Ilwoo Cho and Palle E.T. Jorgensen

Transcription

1 Ouscula Math. 37, no , htt://dx.doi.org/ /omath Ouscula Mathematica SEMICIRCULAR ELEMENTS INDUCED BY -ADIC NUMBER FIELDS Ilwoo Cho and Palle E.T. Jorgensen Communicated by P.A. Couhari Abstract. In this aer, we study semicircular-like elements, and semicircular elements induced by -adic analysis, for each rime. Starting from a -adic number field Q, we construct a Banach -algebra LS, for a fixed rime, and show the generating elements Q, of LS form weighted-semicircular elements, and the corresonding scalar-multiles Θ, of Q, become semicircular elements, for all Z. The main result of this aer is the very construction of suitable linear functionals τ 0, on LS, making Q, be weighted-semicircular, for all Z. Keywords: free robability, rimes, -adic number fields Q, Hilbert-sace reresentations, C -algebras, wighted-semicircular elements, semicircular elements. Mathematics Subect Classification: 05E15, 11R47, 11R56, 46L10, 46L40, 47L15, 47L30, 47L INTRODUCTION The main urose of this aer is to construct weighted-semicircular, and semicircular elements for a fixed rime. Starting from a rime, we consider -adic analysis on the -adic number field Q, and a certain -algebra M of all measurable functions on Q. By establishing suitable C -robabilistic structures on the C -algebra M, generated by M, we focus on a semigrou S in M, generating C -subalgebra S of M. By filterizing, or sectionizing S from a system of linear functionals, we construct-and-study Banach -robabilistic structures, and our associated weighted-semicircular, and semicircular elements. In classical statistics, and in alications of it, one consider Gaussian elements, or Gaussian rocesses by taking suitable measures or suitable robability density functions e.g., [1 3] and [0]. By analogy, we construct our semicircular-like, and semicircular elements by taking a suitable system of linear functionals on a Banach -algebra. c Wydawnictwa AGH, Krakow

2 666 Ilwoo Cho and Palle E.T. Jorgensen Let Q be the -adic number fields for P, where P is the set of all rimes in the natural numbers or the ositive integers N. Then one can naturally understand Q as a measure sace Q, σq, µ, where µ is a both-left-and-right additive invariant Haar measure on the σ-algebra σq, containing the basis elements of the toology for Q, formed by transforming the unit disk Z of Q, satisfying µ Z 1 µ x + Z, for all x Q. The -algebra M, consisting of all µ -measurable functions on Q, is well-determined for P, and we cannot hel emhasizing the imortance of such algebraic structures not only in various mathematical fields modern number theory, geometry with very small distance, and oerator theory, etc, e.g., [15, 16, 18, 19] and [30], but also in other scientific fields quantum hysics, quantum arithmetic chaos theory, etc., e.g., [3, 6, 8, 9, 13, 14] and [9] BACKGROUND AND MOTIVATION The relations between rimes and oerator algebras have been studied in various different aroaches e.g., [3 5, 11, 13, 14, 3, 9] and [3]. For instance, we studied how rimes act on certain von Neumann algebras generated by -adic and Adelic measure saces e.g., [9]. Indeendently, in [7] and [8], we have studied rimes as linear functionals acting on arithmetic functions. i.e., each rime induces a free-robabilistic structure A, g on the algebra A of all arithmetic functions. In such a case, one can understand arithmetic functions as Krein-sace oerators, under certain reresentations See [11]. And, free-robabilistic research on classical Hecke algebras induced by rimes is considered e.g., [10]. Motivated by the main results of [9], we realized that our free-robabilistic settings can be alicable, or used for the alied oerator theory based on number-theoretic information. In articular, one may construct semicircular law, or semicircular-like law from a fixed rime. 1.. MAIN IDEAS In this aer, we study certain oerators of the C -algebras M induced by the -algebra M of µ -measurable functions over a fixed -adic number field Q. In articular, we are interested in mutually-orthogonal roections {P } Z of M induced by generating elements of M. We show that such roections generate a well-defined embedded sub-semigrou S of M. From such a semigrou, the corresonding semigrou C -algebra S is constructed and studied. From the isomorhism theorem of S, we define Banach-sace oerators c and a acting on S, and study fundamental roerties of these oerators. Then we define a new Banach-sace oerator l on S by l c + a, which gives a filterization, or filterings on S.

3 Semicircular elements induced by -adic number fields 667 By fixing a roection P generating S in M, construct a system of oerators {l n P } n1, and study free-distributional data of the elements in the family. We show the family induces free-semicircular law, under additional rocesses. By the semicircularity e.g., [3, 31] and [34], our semicircular elements have the same free-distributional data with any other semicircular elements in free robability theory under identically free-distributedness. So, more interesting results are from our so-called weighted-semicircular elements. We will see that the free-robabilistic information of such semicircular-like elements are determined by the number-theoretic data from M. Constructions of weighted-semicircular elements and semicircular elements, themselves, are the very main results of this aer. It shows that from a -adic analytic data, one can obtain semicircular-like roerty, and semicircularity OVERVIEW In Section, we briefly introduce basic concets for our roceeding works. In Sections 3, free-robabilistic models on M is considered in terms of the basis elements of the toology for Q. In articular, our free-robabilistic structures imly -adic-analytic information under -adic integration. See Theorems 3.7 and 3.8. In Sections 4, the Hilbert-sace reresentations of the free-robabilistic models of M are established, and the corresonding C -algebras M generated by M are constructed. Our Hilbert sace where M act are naturally constructed by defining inner roduct determined by -adic integration of Section 3. Then every element of M is acting on it as a multilication oerator. In Section 5, we build suitable free-robabilistic models of M, and study fundamental free-distributional data on M. See Theorems 5.3 and 5.3, and Corollary 5.4. In Sections 6, we fix certain roections {P } Z in M, and establish the corresonding semigrous S generated by the roections, and semigrou C -algebras S of S in M. The C -subalgebras S give certain filterizations on M. See Theorems 6. and 6.3. In Section 7, based on the constructions of S, we establish weighted-semicircular elements in a certain Banach -robability sace L C S. And then, corresonding semicircular elements are obtained from our weighted-semicircular elements. Of course, one can check our semicircular elements are following the semicircular law, meanwhile, our weighted-semicircular elements followed semicircular-like law determined by a fixed rime. See Theorems 7.5, 7.11, 7.1 and PRELIMINARIES In this section, we briefly mention about backgrounds of our works..1. FUNDAMENTALS Readers can check fundamental analytic-and-combinatorial free robability theory from e.g., [5 7, 31, 33] and [34]. Free robability is understood as the noncommutative and hence, covering commutative oerator-algebraic version of classical robability

4 668 Ilwoo Cho and Palle E.T. Jorgensen theory. The classical indeendence is relaced by the freeness. It has various alications not only in ure mathematics e.g., [, 4] and [3], but also in related mathematical-and-scientific toics e.g., [5, 6, 8, 9, 11, 1, 17] and [3]. In articular, we will use combinatorial free robabilistic aroach of Seicher e.g., [5 7] and [8]. Free moments and free cumulants of oerators will be comuted without introducing in detail... -ADIC NUMBER FIELDS Q Let be a fixed rime in P, and Q, the corresonding -adic number field. Then this set Q is a well-defined ring, which is regarded as a Banach sace equied with the -norm, defined by x k r 1 k, whenever x k r in Q, for some k N 0 N {0}. For instance, , and and , , whenever q P \ {, 3}. q As a toological sace, Q has its basis elements transforming the unit disk Z {x Q : x 1} of Q, consisting of all -adic integers, i.e., Q k Z k Z,.1 where k Z { k y : y Z } Q. Throughout this aer, we write U k k Z in Q, for all k Z, with U 0 Z, for convenience.

