, otherwise, by 5.8 and 5.3.

30 68 Iwoo Cho Then, one can re-write the formua 5.7 as foows: ϕ p,j α Y δ j,y q kq q kq+ q Y,p δ j,y qkq q where in N, and hence, it goes to for a n N. δ j,y q kq+ q Y,p q Y,p q kq δ j,y n Y n p,j n Y q φn p,j, q Y,p q kq, n p,j q + Y,p q Y,p q, q q δ j,y φn p,j, 5.8 n Y n p,j The above two theorems iustrate reations between our C -probabiistic structures, and number-theoretic information by 5.5 and 5.6. Aso, they show a conncetion between the -probabiistic data 3.2, and the C -probabiistic data 5.8, whenever Y + Y in. In the rest parts of this paper, we study +-boundary operators α Y of AN 5.8 in the finite-adeic C -agebra M, and the free distributions of certain operators, generated by these +-boundary operators, in p, j-adeic C -probabiity spaces M p,j for a p and j Z. 6. DISTRIBUTIONS INDUCED BY +-BOUNDARY OERATORS The main purposes of this section is to consider free-distributiona data of +-boundary operators which provide the buiding bocks of operators in M having possibe non-vanishing free distributions. Let M be the finite-adeic C -agebra generated by the finite-adeic probabiity space M, ϕ under the finite-adeic representation H, α, and et M p,j M, ϕ p,j be the p, j-adeic C -probabiity spaces 5.2, for a p, j N 0.

31 Adeic anaysis and functiona anaysis on the finite Adee ring 69 Let Y q S q in σ Q q, where in A Q, and B q q k q be µ-measurabe subsets of the finite Adee ring A Q Y B q k q Z 2 Z 3... q k q q-th Y q : S q q k q } is a finite subset of. Equivaenty, the subsets Y of A Q are in the sense of 5.6. Moreover, assume that Y satisfies the condition 5.9 as in AN 5.8, too, i.e., Y + Y Y in. Then the corresponding +-boundary opeartors α Y are we-determined in M. Reca that if α Y is a +-boundary operator in the sense of AN 5.8, then, as a free random variabe in a p, j-adeic C -probabiity space M p,j, one obtains ϕ α n Y δ j,y n Y n p,j φn p,j for a n N, 6. by 5.5, 5.6 and 5.8, where φ is the Euer totient function 3.4 and where δ j,kq if p Y + Y δ j,y, otherwise, where n Y q Y,p q kq and n p,j q Y,p q in N, Y,p Y p} + Y,p p}, which is a finite subset of. Now, et Y,..., Y N σa Q be in the sense of 5.6 with 5.9, and et α Y be the corresponding +-boundary operators in M, for a,..., N, for N N. Define a new operator T,...,N M by T,...,N By the very construction 6.2 of T,...,N, one can get that T,...,N N α Y N α Y M. 6.2 α N Y n q α B q kq Y, 6.3 in M, under 5.9. Remark that, without the condition 5.9, the reation 6.3 does not hod, in genera, because of the vanishing cases.

32 70 Iwoo Cho The observation 6.3 shows that there exists a µ-measurabe subset Y,...,N of A Q, such that by 6.4, where Y,...,N Y,...,N q Y,...,N B q k q in σa Q, 6.4 T,...,N α Y,...,N in M, 6.5 n Y n + Y + Y,...,N is a finite subset of, where Y are in the sense of AN 5.8, for a,..., N. Therefore, the µ-measurabe subset Y,...,N of 6.4 aso satisfies AN 5.8, and hence, the corresponding operator T,...,N of 6.2 forms a new +-boundary operator α Y,...,N in M, by 6.5. It shows that the products of +-boundary operators become +-boundary operators in M. Lemma 6.. Let T,...,N be the operator product 6.2 of +-boundary operators α Y,..., α YN of AN 5.8 in M, for some N N. Then there exists such that Y,...,N N Y σa Q, where α Y,...,N is a new +-boundary operator in M. T,...,N α Y,...,N M, 6.6 roof. The existence of the µ-measurabe subset Y,...,N is guaranteed by 6.3 and 6.4, and the operator equaity 6.6 is proven by 6.5. Since operator products of +-boundary operators are +-boundary operators in M, by 6.6, we obtain the foowing free-probabiistic information. Theorem 6.2. Let T,...,N be an operator 6.2 in M. As a free random variabe in a p, j-adeic C -probabiity space M p,j, for p, j N 0, we have that ϕ p,j T,...,N n δ j,y,...,n q kq q kq+ q Y,...,N :p 6.7 for a n N, where δ j,y,...,n n Y,...,N n p,j φ n p,j, Y,...,N :p Y,...,N p} + Y,...,N p},

