(a 1,...,a n ) {a,a } n, n N

Transcription

1 METHODS AND APPLICATIONS OF ANALYSIS. c 01 International Press Vol. 19, No. 4, pp , December OPERATORS INDUCED BY PRIME NUMBERS ILWOO CHO Abstract. In this paper, we study operator-theoretic and algebraic structures induced by prime numbers. The Adele ring A Q is an algebraic, topological, and measure-theoretic object induced by the p-adic number fields {Q p} p:prime. By determining a Hilbert space H induced by A Q, we study how prime numbers act on H. In particular, we are interested in the case where they act like shift operators on H.The main purpose of this paper is to understand the properties of such shiftoperators a p induced by prime numbers p. We call them prime multipliers. We characterize the von Neumann algebra vnγ generated by {a p} p:prime. As spectral-theoretic application, we compute the free distributional data of self-adjoint elements a p + a p, for all primes p. Key words. Adele ring, Hilbert spaces, von Neumann algebras, prime operators, free probability, free moments, free cumulants. AMS subject classifications. 05E15, 11G15, 11R04, 11R09, 11R47, 11R56, 46L10, 46L40, 46L53, 46L54, 47L15, 47L30, 47L Introduction. For the Adele ring A Q e.g., [14], we construct the corresponding von Neumann algebra A = C[αA Q ] w acting on the Hilbert space H = L A Q, where α is kind of a shifting action. We are interested in the free distributional data of the operators of A induced by prime numbers in short, primes on H. Such data provide the operator-theoretic, in particular, spectral-theoretic, data of primes on operator-algebraic Adelic structures. To do that, we will use free probability theory Overview. Noncommutative free probability has been studied since mid 1980 s. Voiculescu extend the classical probability theory to that on noncommutative operator algebraic structures e.g., [11]. Speicher established the combinatorial free probability e.g., see [1]. If a given operator a in a topological -algebra A, and if ϕ : A C is a fixed bounded linear functional, then the free-probabilistic information of a is determined by its free moments { ϕa 1 a...a n C a 1,...,a n {a,a } n, n N where X n simply means the Cartesian product of n-copies of X, for all n N. By the Moebius inversion, the free cumulants { k n a 1,...,a n C a } 1,...,a n {a, a } n n N also represent the same or equivalent free-probabilistic information of a See [1]. If a is self-adjoint in A, in the sense that a = a, then the above free-probabilistic data of a represents the spectral information of a. i.e., the free probability measure is identified with the spectral measure of a under self-adjointness e.g., [3], [4], [5], [11] and [1]. Thus, computing the free moments or free cumulants of a provides the spectral data of a. Received May 1, 01; accepted for publication January 11, 013. St. Ambrose University, Department of Mathematics, 41 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 5803, USA choilwoo@sau.edu. 313 },

2 314 I. CHO By Speicher, it is shown that the free cumulant computation characterizes the free structure in A. i.e., two elements a 1 and a are free A, if and only if all mixed cumulants of {a 1, a 1 } and {a, a } vanish in A See Section below. In [1], [5], [6] and [7], we use free probability theory to investigate the structure theorems of certain topological groupoid algebras. Groupoids are algebraic structure with one binary operation having multi-units. For instance, all groups are groupoids with one unit. By characterizing the free blocks of such algebras, we restrict the study of groupoid algebras to the study of free blocks and certain combinatorial connections of those blocks. As application, the free stochastic calculus has been studied See []. By using the same techniques, we characterize how primes as operators act on the Hilbert space generated by the Adele ring. The main purpose of this paper is i to study the spectral information of the primes in A via free probability, and ii to show the inner free-block structures of A, determined by primes. These will provide a bridge between operator theory and number theory. 1.. Acknowledgment. After finishing writing this paper, the author realized that there have been attempts to investigate operator algebra theory with / by number-theoretic objects, results and techniques e.g., [15], [16] and [17]. It is very reasonable but hard approaches by the very historical backgrounds between analysis and number theory, for instance, analytic number theory and L-function theory, etc e.g., [8] and [9]. In particular, in [15] and [16], number-theoretic objects and results are used for studying type III-factors in von Neumann algebra theory. And, in the book [17], various mathematicians provided results and discussions about interconnection among number theory, theoretical physics, and geometry. In [15] and [16], Bost and Connes considered Hecke algebras and type III-factors with help of number-theoretic objects and results. The main purposes of them are to study certain von Neumann algebras. Compared with them, the aim of this paper is to construct von Neumann algebraic models directly from number-theoretic objects, and then study number-theoretic problems in terms of operator-algebraic techniques, especially, those from free probability and dynamical systems under representation theory. Here, we concentrate on establishing the base stone, or the beginning steps of the study. In such senses, one can / may distinguish [15], [16] and this paper. The author also would like to refer [17] to readers for studying connections between number theory and noncommutative-and-commutative operator-algebraic geometry.. Motivation and backgrounds. In this section, we introduce basic definitions and backgrounds of our theory. We first provide the motivation for our study in Section.1. In Section., we introduce the Adele ring A Q. In Sections.3, we briefly review free probability theory, which provides main tools for our study..1. Motivation. The readers can learn why we need to study Adelic structures and p-adic structures in mathematics and in other scientific fields, in particular, in physics, from [14]. To emphasize the importance of Adele-and-p-adic analysis, we put several paragraphs from [14]. In.1.1 and.1. below, the sentences in. are directly from [14]. The main motivation of this paper is introduced in To study small worlds, we need non-archimedean structures and corresponding tools.