5 Semicircular elements induced by -adic number fields 669 Also, the -adic number field Q is a measure sace, Q, σq, µ, equied with the additive left-and-right invariant Haar measure µ on the σ-algebra σ Q. Note that µ U k 1 k µ x + U k, for all k Z,. for all x Q, satisfying, µ U 0 µ Z 1. Remark that, by the very definition, one has the following chain relation, U U 1 U 0 U 1 U,.3 in Q. In conclusion, a -adic number Q is a Banach toological, measure-theoretic ring, satisfying.1,. and.3. For more details, see [9]. Whenever we fix an integer k Z, one can determine so-called the k-th boundary k of U k in Q ; k U k \ U k+1,.4 by.3, where A \ B A B c, for all sets A and B, where B c is the comlement of B in a universal set containing A and B. Remark that, by. and.4, one can get that µ k µ U k µ U k+1 1 k 1 k+1 µ x + k,.5 for all k Z, for all x Q. Also, remark that, by.4, one obtains the artition of Q, Q k Z U k,.6 where means the disoint union. By understanding Q as a measure sace, we have the ure-algebraic -algebra M consisting of all µ -measurable functions over the comlex numbers C, i.e., M {f : Q C : f is µ -measurable},.7 equied with the usual functional addition, and the usual functional multilications. By definition, if f M, it is exressed by f S σq t S χ S, with t S C,

6 670 Ilwoo Cho and Palle E.T. Jorgensen where χ S are the usual characteristic functions for S σq, having its adoint, f t S χ S, S σq where z are the conugates of z, for all z C, and is a finite sum. Indeed, the vector sace M of.7 forms a well-defined -algebra over C. For all f M, one can have the -adic integral of f by fdµ t S µ S. Q S σq Note that, by.6, if S σq, then there exists a subset Λ S of Z, such that Λ S { Z : S },.8 satisfying χ S dµ Q Q Λ S χ S dµ Λ S µ S Λ S µ Λ S , by.4,.6 and.5, i.e., Q χ S dµ Λ S ,.9 for all S σq, where Λ S is subset.8 of Z. More recisely, one can get the following roosition. Proosition.1. Let S σq, and let χ S M. Then there exist r R, such that 0 r 1 in R, for all Λ S,.10 and Q χ S dµ Λ S r Proof. By.9, whenever S σ Q, there exists a subset Λ S of Z, in the sense of.8, such that 1 χ S dµ 1 +1, because for all Λ S. Q Λ S µ S µ ,

7 Semicircular elements induced by -adic number fields 671 and So, for each Λ S, there exists a unique r R, such that Therefore, one can get that 0 r 1, 1 µ S r Q χ S dµ By.10, one obtains that if f Λ S r S σq t S χ S M, then fdµ Q S σq t S Λ S r S ,.11 where r S are in the sense of.10, for all Λ S, for all S σq. The formula.11, obtained from.10, rovides a universal technique to establish -adic calculus. 3. FREE PROBABILITY ON M Throughout this section, fix a rime P, and Q, the corresonding -adic number field, and let M be the -algebra consisting of all µ -measurable functions on Q. In this section, let s establish a suitable free-robabilistic model on the -algebra M. Remark that free robability rovides a universal tool to study free distributions on noncommutative algebras, and hence, it covers the cases where given algebras are commutative. As in Section., let U k be the basis elements of of the toology for Q, with their boundaries k U k \ U k+1. Define a linear functional ϕ : M C by ϕ f U k k Z, for all k Z, 3.1 Q fdµ, for all f M. 3.

8 67 Ilwoo Cho and Palle E.T. Jorgensen Then, by 3., one naturally obtains that 1 ϕ χu, and ϕ 1 χ 1 +1, for all Z. Moreover, by the commutativity on M, ϕ f 1 f ϕ f f 1, for all f 1, f M, and hence, this linear functional ϕ of 3. is a trace on M. Definition 3.1. The free robability sace M, ϕ is called the -adic free robability sace, for P, where ϕ is the linear functional 3. on M. Let U k be in the sense of 3.1 in Q, and χ Uk M, for all k Z. Then χ Uk1 χ Uk χ Uk1 U k χ Umax{k1,k }, by.3, where max{k 1, k } means the maximum in {k 1, k }. Say k 1 k in Z. Then U k1 U k in Q, by.3. Therefore, U k1 U k U k in Q. So, if k 1 k in Z, then χ Uk1 χ Uk χ Uk1 U k χ Uk in M. Lemma 3.. Let U k be in the sense of 3.1 in Q. Then and hence, χ Uk1 χ Uk χ Umax{k1,k } in M, 3.3 ϕ χuk1 χ Uk 1 max{k1,k}. Proof. By the discussion in the very above aragrah, by.3, for k 1, k Z. So, U k1 U k U max{k1,k } in Q, χ Uk1 χ Uk χ Umax{k1,k }, and hence, 1 ϕ χuk1 χ Uk µ Umax{k1,k }. max{k1,k} Inductive to 3.3, we obtain the following result. Proosition 3.3. Let 1,..., N Z N for N N. Then and hence, N χ Ul χ in M Umax{1,..., N }, 3.4 l1 N 1 ϕ χ Ul. max{1,..., N } l1 Proof. The roof of 3.4 is done by induction on 3.3.

9 Semicircular elements induced by -adic number fields 673 Now, let k be the k-th boundary U k \ U k+1 of U k in Q, for all k Z. Then, for k 1, k Z, one obtains that where δ means the Kronecker delta, and hence, χ k1 χ k χ k1 k δ k1,k χ k1, ϕ χ k1 χ k δk1,k ϕ χ k1 δk1,k 1 k1 k1+1. So, we obtain the following comutations. Proosition 3.4. Let 1,..., N Z N, for N N. Then N χ l δ 1,..., N χ 1 in M, 3.6 l1 and hence, where N ϕ χ l δ 1,..., N l1 δ 1,..., N N , δ l, l+1 δ N, 1. l1 Proof. The roof of 3.6 is done by 3.5. Thus, one can get that, for any S σ Q, ϕ χ S ϕ Λ S χ S where Λ S is in the sense of.8 Λ S ϕ Λ S r χs µ S Λ S , 3.7 by.10, where 0 r 1 are in the sense of.10, for all Λ S.