33 Adeic anaysis and functiona anaysis on the finite Adee ring 7 and Y,...,N q : S q q k q } + Y,...,N, are the finite subsets of, whenever Y,...,N S q σa Q, with S q σ Q q, q and where and n Y,...,N δ j,y,...,n δ j,kq if p Y,...,N, otherwise, q kq, and n p,j q, in N. q Y,...,N :p q Y,...,N ::p roof. By 6.6, there exists Y,...,N N Y σa Q, such that T,...,N α Y,...,N, as a new +-boundary operator in M p,j. Since it is a new +-boundary operator, one has T,...,N n n α Y,...,N αy,...,n T,...,N, for a n N. So, one obtains that ϕ p,j T,...,N n ϕ p,j T,...,N ϕ p,j αy,...,n δ j,y,...,n q kq q kq+ q Y,...,N :p by 5.3, where and Y,...,N :p Y,...,N p}, Y,...,N q : S q B q k q } + Y,...,N is a finite subset of, whenever Y,...,N S q, with S q σ Q q, q and hence, it goes to δ j,y,...,n n Y,...,N n p,j φn p,j, where n Y,...,N q kq, and n p,j q q Y,...,N :p q Y,...,N :p in N, by 5.8, where φ is the Euer totient function, for a n N. Therefore, the free-moment formua 6.7 hods.

34 72 Iwoo Cho Now, et B q k be in the sense of 5. in σ A Q, for q, and k N 0, i.e., B q k s S s, with S s q k if s q, Z s otherwise, 6.8 in Q s, for s. Then it provides the corresponding operator α B q in the finite-adeic C -agebra k M, as a +-boundary operator in the sense of AN 5.8, because k 0 in Z, i.e., it induces B q q} B q +, with k q k 0. k k Let B p j,..., B p N jn be the µ-measurabe subsets 6.8 in the finite Adee ring A Q, where p,..., p N are mutuay distinct from each other in, and j,..., j N N 0 which are not necessariy distinct, for N N. So, these sets automaticay satisfy AN 5.8. Now, et α p,j α B p j M, 6.9 be the corresponding +-boundary operators, for a,..., N. Construct now a new operator N S α p,j in M, 6.0 where α p,j are in the sense of 6.9, for a,..., N. Observe that, if S is an operator 6.0 in M, then n S n α ps,j s where,..., n,...,n} n s,..., n,...,n} n α Y,...,n, Y,..., n n s B p s j s σa Q, 6. for a,..., n,..., N} n, for a n N. Remark that the summands α of Y,...,n Sn in 6. form +-boundary operators, for a,..., n,..., N} n, for a n N. Thus, we obtain the foowing free-distributiona data of the operators S of 6.0. Coroary 6.3. Let S be an operator 6.0 in M. Then, as a free random variabe in the p, j-adeic C -probabiity space M p,j, for p, and j N 0, it satisfies that ϕ p,j S n ϕp,j αy,...,n, 6.2,..., n,...,n} n