3 OPERATORS INDUCED BY PRIME NUMBERS 315 If x is an uncertainty in a length measurement, then the inequality hg x takes place. This inequality is stronger than the Heisenberg uncertainty principle. Here, h is the Planck constant, c is the velocity of light, and G is the gravitational constant.... by virtue of the above inequality, a measurement of distances smaller than the Planck length is impossible.... According to the Archimedean axiom, any given large segment on a straight line can be surpassed by successive addition of small segments along the same line. Really, this is a physical axiom which concern the process of measurement.... But as we just discussed, the Planck length is the smallest possible distance that can in principle be measured. So, a suggestion emerges to abandon the Archimedean axiom at very small distance. This leads to a non-euclidean and non-riemannian geometry of space at small distances. Thus, one may need new measuring system, other than the reals R, satisfying non-archimedean measures. The best possible examples are the p-adic numbers Q p, for primes p, and the Adele ring A Q See below..1.. The study of Adelic structures is of great interest from pure mathematical point of view as well as from the physical and hence scientific one. As possible physical applications we note a consideration of models with non- Archimedean geometry of space-time at very small distances, and also in spectral theory of processes in complicated media. Furthermore, it seems to us that an extension of the formalism of quantum theory to the field of p-adic numbers is of great interest even independent of possible new physical applications because it can lead to better understanding of the formalism of usual quantum theory.... the investigation of p-adic quantum theory and field theory will be also useful in number theory, representation theory, and p-adic analysis.... No doubt investigation of p-adic nonlinear interacting systems will provide new deep mathematical results We do know how primes act or play their roles in R which is an Archimedean structure from classical number theory. It is natural to ask how primes act on an Adelic structure which is a non-archimedean structure. In particular, we are interested in how primes act on certain functions on Adelic structures. By considering a topological space induced by Adelic structure, we act primes on the space in terms of operator theory. And then consider the properties of such operators generated by primes via free probability... The Adele Ring A Q. Fundamental theorem of arithmetic says that every positive integer in N except 1 can be expressed as a usual multiplication of primes or prime numbers, equivalently, all positive integers which are not 1 are prime-factorized under multiplication. i.e., the primes are the building blocks of all positive integers except for 1. And hence, all negative integers n, except 1, can be understood as products of 1 and prime-factorizations of n. Thus, primes are playing key roles in both classical and advanced number theory. The Adele field A Q is one of the main topics in advanced number theory connected with other mathematical fields like algebraic geometry and L-function theory, etc.throughout this paper, we denote the set of all natural numbers which are positive integers by N, the set of all integers by Z, and the set of all rational numbers by Q. c 3

4 316 I. CHO Let s fix a prime p. Define the p-norm. p on the rational numbers Q by q p = p ra b p def = 1 p r, whenever q = p r a b Q = Q \ {0}, for some r Z, with an additional identity: 0 p def = 0 for all primes p. For example, 4 5 = = 1 3 = 1 8, 4 5 = = = 1 3, and 1 4 = = = 8, = = 1 = It is easy to check that i q p 0, for all q Q, ii q 1 q p = q 1 p q p, for all q 1, q Q iii q 1 +q p max{ q 1 p, q p }, for all q 1, q Q. In particular, by iii, we verify that iii q 1 +q p q 1 p + q p, for all q 1, q Q. Thus, by i, ii and iii, the p-norm. p is indeed a norm. However, by iii, this norm is non-archimedean. So, the topological pair Q p,. p is a non-archimedean normed space. Definition.1. We define a set Q p by the norm-closure of the normed space Q p, for all primes p. We call Q p, the p-prime field or the p-adic number field. We can check indeed Q p is a field algebraically See below. For a fixed prime p, all elements of the p-prime field Q p are formed by p r a k p k, for 0 a k < p,..1 k=0 for all k N, and for all r Z, where a k N. For example, 1 = p 1p 0 +p 1p+p 1p +. The subset of Q p, consisting of all elements formed by a k p k, for 0 a k < p in N, k=0

5 OPERATORS INDUCED BY PRIME NUMBERS 317 is denoted by Z p. i.e., for any x Q p, there exists r Z, and x 0 Z p, such that x = p r x 0. Notice that if x Z p, then x p 1, and vice versa. i.e., Z p = {x Q p : x p 1 = p 0 }... So, sometimes, the subset Z p of.. is said to be the unit disk of Q p. Remark that Z p pz p p Z p p 3 Z p, moreover, if x p k Z p, then x p 1 p k, for all k N. It is not difficult to verify that and hence Z p p 1 Z p p Z p p 3 Z p, Q p = k= pk Z p, set-theoretically...3 Consider the boundary U p of Z p. By construction, the boundary U p of Z p is identical to U p = Z p \pz p = {x Z p : x p = 1 = p 0 }...4 Similarly, the subsets p k U p are the boundary of p k Z p satisfying p k U p = p k Z p \p k 1 Z p, for all k Z. We call the subset U p of Q p in..4 the unit circle of Q p. And all elements of U p are said to be units of Q p. Therefore, by..3 and..4, we obtain that Q p = k= pk U p, set-theoretically,..5 where means the disjoint union. By [14], whenever q Q p is given, there always exists q a+p k Z p, for a, k Z. Fact See [14]. The p-prime field Q p,. p is a Banach space. And it is locally compact. In particular, the unit disk Z p is compact in Q p. Define now the addition on Q p by a n p n + b n p n = n= N 1 n= N n= max{n 1,N } for N 1, N N, where the summands c n p n satisfies that a n +b n p n if a n +b n < p c n p n def = p n+1 if a n +b n = p s n p n+1 +r n p n if a n +b n = s n p+r n, c n p n,..6

6 318 I. CHO for all n { max{n 1, N },..., 0, 1,,...}. Clearly, if N 1 > N resp., N 1 < N, then, for all j = N 1,..., N 1 N + 1, resp., j = N,..., N N 1 + 1, c j = a j resp., c j = b j. And define the multiplication on Z p by a k1 p k1 b k p k = k 1=0 k =0 n= N c n p n,..7 where c n = rk1,k i k1,k +s k1 1,k i c k 1 1,k k 1+k =n +s k1,k 1i c k 1,k 1 +s k1 1,k 1i c k 1 1, k 1, where by the division algorithm, and and a k1 b k = s k1,k p+r k1,k, i k1,k = { 1 if ak1 b k < p 0 otherwise, i c k 1,k = 1 i k1,k, for all k 1, k N, and hence, on Q p, the multiplication is extended to a k1 p k1 k 1= N 1 b k p k k = N..7 = p N1 p N a k1 N 1 p k1 b k1 N p k. k 1=0 Then, under the addition..6 and the multiplication..7, the algebraic triple Q p, +, becomes a field, for all prime p. Thus the p-prime fields Q p are algebraically fields. Fact. Every p-prime field Q p, with the binary operations..6 and..7 is a field. Moreover, the Banach filed Q p is also a unbounded Haar-measure space Q p, σq p, ρ p, for all primes p, where σq p means the σ-algebra of Q p, consisting of all measurable subsets of Q p. Moreover, this measure ρ p satisfies that ρ p a+p k Z p k =0 = ρ p p k Z p = 1 p k = ρ..8 p k Z p = ρ a+p k Z p,