10 674 Ilwoo Cho and Palle E.T. Jorgensen Also, if S 1, S σ Q, then χ S1 χ S k Λ S1 χ S1 k k, Λ S1 Λ S Λ S χ S χs1 k χ S k, Λ S1 Λ S δ k, χ S1 S Λ S1,S χ S1 S, 3.8 where because k δ k,, for all k, Z. Thus, there exist w R, such that Λ S1,S Λ S1 Λ S, 0 w 1, for all Λ S1,S, 3.9 where Λ S1,S is in the sense of 3.8, and ϕ χ S1 χ S Λ S1,S w , by 3.8 and.10, for all S 1, S σ Q. Lemma 3.5. Let S l σ Q, and χ Sl M, ϕ, for l 1,, and let Λ S1,S Λ S1 Λ S, where Λ Sl are in the sense of.8, for l 1,. Then there exist r R, such that and 0 r 1 in R, for all Λ S1,S, 3.10 ϕ χ S1 χ S Λ S1,S r Proof. The roof of 3.10 is done by 3.8 and Remark 3.6. In fact, the above lemma can be re-formulated as follows. If S 1 and S are given as above, then 1 r 1 ϕ χ S1 χ S if Λ +1 S1,S, Λ S1,S 3.11 µ 0 if Λ S1,S. In the following text, if we mention 3.10, then it means 3.11, recisely.

11 Semicircular elements induced by -adic number fields 675 By the above lemma, we obtain the following general result under induction. Theorem 3.7. Let S l σq, and let χ Sl M, ϕ, for l 1,..., N, for N N. Let N Λ S1,...,S N Λ Sl in Z, where Λ Sl are in the sense of.8, for l 1,..., N. Then there exist r R, such that 0 r 1 in R, for all Λ S1,...,S N, 3.1 l1 and N ϕ χ Sl l1 Λ S1,...,S N r Proof. The roof of 3.1 is done by induction on 3.10 or Similar to 3.10 and 3.11, the above formula 3.1 is refined by N 1 r 1 ϕ χ Sl if Λ +1 S1,...,S N, Λ S1,...,S N l1 0 if Λ S1,...,S N By 3.1 or 3.13, we obtain that if f t S χ S M, ϕ, with t S C, S σq then ϕ f t S ϕ χ S S σq S σq t S Λ S r S , where r S are in the sense of 3.1, for all Λ S. Therefore, one can get the following result. Theorem 3.8. Let f l t Sl χ Sl be elements of our -adic free robability S l σq sace M, ϕ, with t Sl C, for l 1,..., N, for N N. Then N ϕ f l l1 S 1,...,S N σq N N t Sl l1 r S1,...,SN Λ S1,...,S N , 3.14 where r S1,...,S N are in the sense of 3.1, for all Λ S1,...,S N whenever it is nonemty.

12 676 Ilwoo Cho and Palle E.T. Jorgensen Proof. Suose f 1,..., f N be given as above in M, ϕ. Then T in M. Observe that ϕ T N f l l1 S 1,...,S N σq N S 1,...,S N σq N S 1,...,S N σq N N N t Sl χ Sl l1 N N t Sl ϕ χ Sl l1 N t Sl l1 l1 l1 r S1,...,SN Λ S1,...,S N by 3.1 or 3.13, where r S1,...,S N are in the sense of , The above oint free-moment formula 3.14 rovides a universal tool to comute the free distributions of free random variables in our -adic free robability sace M, ϕ. 4. REPRESENTATIONS OF M, ϕ Fix a rime P. Let M, ϕ be the -adic free robability sace. Now, we construct a reresentation of the -algebra M. By understanding Q as a measure sace, construct the L -sace, H def L Q, σq, µ L Q, 4.1 over C, consisting of all square-integrable µ -measurable functions on Q. Then this L -sace is a well-defined Hilbert sace equied with its inner roduct,, f 1, f def Naturally, H is the -norm comletion, where f def where, is the inner roduct 4. on H. Q f 1 f dµ, for all f 1, f H. 4. f, f, for all f H, Definition 4.1. We call the Hilbert sace H L Q of 4.1, the -adic Hilbert sace.

13 Semicircular elements induced by -adic number fields 677 By the very construction 4.1 of the -adic Hilbert sace H, our -algebra M acts on H, via an algebra-action α, αf h fh, for all h H, 4.3 for all f M. i.e., the morhism α of 4.3 is an action of M acting on the Hilbert sace H. i.e., for any f M, the image αf is an oerator on H contained in the oerator algebra BH of all bounded linear oerators on H. Denote αf by α f, for all f M, where α is in the sense of 4.3. Also, for convenience, denote α χs simly by α S, for all S σ Q. For instance, α Uk α χuk αχ Uk, and α k α χ k α χ k, for all k Z, where U k are in the sense of 3.1, and k are the corresonding boundaries of U k in Q, for all k Z. By 4.3, the linear morhism α is a well-determined -algebra action of M acting on H. Indeed, α t1f 1+t f h t 1 f 1 + t f h t 1 f 1 h + t f h t 1 α f1 h + t α f h, for all h H, for all f 1, f M, and t 1, t C; α f1f h f 1 f h f 1 f h f 1 α f h α f1 α f h, for all h H, for all f 1, f M ; and α f h 1, h fh 1, h fh 1 h dµ Q h 1 fh dµ h 1 h f dµ Q Q h 1 f h dµ h 1, α f h, Q for all f M, and for all h 1, h H, which imlies that α f α f, for all f. Proosition 4.. The linear morhism α of 4.3 is a well-defined -algebra action of M acting on H. Equivalently, the air H, α is a well-determined Hilbert-sace reresentation of M.

14 678 Ilwoo Cho and Palle E.T. Jorgensen Proof. By the discussions in the very above aragrahs, the linear morhism α satisfies that α f1f α f1 α f on H, and α f 1 α f 1 on H, for all f 1, f M. i.e., α is a -homomorhism from M into BH. Therefore, the air H, α is a Hilbert-sace reresentation of M. In the above roosition, we showed that the air H, α of the -adic Hilbert sace H and the action α of 4.3 is a Hilbert-sace reresentation of M. Definition 4.3. The Hilbert-sace reresentation H, α is said to be the -adic Hilbert-sace reresentation of M. By the definition 4.3 of the action α of M, it generates multilication oerators α f on the -adic Hilbert sace H of 4.1 with their symbols f, for all f M, ϕ. Definition 4.4. Let M be the oerator-norm closure of M in the oerator algebra BH, i.e., M def α M C [α f : f M ] in BH, 4.4 where X mean the oerator-norm closures of subsets X of BH. Then this C -algebra M is called the -adic C -algebra of M, ϕ. 5. FREE PROBABILITY ON M Throughout this section, we fix a rime P. Let M, ϕ be the corresonding -adic free robability sace, and H, α, the -adic reresentation of M, and let M be the corresonding -adic C -algebra of M, ϕ. In this section, we consider suitable free-robabilistic models on M. In articular, we are interested in a system {ϕ } Z of linear functionals on M, determined by the -th boundaries { } Z of Q. Define a linear functional ϕ : M C by a linear morhism, ϕ def a α a χ, χ, for all a M, 5.1 for all Z, where, is the inner roduct 4. on the -adic Hilbert sace H of 4.1. First, remark that, if a M, then a S σq t S χ S in M, with t S C, where is a finite or an infinite limit of finite sums, under C -toology of M. Thus, the above definition 5.1 is well-defined, and every linear functional ϕ are bounded on M, for all Z.