35 Adeic anaysis and functiona anaysis on the finite Adee ring 73 where α are in the sense of 6., and the summands ϕ Y,...,n p,j are competey determined by 6., or 6.7, for a n N. roof. The proof of the formua 6.2 is done by 6., 6. and FREE RODUCT C -ALGEBRA OF M p,j } αy,...,n of 6.2 In this section, we construct free product C -agebra of the system M p,j } p,j Z, and consider free-distributiona data of free reduced words in the C -agebra. From this, one can not ony study free probabiity induced by the finite Adee ring, but aso appy Adeic anaysis as free-probabiistic objects. In the previous sections, we used concepts and terminoogy from free probabiity theory in extended senses for our commutative structures. In this section, we study traditiona noncommutative free probabiity theory on our structures under free product. Let M be the finite-adeic C -agebra induced by the finite-adeic probabiity space M, ϕ under the representation H, α, and et } M p,j M, ϕ p,j : p, j N 0 7. be the system of p, j-adeic C -probabiity spaces M p,j of 5.2. In this section, we consider free product C -agebra of the system 7.. Reca that, if α q,k are +-boundary operators α B q in M, then k ϕ p,j α n q,k δj,q,p} r kr r kr+ δ j,q,p} φn o, 7.2 n o n p,j for a n N, where in Z, and and n o r q,p} k r δ j,q,p} r q,p} k if r q, j if r p, δ j,k if q p, otherwise, r kr, and n p,j r q,p} r, in N, by 6.7. Remark that the finite subset q, p} in 7.2 satisfies q, p} if q p, q, p} q} p} p} if q p, in.

36 74 Iwoo Cho Coroary 7.. Let α q,k be a +-boundary operator 6.9 in M p,j, for p, q, j, k N 0. Then ϕ p,j α n q,k δj,q,p} r kr r kr+ δ j,q,p} φn p,j, 7.3 n o n p,j r q,p} for a n N, where δ j,q,p}, n o, n p,j are in the sense of 7.2. roof. The formua 7.3 is a coroary of 6.7. See 7.2 above. By 7.3, one aso obtains the foowing two coroaries. Coroary 7.2. Let M p,j be a p, j-adeic C -probabiity space, for p, j N 0. Then for a n N. roof. Observe that ϕ p,j α n p,j p j p j+ φp, 7.4 pj+ ϕ p,j α n p,j ϕp,j α p,j p j p j+ p j+ by 7.3, for a n N. p p φp, pj+ Coroary 7.3. Let M p,j be a p, j-adeic C -probabiity space, for p, j N 0. If p q in, then for a n N. ϕ p,j α n q,k q j q j+ p j p j+ roof. Since p q in, by 7.3, one obtains that for a n N, where ϕ p,j α n q,k ϕp,j α q,k q k q k+ p j q k p j q p q k+ p j+ q q p j+ p q j+ φqp, 7.5 pj+ p q k+ p j+ φq φp q k+ φqp pj+ q k p j qp φqp. n q,p} q k p j, and n p,j qp, in N. Indeed, the free-moment formua 7.3 is refined by both 7.4 and 7.5.

37 Adeic anaysis and functiona anaysis on the finite Adee ring FREE RODUCT C -ROBABILITY SACES Let A k, ϕ k be arbitrary C -probabiity spaces, consisting of C -agebras A k, and corresponding inear functionas ϕ k, for k, where is an arbitrary countabe finite or infinite index set. The free product C -agebra A, A A is the C -agebra generated by the noncommutative reduced words in A, having a new inear functiona ϕ ϕ. with The C -agebra A is understood as a Banach space, n C n as cosed subspaces of A ik, where i,...,i n at n A o i k k A o i k A ik C, for a k,..., n, at n i,..., i n i,..., i n n, i i 2, i 2 i 3,..., i n i n }, 7.6 for a n N, and where the direct product, and the tensor product are topoogica on Banach spaces. In particuar, if an eement a A is a free reduced word, n a i in A, then one can understand a as an equivaent Banach-space vector η a n a i in the Banach space A of 7.6, n contained in a direct summand, A o i k of 7.6. Note that this free reduced word a k and its equivaent vector η a is regarded as an operator n a i in the C -subagebra n n A i C A o i k of A, C k where C means the tensor product of C -agebras. We ca such a C -subagebra n C A i of A, the minima free summand of A containing a. It is denoted by A[a], i.e., A[a] is the minima C -subagebra of A containing a as a tensor product operator.