7 for all a N, and k Z. Also, one has for all a N. Similarly, we obtain that OPERATORS INDUCED BY PRIME NUMBERS 319 ρ p a+u p = ρ p U p = ρ p Z p \ pz p = ρ p Z p ρ p pz p = 1 1 p, ρ p a+p k U p = ρ p k 1 U p = p k 1 pk+1,..9 for all a N, and k Z See Chapter IV of [14]. Fact. The Banach field Q p is an unbounded Haar-measure space, where ρ p satisfies..8 and..9, for all primes p. The computations..8 and..9 are typically important for us, by..3 and..5 See Section 3 below. The above three facts show that Q p is a unbounded Haar-measured, locally compact Banach field, for all primes p. Definition.. Let P = {p : p is a prime} { }. The Adele ring A Q = A Q,+, is defined by the set equipped with and {x p p P : x p Q p, almost all x p U p,x R},..10 x p p +y p p = x p +y p p, x p p y p p = x p y p p, for all x p p, y p p A Q. The Adele ring A Q has its product topology of the usual topology on R and p-adic topologies of Q p, for all primes p. For convenience, we put R = Q. Indeed, the algebraic structure A Q is a ring. Also, by construction, and under the product topology, the Adele ring A Q is also a locally compact Banach space equipped with the product measure. Set-theoretically, A Q = Π p P Q p = R Π p:prime Q p by identifying Q = R. We identify. =., where. means the norm, the absolute value on R = Q. The product measure ρ of the Adele ring A Q is ρ = ρ p, p P by identifying ρ = ρ R, the usual distance-measure induced by. on R. Fact. The Adele ring A Q is a unbounded-measuredlocally compact Banach ring.,

8 30 I. CHO.3. Free probability. We refer to the readers [10] and [11], for more about free probability theory. In this section, we briefly introduce Speicher s combinatorial free probability. The free probability is originally introduced by Voiculescu in analytic way with combinatorial observation e.g., [9]. Since we will study certain group von Neumann algebras, we concentrate on the case where all given operator algebras are von Neumann algebras. Recall that von Neumann algebras are the weak -topology closure of a -algebra. Without loss of generality, a von Neumann algebra A generated by a subset S of a fixed operator algebra BH is C[S] w, containing the identity operator 1 H on H, where Y w means the weak -topology closure of Y BH, where BH is the operator algebra, consisting of all bounded or equivalently, continuous linear operators on a Hilbert space H. Let B A be von Neumann algebras with 1 B = 1 A and assume that there exists a conditional expectation E B : A B satisfying that i E B b = b, for all b B, ii E B b a b = b E B a b, for all b, b B and a A, iii E B is bounded or continuous, and iv E B a = E B a, for all a A. Then the pair A, E B is called a B-valued amalgamated W -probability space with amalgamation over B. For any fixed B-valued random variables a 1,..., a s in A, E B, we can have the B-valued free distributional data of them; i 1,..., i n -th B-valued joint -moments: E B b1 a r1 i 1 b a r i... b n a rn i n j 1,..., j m -th B-valued joint -cumulants: k B m b 1 a t1 j 1, b a t j,..., b ma tm j m, which provide the equivalent B-valued free distributional data of a 1,..., a s, for all i 1,..., i n {1,..., s} n, j 1,..., j m {1,..., s} m, for all n, m N, where b 1,..., b n, b 1,..., b m B are arbitrary and r 1,..., r n, t 1,..., t m {1, }. By the Moebius inversion, indeed, they provide the same B-valued free distributional data of a 1,..., a s. i.e., they satisfy E B b1 a r1 i 1... b n a rn i n = π NCn k B π b1 a r1 i 1,..., b n a rn i n and km B b 1 a r1 j 1,..., b m atm j m = θ NCm E B:θ b 1 a t1 j 1,...,b m atm j µθ,1m m, where NCk is the lattice of all noncrossing partitions over {1,..., k}, for k N, and k B π... ande B:θ... arethe partition-depending cumulant and the partition-depending moment and where µ is the Moebius functional in the incidence algebra I. Recall that the partial ordering on NCk is defined by π θ def blocks V in π, blocks B in θ s.t. V B.

9 OPERATORS INDUCED BY PRIME NUMBERS 31 Under this partial ordering, NCk is a lattice with its maximal element 1 k = {1,..., k} and its minimal element 0 k = {1,,..., k}. The notation... inside partitions {...} means the blocks of the partitions. For example, 1 k is the one-block partition and 0 k is the k-block partition, for k N. Also, recall that the incidence algebra I is the collection of all functionals ξ : k=1 NCk NCk C, satisfying ξπ, θ = 0, whenever π > θ, with its usual function addition + and its convolution defined by ξ 1 ξ π,θ def = π σ θ ξ 1 π,σξ σ,θ, for all ξ 1, ξ I. Then this algebra I has the zeta functional ζ, defined by ζπ,θ def = { 1 if π θ 0 otherwise. The Moebius functional µ is the convolution-inverse of ζ in I. So, it satisfies for all k N \ {1}, and µ0 1,1 1 = 1, and π NCk µ0 k,1 k = 1 k 1 c k 1, µπ,1 k = 0, def m where c m = 1 m+1 is the m-th Catalan number, for all m N. m The amalgamated freeness is characterized by the amalgamated -cumulants. Let A, E B be given as above. Two W -subalgebras A 1 and A of A, having their common W -subalgebra B, are free over B in A, E B, if and only if their mixed -cumulants vanish. Two subsets X 1 and X of A are free over B in A, E B, if vnx 1, B and vnx, B are free over B in A, E B, where vns 1, S means the von Neumann algebra generated by S 1 and S. In particular, two B-valued random variable x 1 and x are free over B in A, E B, if {x 1 } and {x } are free over B in A, E B. Suppose two W -subalgebras A 1 and A of A, containing their common W - subalgebra B, are free over B in A, E B. Then we can construct a W -subalgebra vna 1, A of A generated by A 1 and A. Such W -subalgebra of A is denoted by A 1 B A. If there exists a family {A i : i I} of W -subalgebras of A, containing their common W -subalgebra B, satisfying A = B A i, then we call A, the B-valued i I free product algebra of {A i : i I}. Assume now that the W -subalgebra B is -isomorphic to C = C 1 A. Then the conditional expectation E B becomes a linear functional on A. By ϕ, denote E B. Then, for a 1,..., a n A, ϕ, k n a 1,...,a n = π NCn ϕ π a 1,...,a n µπ,1 n