15 Semicircular elements induced by -adic number fields 679 Definition 5.1. Let Z, and let ϕ be the linear functional 5.1 on the -adic C -algebra M. Then the C -robability sace M, ϕ is said to be the -th -adic C -robability sace. So, one can get the system {M, ϕ : Z} of C -robability saces for a fixed C -algebra M. Now, fix Z, and take the corresonding -th C -robability sace M, ϕ for S σ Q, and an element χ S M, one has that ϕ χ S α S χ, χ χ S, χ χ S χ dµ χ S χ dµ Q Q Q χ S dµ µ S r S , for some 0 r S 1 in R, i.e., there exists 0 r S 1, such that ϕ 1 χ S µ S r S 1 +1, 5. for any S σ Q. Proosition 5.. Let S σ Q, and α S α χs M, ϕ, for a fixed Z. Then there exists r S R, such that 0 r S 1 in R, 5.3 and 1 ϕ αs n r S 1 +1, for all n N. Proof. Remark that the element α S is a roection in M in the sense that since and So, α S α S α S in M, α S αχ S αχ S αχ S α S, α S αχ S αχ S α S in M. α n S α S, for all n N. Thus, for any n N, we have ϕ αn S ϕ 1 α S r S 1 +1, for some 0 r S 1 in R, by 5..

16 680 Ilwoo Cho and Palle E.T. Jorgensen The free-moment formula 5.3 characterizes the free distributions of χ S in the -th C -robability sace M, ϕ, for all S σ Q, for all Z. More recisely, we obtain the following theorem. Theorem 5.3. Let S l σ Q, and α Sl α χ Sl M, ϕ, for a fixed Z, for l 1,..., N, for N N. Then there exists r S1,...,S N R, such that and for all n N. 0 r S1,...,S N 1 in R, 5.4 N n ϕ α Sl r S1,...,S N l , Proof. Let S 1,..., S N be µ -measurable subsets of Q, for N N, and let Then, one has that satisfying S α S N S l σ Q. l1 N α Sl in M, l1 α S α S α S in M. Indeed, if S, then the above roection-roerty holds in M, and if S, then χ S 0 M, the zero element of M, which is a roection, too. So, Therefore, α n S α S, for all n N. ϕ αn S ϕ 1 α S r S 1 +1, for some 0 r S 1 in R, by 5.3, for all n N. The above oint free-moment formula 5.4 characterizes the free-distributions of finitely many roections α S1,..., α SN in the -th C -robability sace M, ϕ, for Z. As corollaries of 5.4, we obtain the following results. Corollary 5.4. Let U k be in the sense of 3.1, and k, the k-th boundaries of U k in Q, for all k Z. Then ϕ α n Uk { if k, 0 otherwise, 5.5

17 Semicircular elements induced by -adic number fields 681 and for all n N, for k Z. ϕ α n 1 k δ,k 1 +1 Proof. Observe first that, for any k Z, ϕ α n k ϕ 1 α k µ k δ,k µ δ,k 1 +1, for all n N, because k δ,k in Q. Consider now that, for an arbitrarily given k Z, one has U k { l if k, if k >. l k in Z 5.6 So, ϕ α n Uk ϕ α U k µ U k { µ 1 1 if k, +1 µ 0 otherwise, by 5.6, for all n N. Now, let S l σ Q, and a l χ Sl M, ϕ, for Z, for l 1,. Then, for i 1,..., i n {1, } n, for n N, we have the oint free cumulant in terms of ϕ, kn, a i1,..., a in a il µπ, 1 n π NCi 1,...,i n ϕ i l V by the Möbius inversion of [6] π NCi 1,...,i n π NCi 1,...,i n µ S il µπ, 1 n i l V 1 r π,v 1 +1 µπ, 1 n, 5.7 by 5.4, where 0 r π,v 1 are in the sense of 5.4. Therefore, by the free cumulant formula 5.7, we obtain the following freeness condition on the -th C -robability sace M, ϕ.

18 68 Ilwoo Cho and Palle E.T. Jorgensen Theorem 5.5. Let S l σ Q, and a l α Sl M, ϕ, for Z, for l 1,. If / Λ Sl in Z, for all l 1,, where Λ Sl are in the sense of.8, for l 1,, then two free random variables a 1 and a are free in M, ϕ. i.e., Proof. Assume that / Λ Sl, l 1 or l α S1 and α S are free in M, ϕ. 5.8 / Λ S1, and / Λ S in Z, where Λ Sl {k Z : S l k }, for l 1,. Then, by 5.3 or, by 5.4, α n Sl 0, for all l 1,. ϕ Moreover, by the above assumtion, we have So, if / Λ S1,S Λ S1 Λ S. i 1,..., i N {1, } N is mixed in {1, }, for N N \ {1}, then N α Sit 0, ϕ t1 by 5.4. It shows that, the self-adoint elements χ S1 and χ S have not only vanishing free moments, but also vanishing mixed free moments. So, by 5.7, we obtain that k, n αsi1,..., α S in π NCi 1,...,i n π NCi 1,...,i n r π,v µπ, 1 n 0, µπ, 1 n for all mixed n-tules i 1,..., i n {1, } n, for all n N \ {1}. It guarantees that two random variables χ S1 and χ S are free in M, ϕ. 6. PROJECTIONS {P } Z AND THE SEMIGROUP S IN M Let s fix a rime P. In Section 5, we considered the free robability on the -th C -robability sace M, ϕ, where M is the -adic C -algebra and ϕ is the linear

19 Semicircular elements induced by -adic number fields 683 functional 5.1, for Z. In articular, we observed fundamental free distributions of self-adoint generating elements of M. In this section, we concentrate on a system of roections, {P α α χ : Z} 6.1 in M. Remark that these roections are mutually orthogonal from each other in the sense that P 1 P δ 1, P 1 in M, for all 1, Z, because χ 1 χ δ 1, χ 1 in M, for all 1, Z. Let P k α k be a roection 6.1 in the -adic C -algebra M, for k Z. Then, as we have seen in 5.5, ϕ 1 P k δ,k µ δ,k 1 +1, 6. for all, k Z. Now, from the system {P } Z of mutually orthogonal roections, let s construct the multilicative semigrou S in M, S def {P } Z in M, 6.3 under the inherited oerator multilication on M. Then this algebraic structure S of 6.3 is a well-determined commutative semigrou in M, which is not a grou. Indeed, the inherited oerator multilication is associative on S, but it has no identity in S. We call S, the roection semigrou of the system {P } Z of 6.1. Definition 6.1. Define the C -subalgebra S of the -adic C -algebra M by the C -algebra generated by the roection semigrou S of 6.3. We call S the roection-semigrou C -subalgebra of M. Our roection-semigrou C -subalgebra S of M has the following structure theorem. Theorem 6.. Let S be the roection-semigrou C -subalgebra of M. Then -iso S C P k -iso C Z in M, 6.4 k Z where -iso means being -isomorhic, and where means toological direct roduct of C -algebras. Proof. Let S be the roection semigrou of the system {P } Z of mutually orthogonal roections P α, for all Z, and let S C S be the corresonding

20 684 Ilwoo Cho and Palle E.T. Jorgensen roection-semigrou C -subalgebra of the -adic C -algebra M. Then, by the very definition 6.3, S C S C {P } Z C [{P } Z ], where X are the C -norm closures of the subsets X of M Z C P by the mutually orthogonality of {P } Z, and the roection-roerty of all P s, where means ure-algebraic direct sum of algebras Z C P C Z. The above structure theorem 6.4 of the roection-semigrou C -subalgebra S shows that the embedded structure S rovides a certain filterization, or diagonalization in the -adic C -algebra M. Let M, ϕ be the -th C -robability saces of Section 5, for all Z, and let S be our roection-semigrou C -subalgebra of M. Then, naturally, one obtains the system of C -robability saces, { S, ϕ } : Z, 6.5 by restricting the linear functionals ϕ on M to those on S, for all Z. And free-distributional data on S is comletely determined by the following result. Theorem 6.3. Let S, ϕ be a C -robability sace in 6.5, for any Z. For an element T k Zt k P k S, with t k C, we have for Z. ϕ T n t n , for all n N, 6.6 Proof. Let T k Zt k P k S, with t k C. Then, by the structure theorem 6.4, T is equivalent to T equi t k P k in C Z S, k Z satisfying that in S, for all n N. T n equi n t k P k t n kpk n k Z k Z equi k Zt n kp k,

21 Semicircular elements induced by -adic number fields 685 Therefore, for any fixed Z, one can get that ϕ T n ϕ t n kp k ϕ by 6., for all n N. k Z t n P t n , 7. WEIGHTED-SEMICIRCULARITY INDUCED BY M Let be a fixed rime in P, and let M be the -adic C -algebra induced by the -algebra M, under the -adic reresentation H, α of M. Let S be the roection-semigrou C -subalgebra of M, satisfying the structure theorem 6.4; -iso S C P, Z where P α are the mutually-orthogonal roections of 6.1, for all Z. Also, let { S, ϕ } be the system 6.5 of -th k Z C -robability saces of S. Recall again that ϕ 1 α k δ,k 1 k+1, for all, k Z. 7.1 Recall now that an arithmetic function φ : N C is a Euler totient function, if φn def {k N 1 k n and gcdk, n 1}, 7. for all n N, where gcd means the greatest common divisor, and where x mean the cardinalities of sets X. It is a well-determined multilicative arithmetic function in the sense that whenever gcdn 1, n 1, because φn n φn 1 n φn 1 φn, P, n 1 1, for all n N. 7.3 By 7. and 7.3, the Euler totient function φ satisfies φ 1 1 1, for all P. So, our free-moment comutation 7.1 can be re-stated as follows: ϕ α 1 k δ,k δ,k φ, for all, k Z.

22 686 Ilwoo Cho and Palle E.T. Jorgensen Define now a morhism τ τ on S by a linear functional satisfying def 1 φ ϕ, for all Z. 7.5 Then the airs S, τ are well-determined C -robability saces, satisfying τ P k δ,k φ ϕ P δ,k 1 φ 1 +1 δ,k δ 7.6,k +1, for all, k Z. Definition 7.1. We call the C -robability saces S, τ, the -th roection -semigrou C -robability saces, for all Z. Free distributional data on the -th roection robability saces S, τ is characterized as follows. Proosition 7.. Let S, τ be the -th roection robability sace for Z, where τ is the linear functional 7.5, for a fixed Z. Then for all n N. τ P k n δ,k, for all, k Z, Proof. The free distribution 7.7 of a roection P k is obtained by 7.6, for all k, Z. Now, we have all ingrediants to construct semicircular-like roerty, and semicircularity induced by M WEIGHTED-SEMICIRCULARITY AND SEMICIRCULARITY Let A, ϕ be an arbitrary toological, or ure-algebraic -robability sace of a -algebra A, and a linear functional ϕ on A. Remember that -algebra is an algebra equied with the adoint on A. An element a of a -algebra A is said to be self-adoint, if a a in A, where a is the adoint of a. Definition 7.3. Let a be a free random variable in a -robability sace A, ϕ, and let k n... be the free cumulant on A in terms of ϕ e.g., see [6]. The given free random variable a is said to be semicircular in A, ϕ, if i a is self-adoint, and ii a satisfies for all n N. k n a, a,..., a }{{} n-times { 1 if n, 0 otherwise, 7.8

23 Semicircular elements induced by -adic number fields 687 Formore about free moments and free cumulants, see [6] and cited aers therein. By the Möbius inversion of [6], one can get the equivalent definition of the semicircularity 7.8 as follows: A free random variable a is semicircular in A, ϕ, if and only if i a is self-adoint, and ii all odd free-moments of a vanish, equivalently, and iii all even free-moments of a satisfy where c n are the n-th Catalan number, ϕ a n 1 0, for all n N, 7.9 ϕa n c n, for all n N, 7.10 c n 1 n n! n + 1 n n!n + 1!, for all n N see [6]. So, the free-moment formulas 7.9 and 7.10 characterize the semicircularity 7.8 under self-adointness. Motivated by the definition 7.8 of semicircularity, we define the following semicircular-like roerty, called the weighted-semicircularity as follows. Definition 7.4. A free random variable a A, ϕ is said to be weighted-semicircular in A, ϕ with weight t 0 C \ {0} in short, t 0 -semicircular, if a is self-adoint in A, and { k a, a t 0 if n, k n a,..., a otherwise, for all n N. Of course, if t 0 1, then 1-semicircularity of 7.11 is the same as the semicircularity 7.8. By the Möbius inversion of [6], if a free random variable a is t 0 -semicircular in A, ϕ, then ϕa m 1 0, and ϕa m π NCm π NC m k V a,..., a }{{} V -times k V a,..., a }{{} V -times

24 688 Ilwoo Cho and Palle E.T. Jorgensen where NC m {θ NCm : V π, V } by 7.11 π NC m π NC m k a, a t π 0 π NC m t 0 where π means the number of blocks of the artition π π NC m t m 0 since all noncrossing artitions π in NC m has m t m 0 1 t m 0 c m, π NC m -many blocks where c m means the m-th Catalan number, for all m N. So, by the definition 7.11, one obtains that: if a ist 0 -semicircular in A, ϕ, then it is self-adoint, and there exists t 0 C, such that for all n N. ϕa n { t n 0 c n if n is even, 0 if n is odd, 7.1 Theorem 7.5. Let a A, ϕ be a self-adoint non-zero free random variable. Then a is t 0 -semicircular in A, ϕ for some t 0 C, if and only if for all n N. ϕa n { t n 0 c n if n is even, 0 if n is odd, 7.13 Proof. The roof of the free-moment characterization 7.13 of the t 0 -semicircularity is done by 7.11 and 7.1, via the Möbius inversion of [6]. If a is t 0 -semicircular in A, ϕ, then the free-moment formula 7.13 holds by 7.1.