10 3 I. CHO by the Moebius inversion = π NCn Π ϕ V a 1,...,a n µπ,1 n V π since the image of ϕ are in C For example, if π = {1, 3,, 4,5} in NC5, then ϕ π a 1,...,a 5 = ϕa 1 ϕa a 3 ϕa 4 a 5 = ϕa 1 a 3 ϕa ϕa 4 a 5. Remark here that, if ϕ is an arbitrary conditional expectation E B, and if B C 1 A, then the above second equality does not hold in general. = Π ϕ V a 1,..., a n µ0 V, 1 V V π π NCn by the multiplicativity of µ. i.e., If A, ϕ is a C-valued W -probability space, then k n a 1,...,a n = Π ϕ V a 1,...,a n µ 0 V, 1 V, V π π NCn for all n N, where a 1,..., a n A, ϕ. 3. Adele von Neumann algebra A. In this section, we consider the von Neumann algebraa generatedby the Adele ring A Q. Since the Adele ring A Q is a measure space, naturally, one can construct the L -space H induced by this measure space, consisting of all square integrable function, as a Hilbert space. i.e., let H = L A Q,ρ. 3.1 Notice that all elements g of H has its form g = t S χ S, with t S C, 3. S Suppg where χ S is the characteristic functions, where Suppg means the support of g, Suppg def = {S σa Q : t S 0}. 3.3 Now, define an action α of A Q acting on H by satisfying α : x A Q αx denote = α x BH, 3.4 α x χ S = χ xs, for all S σa Q, where xs = {xs : s S}, for all S A Q, and where BH means the operator algebra consisting of all bounded linear operators on the Hilbert space H. Definition 3.1. The L -Hilbert space H = L A Q, ρ of 3.1 is called the Adele Hilbert space. Let α be a action 3.4 of the Adele ring A Q, acting on H. Then the

11 OPERATORS INDUCED BY PRIME NUMBERS 33 corresponding von Neumann algebra A = C[αA Q ] w induced by the Adele ring A Q is called the Adele von Neumann algebra. By the very definition, if a is an element of the Adele von Neumann algebra A, then it has its expression, a = x A Q t x α x, with t x C, where x = x p p A Q. Extend the action α of the Adele ring A Q acting on the Adele Hilbert space H to the representation of the von Neumann algebra A acting on H. Let s denote this extended action of A on H by α, too. i.e., α a r S χ S = t x r S χ xs, 3.5 for all a = S σa Q x A Q t x α x A, where x A Q S σa Q α x χ S = x p p S = x p p Π p S p = Π p x p S p. 3.6 Definition 3.. The pair H, α of the Adele Hilbert space H and the von Neumann algebra representation α of 3.5 of the Adele von Neumann algebra A is called the canonical representation of A acting on H. Define now elements a p of the Adele von Neumann algebra A by for all primes p, where a p = α 1, 1,..., 1, p, 1, 1,... p-th entry 1,1,...,1,p,1,... A Q. A, 3.7 Then this element a p of A satisfies that a p t S χ S = S σa Q for all t S χ S H, where S σa Q = S σa Q t S χ ps S σa Q t S χ Π q P S q, { S q = Sq if q p ps p if q = p, for primes q. Notice here that if S is a measurable subset of A Q, then S is of the form S = Π q P S q, for S q σq q,

12 34 I. CHO for all q P, but for almost of all primes q, S q Z q by the very definition of the Adele ring A Q. Definition 3.3. The elements a p of the Adele von Neumann algebra A defined in 3.7 are called the p-prime multiplier, for all primes p. The family {a p } p:prime in A is called the prime multipliers. Remark that e def = 1,1,1,... A Q A, is the unity of A Q and hence, the αe = α e is the identity operator in A. Let a p be the p-prime multiplier for a fixed prime p. Then the adjoint a p of a p is identified with a p 1, i.e., a p = α 1, 1,..., 1, p 1, 1, 1,... p-th, 3.8 for all primes p. Indeed, for any fixed S k = Π q P S q:k σa Q, with S q:k σq q for almost of all S q:k Z q, for k = 1,, < a p χ S1,χ S > = < pχ S1,χ S > = < χ ps1,χ S >= χ ps1 χ S dρ A Q = χ ps1 S dρ = ρps 1 S A Q = ρ S 1 1 p S = χ S1 1Sdρ p A Q = χ S1 χ 1 Sdρ =< χ p S 1, 1 A Q p χ S >, and hence the adjoint a p is identical to the right-hand side of 3.8. We denote the element which is represented by the right-hand side of 3.8 by a p 1 or a 1, for primes p p. Then the relation 3.8 can be re-written by a p = a p 1 in A, for all primes p. Now, let p and q be arbitrarily fixed primes, and let x {p, p 1 }, and y {q, q 1 }. If p q in primes, then a x a y = α 1, 1,..., 1, x, 1, 1,... α 1, 1,..., 1, y, 1, 1,... p-th q-th α 1,..., 1, x, 1,..., 1, y, 1,... if p < q p-th q-th = α 1, 1,..., 1, xy, 1, 1,... if p = q. p-th 3.9

13 OPERATORS INDUCED BY PRIME NUMBERS 35 If a p is the p-prime multiplier in the Adele von Neumann algebra A, for a prime p, then a n p = α1, 1,..., 1, p, 1, 1,... n = α1, 1,..., 1, p n, 1, 1,..., 3.10 by 3.9, for all n N. By a little abuse of notation, we denote the resulted operator α1, 1,..., 1, p n, 1, 1,... of 3.10 by a p n, alternatively. Define now new elements T p in the Adele von Neumann algebra A by T p = a p +a p = a p +a p 1, for all prime p. Then T p is a well-defined self-adjoint element of A. Definition 3.4. The element T p = a p + a p 1 of the Adele von Neumann algebra A induced by the p-prime multiplier a p and its adjoint a p is called the p-prime operator of A, for all prime p. Notice that a p a p 1 = α e = 1 A = a p 1a p, 3.11 the identity element of A. By 3.11 we have the following lemma. Lemma 3.1. Every n-power of the p-prime multiplier a n p is unitary on the Adele Hilbert space H, for all n N, for all primes p. Proof. If n = 1, the p-prime multiplier a p is unitary in A, by Now, assume that n > 1. Then, by 3.10, a n p = a pn, for all n N. So, it is not difficult to check by 3.7 Therefore, a p n = a p n, for n N. a n p a n p = a p n a p n = a p na p n = a p n p n = α e = 1 A = a p n p n = a p na p n = a p na p n = a n pa n p. Therefore, a n p is unitary in A, for all n > 1. So, the elements an p are unitaries in A, for all n N, for primes p. By the above lemma, we obtain the following theorem. Theorem 3.. Let {a p } p:prime be the prime multipliers in the Adele von Neumann algebra A. Then the subfamily {a p, a p } generates a group Γ p under operator multiplication on A, for each prime p. Moreover, the group Γ p is group-isomorphic to the infinite cyclic abelian group Z. i.e., Γ p Group = Z, 3.1