25 Semicircular elements induced by -adic number fields 689 If a self-adoint free random variable a satisfies the free-moment comutation 7.13 in A, ϕ, then k n a,..., a ϕ a V µπ, 1 n, 7.14 π NCn by the Möbius inversion, for all n N. Since all odd free-moments of a vanish by 7.13, whenever a block V of any noncrossing artition contains odd-many elements, then ϕ a V 0, and hence, if a artition π contains a block V 0 with odd-many elements, then ϕ a V ϕ a V0 V a 0.,V V 0 ϕ Notice now that all noncrossing artitions π of NCn contains at least one odd block in π, whenever n is odd in N. So, by 7.14, one obtains that k n a,..., a 0, whenever n is odd in N Now, let k N, and observe k k a,..., a π NCk π NC ek ϕ a V µπ, 1 k ϕ a V µπ, 1 k, 7.16 where NC e k {θ NCk : B θ B is even}. It is not difficult to check that the sub-lattice NC e k of the lattice NCk is equivalent to NCk, for all k N. Thus, the formula 7.16 goes to k k a,..., a ϕ a B µθ, 1 k θ NCk θ NCk t k 0 θ NCk B θ t k 0 θ NCk B θ t B 0 c B µθ, 1 k c B µθ, 1 k B θ c B µθ, 1 k B θ { t 0 if k 1, 0 otherwise, 7.17

26 690 Ilwoo Cho and Palle E.T. Jorgensen by 7.9 and 7.10 which are equivalent to the semicircularity 7.8. Indeed, c B µθ, 1 k 0, θ NCk B θ whenever k 1, because of the semicircularity. Therefore, if the free-moment comutation 7.13 holds, then a is t 0 -semicircular in A, ϕ, by 7.15 and Thus, by and, a self-adoint elemnet a is t 0 -semicircular in A, ϕ, if and only if it satisfies { ϕa n t n 0 c n if n is even, 0 if n is odd, for all n N. So, by the above free-momental characterization 7.13, our t 0 -semicircularity 7.11 ce be re-stated. 7.. TENSOR PRODUCT BANACH -ALGEBRA LS Let M be the -adic C -algebra containing its -adic roection-semigrou C -subalgebra S, and let τ k be linear functionals 7.5 on S, for all k Z. Throughout this section, we fix k in Z, and the corresonding k-th C -robability sace S, τ k. The formula 7.6 says that Recall that τ k P δ k,, for all Z k+1 -iso S C P -iso C Z in M, 7.19 Z by the structure theorem 6.4. By 7.19, one can define a Banach-sace oerators c and a acting on S by linear transformations satisfying c P P +1, and a P P 1, 7.0 acting on S, for all Z. By the very definition 7.0, these linear oerators c and a are bounded or continuous under the oerator-norm of S, inherited from the C -norm on the -adic C -algebra M. So, they are well-defined Banach-sace oerators acting on S. Definition 7.6. The Banach-sace oerators c and a on S in the sense of 7.0 are called the -creation, resectively, the -annihilation on S. Define a new Banach-sace oerator l on S by l c + a on S. 7.1 We call this oerator l of 7.1, the -radial oerator on S.

27 Semicircular elements induced by -adic number fields 691 Let l be the radial oerator c + a of 7.1 on S, where c and a are the creation, resectively, the annihilation of 7.0. Construct a Banach algebra L by L C[l ] in BS, 7. where BS means the toological oerator sace, consisting of all bounded equivalently, continuous linear transformations on S, equied with its oerator-norm, defined by T su{ T x S : x S, x S 1}, where x S su{ xh : h H, h 1}, giving the C -norm toology on M and hence, on S, where means the Hilbert-sace norm on the -adic Hilbert sace H. By the definition 7. of the Banach algebra L, every element x of L is exressed by x t k l, k with t k C, k0 in L, with identity: l 0 1 L, the identity oerator on L, satisfying that: 1 L P P, for all Z. Now, define the adoint on L by x t k l k def k0 t k l. k k0 Then the Banach algebra L forms a Banach -algebra. Definition 7.7. Let L be the Banach -algebra 7. in the oerator sace BS. We call it the -adic radial Banach- -algebra on S. Let L be the radial algebra on S. Construct now the tensor roduct Banach -algebra LS by LS L C S, 7.3 where C means the tensor roduct of Banach -algebras. Consider elements l k P of the tensor roduct Banach -algebra LS of 7.3, for k N 0 N {0}, and Z. By the very definition 7.3, such elements l k P generate LS. We concentrate on such generating oerators l k P of LS, later. Define a morhism E : LS S by a linear transformation satisfying that: E l k P +1 k+1 [ k ] + 1 lk P, 7.4

28 69 Ilwoo Cho and Palle E.T. Jorgensen for all k N 0, Z, where [ k ] means the minimal integer greater than or equal to k, for instance, [ 3 [ 4. ] ] By 7.19, if T N n1 tn l n k s n P LS with t n, s n C, for N N, then and hence, T N n1 t n s n l n k P, E T N n1 t n s n E l n k N +1 n k +1 P t n s n [ n k ] + 1 ln k P, 7.5 n1 in S, by 7.4. Note that, if Q l l k l P l LS, for 4k l N 0, l Z, for l 1,, then E Q 1 Q E l k 1 l k P 1 P E l k 1+k δ 1, P 1 δ 1, E l k 1+k P 1 +1 k 1+k +1 δ 1, ] l k1+k P [ k1+k 7.6 Proosition 7.8. Let Q l l k l P l LS, for k l N 0, l Z, for l 1,. If E is the morhism in the sense of 7.4, then +1 k 1+k +1 E Q 1 Q δ 1, ] l k1+k P [ k1+k Proof. The roof of 7.7 is directly done by 7.6. Now, consider how our radial oerator l c + a acts on S. First, observe that if c and a are the creation, resectively, the annihilation on S, then c a P c a P c P 1 P, 7.8 and for all Z. a c P a c P a P +1 P,

29 Semicircular elements induced by -adic number fields 693 Lemma 7.9. Let c, a be the creation, resectively, the annihilation on S. Then where 1 S is the identity oerator on S. c a 1 S a c, 7.9 Proof. Since the C -algebra S is -isomorhic to C P by 7.19, the formula 7.9 holds by 7.8, under the linearity of c and a on Z S. The formula 7.9 shows that the Banach-sace oerators c and a are commutative on S. Therefore, one can get that n l n c + a n n c k k a n k, 7.30 with identities: k0 c 0 1 S a 0, for all n N, where n n! k k!n k!, for all n N, k N 0. Consider now the formulas 7.9 and 7.30 together. Assume first that n m 1 is odd, for m N. Then m 1 l n l m 1 m 1 c k k a m 1 k, k0 k0 by Thus, we can realize that l m 1 has vanishing 1 S -terms by 7.9, for all m N. i.e., l m 1 does not contain 1 S -terms, for m N Now, suose that n m is even, for m N. Then m l m m c k k a m k by 7.30 by 7.9 m c m m a m + k m {0,1,...,m} m 1S m + m k m {0,1,...,m} m m 1 S + [non-1 S -terms], m k m k c k a m k c k a m k