14 36 I. CHO where Group = means being group-isomorphic, for all primes p. Proof. Bythe previouslemma, all powersofp-primemultipliers a p areunitaries, for all primes p, on the Adele Hilbert space H. Therefore, their adjoints a p = a p 1 are unitaries on H, too, as inverses of a p, for all primes p. So, the family {a p, a p 1} is a unitary family in the Adele von Neumann algebra A. Moreover, the set Γ p becomes a group, with identification Γ p def = {a p n,a p n : n Z} a p 0 = α e = 1 A = a p 0, under the operator multiplication on A. Indeed, the operator multiplication on A is closed in Γ p. i.e., whenever a p n 1, a p n Γ p, for n 1, n Z, a p n 1a p n = a p n 1p n = a p n 1 +n Γ p, too. Also, the operation is associative, and commutative. The operation-identity α e = a p 0 exists in Γ p ; a p nα e = a p n+0 = a p n, for all n Z. For n Z \ {0}, there always exists n Z, such that a p na p n = a p n n = α e. So, for any a p n Γ p \ {α e }, there always exists a p n = a p, the operation-inverse. Therefore, the set Γ p, equipped with the operator multiplication, forms a group. For convenience, we denote this group Γ p, simply by Γ p. Define a map by Φ : Γ p Z Φa p n = n, for all n Z, where Z = Z, + is the infinite cyclic abelian group, consisting of all integers. Then it is bijective, by the very definition of Γ p. Moreover, Φa p n 1a p n = Φ a p n 1 +n = n1 +n = Φa p n 1+Φa p n, for all n 1, n Z. Thus, the map Φ is a group-homomorphism. By the bijectivity of Φ, it is in fact a group-isomorphism. i.e., Γ p Group = Z. The above theorem shows that in the Adele von Neumann algebra A, each p- prime multiplier a p, which is a unitary, generates a group Γ p, under the operator multiplication on A, and the corresponding subgroup Γ p of A is group-isomorphic to Z. It shows that there are infinite copies of embedded cyclic abelian group Z in A.

15 OPERATORS INDUCED BY PRIME NUMBERS 37 Definition 3.5. Let Γ p be a group generated by the p-prime multiplier a p, for all primes p. Then we call Γ p, the p-prime subgroup of A, for all primes p. Sometimes, to emphasize the generator of the p-prime subgroup Γ p, we write Γ p = < a p >, for all primes p. Now, notice that all prime subgroups Γ p = < a p >, for all primes p, share the group-identity α e in the Adele von Neumann algebra A. So, one obtains the following corollary. Corollary 3.3. Let {Γ p } p:prime be the family of all prime subgroups of A. Then the group Γ generated by {Γ p } p:prime, under the operator multiplication on the Adele von Neumann algebra A, is group-isomorphic to the free group F with countably infinite generators. Proof. Let Γ p be p-prime subgroups of A, and let Γ be the group generated by {Γ p } p:prime, under the operator multiplication on A. It is true that, then Γ is the group generated by all p-prime multipliers {a p } p:prime, i.e., Γ =< {a p } p:prime >=< {a p,a p 1} p:prime >. Let F be the free group with -generators {g n } n=1, with their inverses {gn 1} n=1. It is well-known that there are countably infinitely many primes. i.e., there exists a bijection from {p} p:prime onto N. Say h. Define now a map satisfying Ψ a k def p Ψ : Γ F = g k hp and Ψ a k p = g k hp, for all k N, and for all primes p, where g k hp means the group-inverse of gk hp. Notice that if p, q are primes and k 1, k Z, then the operator a p k 1 a q k satisfies that ap k 1 a q k ap k 1 a q k = a q k a p k 1a p k 1 a q k = a q k a p k 1 a p k 1 a q k = a q k ap k 1 +k 1 aq k = a q k +k = α e, and hence a p k 1 a q k is a unitary in A, too. Thus, the products N Π a n=1 p kn n are unitaries, too. i.e., all elements of Γ are unitaries on the Adele Hilbert space H. So, the map Ψ is a well-defined group-homomorphism from Γ to F. Moreover, by the bijectivity of h, it is bijective, too. So, Ψ is a group-isomorphism. Therefore, Γ Group = F Let F n be the free group with n-generators, for n N { }. The group von Neumann algebra LF n acting on the group Hilbert space L F n is called the free group factor. By the above corollary, one can recognize that the Adele von Neumann algebra A contains free group factors as its W -subalgebras.

16 38 I. CHO Corollary 3.4. Let A be the Adele von Neumann algebra, and let LF n be a free group factor, for n N \ { }. Then the free group factor LF n is -isomorphic to a W -subalgebra of A. Proof. Fix n N { }, and take X = {p 1,..., p n }, a set of distinct n-primes. Then we have {a p1,...,a pn } A. Consider the subgroup Γ X, generated by {a p1,..., a pn }. By the above corollary, Group Γ X = n Group Group Γ pk = Z... Z k=1 }{{} = F n. n-times Therefore, the W -subalgebra vnγ X satisfies vnγ X -iso = LF n in A The above corollary provides the motivation of this paper. i.e., our Adele von Neumann algebra A has certain inner free structure, determined by primes. In Sections 4, 5, and 6, we consider such free structure of A in detail. It is well-known that, if Γ 1 and Γ are groups and Γ = Γ 1 Γ is the group- free product group of Γ 1 and Γ, then the group von Neumann algebra vnγ is -isomorphic to the operator-algebraic-free product von Neumann algebra vnγ 1 C vnγ of the group von Neumann algebras vnγ 1 and vnγ. So, the free group factors LF n is -isomorphic to LZ C... C LZ, }{{} n-times for all n N { } See [1], [1], [], and [4]. As we have seen above, the Adele von Neumann algebra A contains LF n, for n N { }. It shows that there does exist the free structures in A, induced by LZ -iso = vn Γ p, for all primes p. Now, we provide the following proposition which gives us a useful tool for studying free structure of A, in Sections 4, 5, and 6. Proposition 3.5. Let p be a prime, and let T p = a p + a p be the p-prime operator of the Adele von Neumann algebra A. Then n for all n N, where k T n p = a p +a p n = n n k k=0 n! = k!n k!, for all k n N. a p k n, 3.15 Proof. The proof of 3.13 is straightforward, by the binomial expansion. In fact, it is enough to show that a p na p m = a p ma p n, 3.16

17 OPERATORS INDUCED BY PRIME NUMBERS 39 for all n, m Z, with identification a p 0 = α e = 1 A. Since p n p m = p n+m = p m+n = p m p n in Q p, for all primes p, the relation 3.14 holds true. So, n Tp n = a p +a p 1 n n = a k p ka p n k. k=0 Definition 3.6. Let T p be the p-prime operator in the Adele von Neumann algebra A. The element def T o = p:prime of A is called the prime radial operator of the Adele von Neumann algebra A.Let X be a subset of the set of all primes. Then the X-radial operator T X is defined by T p T X def = p XT p. 4. Free Moments of Prime Operators. Let A Q be the Adele ring and let A be the Adele von Neumann algebra C[αA Q ] w. And let {a p = α1, 1,..., 1, p, 1, 1,...} p:prime be the prime multipliers in the sense of 3.5, which are the unitaries on the Adele Hilbert space H = L A Q, ρ, and {T p = a p +a p 1} p:prime, the prime operators, which are self-adjoint operators on H. Also, let T o = p:prime be the prime radial operator of A. Define now the bounded or continuous linear functional ϕ on A by ϕ t x α x def = t e, 4.1 x A Q where e = 1, 1,... is the unity of the Adele ring A Q. Then it is a well-defined linear functional on A. Indeed, ϕ t x α x + s y α y = ϕ t x +s x α x x A Q y A Q x A Q T p = t e +s e = ϕ t x α x +ϕ s y α y, x A Q y A Q

18 330 I. CHO and ϕ t t x α x = ϕ tt x α = tt e = tϕ t x α x, x A Q x A Q X A Q for all t x α x, s y α y A. And x A Q y A Q where and ϕ t x α x = ϕ t x α x 1, X A Q x A Q x 1 = x p p 1 = x 1 p p in A Q, α x = α x 1, forall x=x p p A Q =t e = ϕ t x α x, for all t x α x A.Moreover,it x A Q x A Q is bounded by the verydefinition 4.1. So, the pair A, ϕ is a well-defined C-valued W -probability space in the sense of [1]. Definition 4.1. Let A be the Adele von Neumann algebra and let ϕ be a bounded linear functional on A, introduced in 4.1. Then the W -probability space A, ϕ is called the Adele W -probability space. So, all elements of the Adele von Neumann algebra A are understood as free random variables in the Adele W -probability space A, ϕ. In this section, we are interested in the free distributional data of the prime operators {T p } p:prime and the prime radial operator T o. By 3.13, if T p = a p + a p 1 is the p-prime operator in A, then for all n N, and for all prime p. n Tp n = n k k=0 a p k n, 4. Theorem 4.1. For a fixed prime p, let T p = a p +a p be the p-prime operator in the Adele von Neumann algebra A, where a p is the p-prime multiplier and a p = a p 1 is the adjoint of a p. Then n ϕtp n n = n! n =! if n is even if n is odd, for { all n } N. Therefore, the p-prime operator T p has its free distributional data n n. n=1

19 OPERATORS INDUCED BY PRIME NUMBERS 331 Proof. By 4., we have that n Tp n = n k k=0 a p k n, for all n N. Thus, n k n = = ϕtp n n if n = k k k = 0 otherwise, for all n N. By 4.3, we have ϕt n p = n n = n! n!, for all n N, and ϕt n 1 p = 0, for all n N. Definition 4.. Let A, ψ be an arbitrary W -probability space, and let a 1, a A, ψ be free random variables. Assume more that two free random variables a 1 and a are self-adjoint in the von Neumann algebra A. These two free random variables a 1 and a are identically free-distributed, if ψa n 1 = ψa n, for all n N. i.e., the free random variables a 1 and a have equivalent free-moment data {ψa n 1 } n=1 and {ψa n } n=1. By 4.3, we obtain the following corollary. Corollary 4.. Let A, ϕ be the Adele W -probability space and let {T p } p:prime be the prime operators in the Adele von Neumann algebra A. Then they are identically distributed. Proof. By 4.3, for all prime p, ϕt n p = n n, and ϕt n 1 p = 0, for all n N. Therefore, the p-prime operators T p s are all identically distributed. Now, let p 1,..., p k be primes, for some k N. For a mixed n-tuple p i1,..., p in of {p 1,..., p k }, we can compute the mixed free moments ϕ T pi1 T pi... T pin. Before computing the mixed free moments of T p1,..., T pk in the Adele W - probability space A, ϕ, we consider the following lemma, first.

20 33 I. CHO Lemma 4.3. Let p, q be distinct prime, and let T p and T q be corresponding prime operators in A, ϕ. Then T p T q = T q T p. 4.4 Proof. Let T p and T q be prime operators in the Adele W -probability space A, ϕ. Then T p T q = a p +a p 1 aq +a q 1 = a p a q +a p a q 1 +a p 1a q +a p 1a q 1 = αx p αx q +αx p αx q 1+αX p 1αX q +αx p 1αX q 1 by the definition of prime multipliers, where in A Q X r = 1,...,1, r,1,1,..., for all primes r r-th = αx p X q +αx p X q 1+αX p 1X q +αx p 1X q since α is an action. Assume that p < q in primes. Then the formula 4.5 becomes that: α[1,..., 1, p, 1,..., 1, q, 1, 1,...] + α [1,..., 1, p, 1,..., 1, q 1, 1, 1,...] + α [1,..., 1, p 1, 1,..., 1, q, 1, 1,...] + α [1,..., 1, p 1, 1,..., 1, q 1, 1, 1,...] = αx q X p + α X q 1X p + α Xq X p 1 + α Xq 1X p 1 = a q a p + a q 1a p + a q a p 1 + a q 1a p 1 = a q +a q 1 ap +a p 1 = T q T p. Therefore, we obtain that T p T q = T q T p. By the commutativity of {T p } p:prime, as in 4.5, one can verify that T n p T m q = T m q T n p, 4.6 for all n, m N, and for all primes p, q. Therefore, we obtain the following theorem. Theorem 4.4. Let p 1,..., p k be arbitrarily fixed primes, for some k N \ {1}. Let p i1,..., p in be the mixed n-tuple of {p 1,..., p k }, for n N \ {1}, and assume that #p j means the number of p j s in the given n-tuple p i1,..., p in in N {0}, for j = 1,..., k. Then we obtain that: if #p j are even or 0, for all n N, and if there exists at least one j such that #p j 0, then ϕ T pi1... T pin = ϕ T p #p1 1 T p #p... T #p k p k #p = Π k j 4.7, j=1 #p j