30 694 Ilwoo Cho and Palle E.T. Jorgensen i.e., l m contains the term m 1 m S, for m N. 7.3 Proosition Let l L be the radial oerator on S. Then for all m N. l m 1 does not contain a 1 S -term, and 7.33 l m m contains its 1 S -term, 1 m S, 7.34 Proof. The roofs of 7.33 and 7.34 are done by 7.31, resectively 7.3, with hel of 7.9 and Now, we have all ingredients to construct weighted-semicircular elements in LS WEIGHTED-SEMICIRCULAR ELEMENTS Q INDUCED BY HG Let LS be the tensor roduct Banach -algebra L C S in the sense of 7.3, and let E : LS S be the linear transformation 7.4. Throughout this section, Fix an element Q l P LS, 7.35 for some Z. Observe that Q n l P n l n P n l n P, 7.36 for all n N, because P are roections in S, for all Z. Consider that, if Q LS is in the sense of 7.35, for Z, then E Q n E l n P +1 n+1 [ n ] + 1 l n P 7.37 in S, by 7.36, for all n N. Now, for each fixed Z, define a linear functional τ 0 : on LS by τ 0 : τ E on LS, 7.38 where τ is in the sense of 7.5 on S. By the linearity of both τ and E, the morhism τ: 0 of 7.38 is a linear functional on LS. i.e., LS, τ: 0 forms a Banach -robability sace in the sense of [6] and [31]. By 7.37 and 7.38, one has that: if Q is in the sense of 7.35, then for all n N. τ 0 : Q n τ E Q n +1 n+1 [ n ] + 1 τ l n P, 7.39

31 Semicircular elements induced by -adic number fields 695 Theorem Let Q l P LS, τ: 0, for a fixed Z. Then Q is +1 -semicircular in LS, τ 0. Moreover, τ 0 Q n { +1 n c n if n is even, 0 if n is odd, 7.40 for all n N. Proof. Let us fix Z, and the corresonding Banach -robability sace LS, τ 0 :, and let Q be a generating oerator l P of LS. Then it is trivial that Q is self-adoint in LS. Indeed, one has Q l P l P l P Q. Observe now that τ 0 : Q m 1 +1 m 1+1 ] τ + 1 [ m 1 l m 1 P by m ] τ + 1 [ m 1 +1 m [ m 1 ] τ + 1 m 1 k0 m 1 k0 m 1 c k k a m 1 k P m 1 k c k a m 1 k P, 7.41 for all m N. Remark that the embedded arts c k a m 1 k P P m+1+k of the summands in 7.41 cannot be P -terms by 7.33, for all k 0, 1,..., m 1. Therefore, the formula 7.41 vanishes, for all m N.

32 696 Ilwoo Cho and Palle E.T. Jorgensen Now, consider that τ 0 : Q m +1 m+1 ] τ + 1 [ m l m P by 7.39 by m+1 m m+1 m + 1 τ τ m m c k k a m k P k0 m m P m + k k m {0,1,...,m} c k a m k P +1 m+1 m m+1 m m+1 m + 1 τ τ m m m P m P + [non-p -terms] m 1 m +1 1 m + 1 c m +1 m +1 m cm, where c m means the m-th Catalan number, i.e., τ 0 : +1 m+1 m + 1 m m m m τ P +1 m Q m cm +1 m cm +1 m, 7.4 for all m N. So, by the free-moment comutations 7.41 and 7.4, the self-adoint free random variable Q of our Banach -robability sace LS, τ 0 : is a weighted-semicircular element with its weight +1. i.e., there exists C, such that and τ 0 : Q m cm +1 m, τ 0 : Q m 1 0, for all m N. Therefore, the oerator Q is +1 -semicircular in LS, τ 0 :, by 7.11 and 7.13.

33 Semicircular elements induced by -adic number fields 697 One can construct the system LS { LS, τ: 0 } 7.43 Z of Banach -robability saces. Then, every Banach -robability sace LS, τ: 0 in the family LS of 7.43 has its +1 -semicircular element Q l P, for all Z. i.e., we have family WS { Q l P LS, τ, 0 } 7.44 of weighted-semicircular elements in the family LS of So, if k Z, then one obtains a corresonding k+1 -semicircular element Q k WS in a Banach -robability sace LS, τ:k 0 LS, for all P, satisfying { τ:k 0 Q n c n k +1 n if n is even, 0 if n is odd, equivalently, kn.,0 Q, Q,..., Q }{{} n-times Z { +1 if n, 0 otherwise, for all n N, where kn,,0... is the free cumulant with resect to the linear functional τ: 0 on LS in the sense of [6]. The following theorem re-rove an equivalent free-distributional data of 7.40, in terms of free cumulant. In fact, the following theorem must hold true by 7.40, and by the Möbius inversion of [6]. However, we rovide an indeendent roof of the theorem below. Theorem 7.1. Let Q l P LS, τ: 0 be given as in 7.35, for Z. Then { kn.,0 +1 if n, Q, Q,..., Q 7.45 }{{} 0 otherwise, n-times for all n N. Therefore, Q is a +1 -semicircular in LS, τ 0,, for Z. Proof. Let Q be a self-adoint free random variable 7.35 of the Banach -robability sace LS, τ, 0, for a fixed Z. Then τ: 0 Q m cm +1 m, and τ 0 Q m 1 0, 7.46 for all m N, by By the Möbius inversion of [6], one has for all n N. k,,0 n Q, Q,..., Q π NCn τ 0 : Q V µπ, 1 n 7.47

34 698 Ilwoo Cho and Palle E.T. Jorgensen Suose now that n in 7.47 is odd. Then every artition π in the lattice NCn consisting of all noncrossing artitions over {1,..., n} contains at least one odd block V 0 in π, i.e., there always exists at least one block V 0 of π has odd-many elements. Then and hence, for a artition π, τ 0 : Q V τ 0 : τ: 0 Q V0 Q V0 0,,V V 0 τ 0 Q V 0, whenever n is odd in N. Since π is arbitrary in NCn, the formula 7.47 vanishes, whenever n is odd, i.e., Consider now that, if n, then k,,0 n Q,..., Q 0, if n is odd k,,0 Q, Q τ: 0 Q τ 0 : Q τ: 0 Q τ: 0 l P τ 0 : Q τ 0 l P τ E l P τ P by 7.47, i.e., one obtains that +1 +1, k,,0 Q, Q Let m > 1 in N. Then k,,0 m Q,..., Q π NC em π NC em π NC em +1 m τ 0 : Q V µπ, 1 m c V +1 V µπ, 1 m +1 c m V µπ, 1 m π NC em c V µπ, 1 m, 7.50 by 7.47 and 7.49, where NC e m {θ NCm : B θ B is even}.

35 Semicircular elements induced by -adic number fields 699 By the semicircularity 7.9 and 7.10, we know that c V π NC em µπ, 1 m 0, 7.51 whenever m > 1 in N. Thus, the formula 7.50 vanishes whenever m > 1. Thus, we have if m > 1 in N, then k,0 m Q,..., Q 0, 7.5 by 7.50 and Therefore, by 7.48, 7.49 and 7.5, we obtain k,,0 n Q,..., Q { +1 if n, 0 otherwise, for all n N. It guarantees that the element Q is +1 -semicircular in LS, τ 0, for all Z, by 7.11 and SEMICIRCULAR ELEMENTS INDUCED BY M In this section, we consider semicircular elements in the Banach -robability saces LS, τ: 0, for Z. We will use the same notations used in Section 7.3. As we have seen in 7.40 and 7.45, the generating oerators Q l P of LS are +1 -semicircular in the Banach -robability sace LS, τ 0 :, for each Z. Throughout this section, let s fix Z, and the corresonding Banach -robability sace LS, τ 0 :. Define now an oerator Θ of LS by a free random variable, Θ 1 +1 Q LS, τ: Since +1 1 Q, the quantity Q, too, in R, and hence, the oerator Θ +1 is self-adoint in LS, by the self-adointness of Q.