21 OPERATORS INDUCED BY PRIME NUMBERS 333 where #p j #p j def = #p j #p j if #p j N 0 otherwise, 4.8 for j = 1,..., k. By 4.7, if n is odd in N, then ϕ T pi1... T pin = 0. Proof. Let p i1,..., p in be the mixed n-tuple of {p 1,..., p k }, for some k N \ {1}, for all n N \ {1}. Assume first that there exists at least one nonzero #p j, and suppose all #p 1,..., #p k in p i1,..., p in are either even or 0. Then ϕ T pi1... T pin = ϕ T p #p T #p k p k by 4.6 where #p = ϕ Π k j α e + j=1 j=1 #p j #p = ϕ Π k j #p j denote = k Π j=1 #p ij k=1 α e + #p j k=1 #pj k #pij k a p k #p j j a p k #p ij i j then #p = ϕ Π k j j=1 #p j α e + 0 since a p n 1 a q n does not becomes α e, whenever p q in primes #p = Π k j. where j=1 #p j #p j #p j def = #p j #p j, whenever #p j N, and it becomes 0, otherwise, for all j = 1,..., n. Suppose now n is odd in N. Then there exists at least one j in {1,..., k}, such that #p j is odd. Therefore, T pi1...t pin = T #p1 p 1...T #p k p k

22 334 I. CHO does not contain the α e -terms, and hence ϕ T pi1... T pin = 0. The above theorem characterizes the mixed moment computations of prime operators. Example Let p, q, q, q, q, p be a mixed 6-tuple of fixed distinct primes {p, q}. Then ϕt p T q T q T q T q T p = ϕt p T4 q 4 = ϕ α 1 e 4 = = 1. 1 Let p, q, q, p, q be a mixed 5-tuple of fixed distinct primes {p, q}. By the above theorem, one can verify that ϕt p T q T q T p T q = 0, since 5 is odd in N. 3 Let p, p, q, q, q, p, q, q be a mixed 8-tuple of fixed distinct primes {p, q}. Then ϕt p T p T q T q T q T p T q T q = ϕ T 3 p T5 q = 0, by Let p, q, p, p, r, q, p, r be a mixed 8-tuple of fixed distinct primes {p, q, r}. Then ϕt p T q T p T p T r T q T p T r = ϕ Tp 4 T q T r 4 = 1 1 = Freeness of prime operators. In this section, we also let A Q be the Adele ring, a locally compact measured topological ring, and A, ϕ, the Adele W - probabilityspace. And leta p bethe p-primemultipliersandt p =a p +a p,the p-prime operators, for all prime p. In Section 4, we computed the free moments and mixed free moments of p-prime operators T p. In particular, we obtain that the computation 4.3: ϕ Tp n n =, and ϕ T n p n 1 = 0, for all n N, for all primes p. Moreover, for any mixed n-tuple p i1,..., p in of a fixed family {p 1,..., p k } of distinct primes, the computation 4.7 shows that ϕ #p k j T pi1... T pin = Π j=1 #p j,

23 OPERATORS INDUCED BY PRIME NUMBERS 335 whenever n is even in N, where k = k k k if k is even 0 if k is odd, and it vanishes whenever n is odd in N. In this section, we will compute the equivalent free distributional data determined by free cumulants. By the Moebius inversion, for any a A, ϕ, we have that k n a,...,a = Π V π ϕa V µ 0 V, 1 V, 5.1 π NCn for all n N See Section. or [1]. Theorem 5.1. Let T p = a p + a p 1 be the p-prime operator in A, for a fixed prime p. Then k n T p,..., T p = Π V! V π µ 0 V, 1 V, 5. V π NCEn! for all n N, where NCEn def = {π NCn : V N, V π}. 5.3 We call the subset N CEn, the even noncrossing partition set. In particular, if NCEn is empty in the noncrossing partition set NCn, then k n T p,...,t p = 0, whenever n is odd. Especially, if n is odd, then the free cumulant vanishes. Proof. For n N, we have that k n T p,...,t p = π NCn Π ϕ V π T p V µ 0 V, 1 V by 5.1 = π NCn Π ϕ V π, V N T p V µ 0 V, 1 V by 4.3 = π NCEn Π ϕ V π T p V µ 0 V, 1 V, 5.4 by the definition of the even noncrossing partition set N CEn. Therefore, we obtain the desired computation 5..

24 336 I. CHO It is clear that, if the subset NCEn of NCn is empty, then the formula 5.4 vanishes, and hence the free cumulant vanishes. As a special case, assume now that n is odd. Then NCEn is empty. Indeed, for any π NCn, there always exists at least one block V o π, such that V o is odd in N. Therefore, for all π NCn, the corresponding block-depending summand Π ϕ T p µ V 0 V, 1 V V π of k n T p,..., T p satisfies that: Π ϕ T p µ V 0 V, 1 V V π = ϕ µ 0 Vo, 1 Vo = 0, T Vo p and hence, if n is odd in N, then Therefore, only if n is even in N, k n T p,...,t p = Π V π, V V o ϕ k n T p,...,t p = 0. π NCEn Π ϕ V π T p V T V p µ 0 V, 1 V µ 0 V, 1 V, by 5.4 = π NCEn Π V π V V µ 0 V, 1 V, by 4.3. In Section 4, we defined the identically free-distributedness of self-adjoint elements in W -probability spaces. And it is shown that all prime operators T p are identically distributed by 4.3. The above theorem or 5. again demonstrates that T p are identically distributed via the Moebius inversion. i.e., the free moments and the free cumulants contain the equivalent free-distributional data. So, the computation 5. for all prime p guarantees the identically distributedness of prime operators, too. In the rest of this section, let s consider the mixed cumulants of T pi1,..., T pin, where p i1,...,p in {p 1,...,p k } n, for some fixed k N \ {1}, for all n N \ {1}. Let p q be fixed distinct primes, and let T p and T q be corresponding prime operators in the Adele W -probability space A, ϕ. Then Observe that k T p,t q = k T p,t q = π NC ϕ π T p,t q µπ,1 = ϕt p T q ϕt p ϕt q = 0.

25 Also, we get that OPERATORS INDUCED BY PRIME NUMBERS 337 k 4 T p, T q, T p, T q = k 4 T p, T p, T q, T q since T p T q = T q T p, by 4.4 = k T p, Tq by Speicher See [11] and [1], and by 4.7 = ϕ π T p, Tq µ0π, 1 π NC = ϕ TpT q ϕ T p ϕ T q = = 0. Thus, by the commutativity of {T p } p:prime, k 4 T r1,t r,t r3,t r4 = 0, 5,5 whenever r 1, r, r 3, r 4 are the mixed quadruples of {p, q} satisfying #p = = #q. Remark 5.1. By 5.5, we have ϕ T pt q = ϕ T p ϕ T q. So, it is possible that one may be tempted tobelieve ϕ is acharacter on {T p } p:prime, in the sense that ϕt n1 p T n q = ϕt n1 p ϕt n q, for all primes p, q, and n 1, n N. However, it does not hold in general. For instance, if the above multiplicativity of ϕ were true, then we must have ϕ Tp 6 6 = = 0 = ϕt 3 p ϕtp 5, which is definitely not true. Also, if ϕ were a character, then ϕ 6 Tp 6 = = = 1 = 1 = ϕ Tp ϕ T 4 p, which is not true, either. So, of course, the linear functional ϕ cannot be a character on {T p } p:prime. However, we remark that ϕ T n1 p n1 n Tq n = n 1 n = ϕ T n1 p ϕ T n q,

26 338 I. CHO for all n 1, n N, and for all distinct primes p q. Motivated by 5.5 and 5.5, we obtain the following computations. In fact, the proof of the following theorem is not directly related to free cumulant computations as usual. It is proven theoretically based on the results from the structure theorems in Section 3 See 3.13 and Theorem 5.. Let p 1,..., p k be fixed distinct primes, for k N \ {1}, and let p i1,..., p in be a mixed n-tuple of {p 1,..., p k }, for all n N \ {1}. Then for all n N. k n Tpi1,..., T pin = 0, 5.6 Proof. By 3.13, the group Γ generated by the p-prime subgroups Γ p = < a p > of the Adele von Neumann algebra A is group-isomorphic to the free group F. And hence, by 3.14, the free group factor LF satisfies LF -iso = C LZ -iso = C vnγ p -iso = vnγ 5.7 n=1 p:prime is a W -subalgebra of A, where vnγ p and vnγ mean the group von Neumann algebras generated by Γ p and Γ, respectively. The first -isomorphic relation of 5.7 is came from [11] and [1], under the canonical group-von Neumann-algebra linear functional τ τ t g g def = t e, for all t g g LF. g F g F The second -isomorphic relation of 5.7 holds, because our linear functional ϕ and the canonical group-von Neumann-algebra linear functional τ are equivalent on vnγ, respectively on LF See [1], [3], and [4]. Thus, the third -isomorphic relation of 5.7 holds because of i.e., vnγ p and vnγ q are free in vnγ, whenever p q. Since p 1,..., p k are distinct from each other, there exist W -algebras vnγ 1,..., vnγ k, such that they are free in A, ϕ. By definition, the corresponding p-prime operators T pj are contained in vnγ pj, for j = 1,..., k. Therefore, they are free from each other in A, ϕ, too. Equivalently, all mixed free cumulants of them vanish. As an inner structure, one can verify that there is an inner free structure induced by p-prime operators {T p } p:prime. Corollary 5.3. Let T p be prime operators in the Adele von Neumann algebra A, and let A p be the W -subalgebras C[{T p }] w of A, generated by T p, for all primes p. Then A p s are free from each other, for all prime p, in the Adele W -probability space A, ϕ. Therefore, the von Neumann subalgebra A prime of A, generated by {T p : all primes p}, is -isomorphic to -iso A prime = C A p. 5.8 p:prime Proof. It suffices to check A p are W -subalgebra of vnγ p, for all primes p. But, by the very definition, it is trivial. Therefore, by 5.7, we obtain the relation 5.8.

27 OPERATORS INDUCED BY PRIME NUMBERS 339 REFERENCES [1] I. Cho, Graph von Neumann Algebras, ACTA Appl. Math., , pp [] I. Cho, Direct Producted W -Probability Spaces and Corresponding Amalgamated Free Stochastic Integration, Bull. Korean Math. Soc., 44:1 007, pp [3] I. Cho, The Moments of Certain Perturbed Operators of the Radial Operator in the Free Group Factor LF N, J. Anal. Appl., 5:3 007, pp [4] I. Cho, Group Freeness and Certain Amalgamated Freeness, J. Korean Math. Soc., 45:3 008, pp [5] I. Cho, Group-Framing and Corresponding von Neumann Algebras, Adv. Appl. Math. Sci., :5 010, pp [6] I. Cho and P. E. T. Jorgensen, C -Subalgebras Generated by Partial Isometries, J. Math. Phy., DOI: / , 009. [7] I. Cho and P. E. T. Jorgensen, Free Probability Induced by Electric Resistance Networks on Energy Hilbert Spaces, Opuscula Math., 011, to appear. [8] T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, 010 PhD Thesis. [9] T. Gillespie, Prime Number Theorems for Rankin-Selberg L-Functions over Number Fields, Sci. China Math., 54:1 011, pp [10] F. Radulescu, Random Matrices, Amalgamated Free Products and Subfactors of the C - Algebra of a Free Group of Nonsingular Index, Invent. Math., , pp [11] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator- Valued Free Probability Theory, Amer. Math. Soc. Mem., 13:67, [1] D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, vol 1., 199. [13] D. Bump, Automorphic Forms and Representations, Cambridge Studies in Adv. Math., 55, ISBN: , 1996 Cambridge Univ. Press. [14] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol 1, ISBN: , 1994 World Scientific. [15] J.-B Bost and A. Connes, Hecke Algebras, Type III-Factors and Phase Transitions with Spntaneous Symmetry Breaking in Number Theory, Selecta Math., 1:3 1995, pp [16] J.-Beno Bost and A. Connes, Products eulériens et Facteurs de Type III, C. R. Acad Sci. Paris Ser. Math., 315:3 199, pp [17] P. Cartier, B. Julia, P. Moussa, and P. Vanhove Editors, Frontier in Number Theory, Physics, and Geometry II, ISBN: , 007 Springer-Verlag Berlin.

28 340 I. CHO