36 700 Ilwoo Cho and Palle E.T. Jorgensen Observe now that, if Θ is a self-adoint oerator 7.53 in LS, then kn,,0 τ: 0 µπ, 1 n Θ, Θ,..., Θ }{{} n-times π NCn π NCn Θ V V 1 +1 τ: 0 n 1 +1 π NCn n 1 +1 π NCn τ 0 : τ 0 : Q V Q V Q V n 1 +1 kn,,0 Q, Q,..., Q, }{{} n-times µπ, 1 n µπ, 1 n µπ, 1 n 7.54 for all n N. Remark that the above formula 7.54 can be directly obtained by the bimodule-ma roerty of free cumulants. i.e., for all n N e.g., see [6]. kn,,0 X,..., X kn,, Q 1,..., +1 Q n kn,,0 Q,..., Q, Lemma Let Θ 1 Q +1 1 l +1 P be in the sense of 7.53 in our Banach -robability sace LS, τ: 0, for a fixed Z. Then for all n N. k,,0 n Θ,..., Θ 1 +1 n k,,0 n Q,..., Q, 7.55 Proof. The roof of the free-cumulant formula 7.55 is done by The above free-cumulant formula 7.55 shows that the free-distributional data of Θ are determined by those of Q. Theorem Let Θ be in the sense of 7.53 in LS, for Z. Then it is semicircular in LS, τ: 0, for Z. Proof. Observe that k,,0 n Θ,..., Θ n 1 +1 kn,,0 Q,..., Q

37 Semicircular elements induced by -adic number fields 701 by k,,0 +1 Q, Q if n, n otherwise, +1 for all n N, by the +1 -semicircularity 7.45, or 7.40 of Q, for Z. So, we obtain that kn,,0 1 Θ,..., Θ +1 1 if n, +1 0 otherwise, 7.56 for all n N. Therefore, by 7.8 and 7.56, the self-adoint oerators Θ in LS, τ: 0, for all Z. are semicircular REFERENCES [1] S. Albeverio, P.E.T. Jorgensen, A.M. Paolucci, Multiresolution wavelet analysis of integer scale Bessel functions, J. Math. Phys , , 4. [] S. Albeverio, P.E.T. Jorgensen, A.M. Paolucci, On fractional Brownian motion and wavelets, Comlex Anal. Oer. Theory , [3] D. Alay, P.E.T. Jorgensen, Sectral theory for Gaussian rocesses: reroducing kernels, boundaries, and L -wavelet generators with fractional scales, Numer. Funct. Anal. Otim , [4] D. Alay, P.E.T. Jorgensen, D. Kimsey, Moment roblems in an infinite number of variables, Infin. Dimens. Anal. Quantum Probab. Relat. To. Prob , [5] D. Alay, P.E.T. Jorgensen, D. Levanony, On the equivalence of robability saces, J. Theo. Prob. 016, to aear. [6] D. Alay, P.E.T. Jorgensen, G. Salomon, On free stochastic rocesses and their derivatives, Stochastic Process. Al , [7] I. Cho, Free distributional data of arithmetic functions and corresonding generating functions, Comlex Anal. Oer. Theory 8 014, [8] I. Cho, Dynamical systems on arithmetic functions determined by rims, Banach J. Math. Anal , [9] I. Cho, Free roduct C -algebras induced by -algebras over -adic number fields 016, submitted. [10] I. Cho, T. Gillesie, Free robability on the Hecke algebra, Comlex Anal. Oer. Theory 9 015,

38 70 Ilwoo Cho and Palle E.T. Jorgensen [11] I. Cho, P.E.T. Jorgensen, Krein-Sace Oerators Induced by Dirichlet Characters, Secial Issues: Contem. Math.: Commutative and Noncommutative Harmonic Analysis and Alications, Amer. Math. Soc. 014, [1] A. Connes, Noncommutative Geometry, Academic Press, San Diego, CA, [13] A. Connes, Hecke algebras, tye III-factors, and hase transitions with sontaneous symmetry breaking in number theory, Selecta Math. New Series , [14] A. Connes, Trace formula in noncommutative geometry and the zeroes of the Riemann zeta functions, arxiv:math/ [math.nt] [15] T. Gillesie, Suerosition of zeroes of automorhic L-functions and functoriality, University of Iowa, PhD Thesis 010. [16] T. Gillesie, Prime number theorems for Rankin-Selberg L-functions over number fields, Sci. China Math , [17] P.E.T. Jorgensen, Oerators and Reresentation Theory: Canonical Models for Algebras of Oerators Arising in Quantum Mechanics, nd ed., Dover Publications, 008. [18] P.E.T. Jorgensen, A.M. Paolucci, Wavelets in mathematical hysics: q-oscillators, J. Phys. A , [19] P.E.T. Jorgensen, A.M. Paolucci, States on the Cuntz algebras and -adic random walks, J. Aust. Math. Soc , [0] P.E.T. Jorgensen, A.M. Paolucci, q-frames and Bessel functions, Numer. Funct. Anal. Otim , [1] P.E.T. Jorgensen, A.M. Paolucci, Markov measures and extended zeta functions, J. Al. Math. Comut , [] F. Radulescu, Random matrices, amalgamated free roducts and subfactors of the C -algebra of a free grou of nonsingular index, Invent. Math , [3] F. Radulescu, Conditional exectations, traces, angles between saces and reresentations of the Hecke algebras, Lib. Math , [4] F. Radulescu, Free grou factors and Hecke oerators, notes taken by N. Ozawa, Proceedings of the 4th Conference in Oerator Theory, Theta Advanced Series in Math., Theta Foundation, 014. [5] R. Seicher, Multilicative functions on the lattice of non-crossing artitions and free convolution, Math. Ann , [6] R. Seicher, Combinatorial theory of the free roduct with amalgamation and oerator-valued free robability theory, Amer. Math. Soc. Mem [7] R. Seicher, A concetual roof of a basic result in the combinatorial aroach to freeness, Infin. Dimens. Anal. Quantum Probab. Relat. To , 13. [8] R. Seicher, P. Neu, Physical Alications of Freeness, XII-th International Congress of Math. Phy. ICMP 97, International Press, 1999, [9] V.S. Vladimirov, -adic quantum mechanics, Comm. Math. Phys ,

39 Semicircular elements induced by -adic number fields 703 [30] V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, -Adic Analysis and Mathematical Physics, Ser. Soviet & East Euroean Math., vol. 1, World Scientific, [31] D. Voiculescu, Free robability and the von Neumann algebras of free grous, Re. Math. Phys , [3] D. Voiculescu, Symmetries arising from free robability theory, Frontiers in Number Theory, Physics and Geometry 006, [33] D. Voiculescu, Asects of free analysis, Jn. J. Math , [34] D. Voiculescu, K. Dykemma, A. Nica, Free Random Variables, CRM Monograh Series, vol. 1, 199. Ilwoo Cho choilwoo@sau.edu St. Ambrose University Deartment of Mathematics and Statistics 41 Ambrose Hall, 518 W. Locust St. Davenort, Iowa, 5803, USA Palle E.T. Jorgensen alle-orgensen@uiowa.edu The University of Iowa Deartment of Mathematics 14 MacLean Hall Iowa City, IA , USA Received: October 10, 016. Acceted: December 4, 016.