BANACH -ALGEBRAS GENERATED BY SEMICIRCULAR ELEMENTS INDUCED BY CERTAIN ORTHOGONAL PROJECTIONS. Ilwoo Cho and Palle E.T. Jorgensen

Transcription

1 Opuscula Math. 38, o , Opuscula Mathematica BANACH -ALGEBRAS GENERATED BY SEMICIRCULAR ELEMENTS INDUCED BY CERTAIN ORTHOGONAL PROJECTIONS Ilwoo Cho ad Palle E.T. Jorgese Commuicated by P.A. Cojuhari Abstract. The mai purpose of this paper is to study structure theorems of Baach -algebras geerated by semicircular elemets. I particular, we are iterested i the cases where give semicircular elemets are iduced by orthogoal projectios i a C -probability space. Keywords: free probability, orthogoal projectios, weighted-semicircular elemets, semicircular elemets. Mathematics Subject Classificatio: 46L10, 46L54, 47L15, 47L30, 47L INTRODUCTION The mai purposes of this paper are i to itroduce a way to costruct weighted-semicircular, ad semicircular elemets geerated by orthogoal projectios i a certai Baach -probability space, ii to study operator-algebraic properties of Baach -algebras geerated by our weighted-semicircular or semicircular elemets, ad iii to apply the results of ii to operator-theoretic or spectral data of operators i such Baach -algebras via free probability theory. There are differet approaches to establish semicircular elemets e.g., [1,11,13] ad [17] i topological -probability spaces e.g., C -probability spaces, or W -probability spaces, or Baach -probability spaces, etc. Our costructio of semicircular elemets is differet from those already kow. It is highly motivated by weighted-semicircularity ad semicircularity i the sese of [3, 4] ad [5] MOTIVATION Semicircularity is extremely importat ot oly i mathematical classical, or operator-valued probability theory ad statistics, but also i mathematical physics quatum physics, quatum chaos theory, etc, e.g., see [8] ad [10]. I particular, c Wydawictwa AGH, Krakow

2 50 Ilwoo Cho ad Palle E.T. Jorgese i free probability theory, studyig semicircularity of free radom variables is oe of the mai braches e.g., [1, 9, 11 14, 16] ad [17]. Idepedetly, p-adic ad Adelic aalysis provide a fudametal tools, ad play importat roles ot oly i umber theory ad geometry at very small distace e.g., [7] ad [15], but also i related mathematical or scietific fields e.g., [ 4] ad [5]. So, we caot help emphasizig their importaces ad valuable applicatios. I [5], the we studied certai semicircular-like elemets, called weighted-semicircular elemets, ad correspodig semicircular elemets iduced from p-adic umber fields p, motivated by the earlier studies of free-probabilistic structures over p e.g., see [] ad cited papers therei. I [4], the author exteded the p-adic local weighted-semicircular laws ad the semicircular law of [5] to Adelic uiversal weighted-semicircular laws ad the semicircular laws uder free product of Baach -probability spaces i the sese of [5], over primes p. As applicatio of the mai results i [4], we studied free stochastic itegratio for free stochastic processes determied by the weighted-semicircular laws ad the semicircular law of [4] i [3]. Also, see [6] ad [1]. I this paper, we apply the costructio of weighted-semicircular elemets, ad that of semicircular elemet of [3,4] ad [5] to costruct such free radom variables i certai Baach -probability spaces iduced by Z -may orthogoal projectios i a C -algebra. 1.. OVERVIEW Our study starts from the assumptio that there exists iteger-may orthogoal projectios {q j } j Z i a arbitrarily give C -algebra A. Oe ca have such cases artificially e.g., see Sectio 3 below, or aturally e.g., [3,4] ad [5]. Defie a C -subalgebra of A geerated by the family. Ad the costruct a suitable radial operator l actig o. Defie a cyclic Baach -algebra L C[{l}] geerated by {l}. The, each elemet of L is a Baach-space operator actig o. The costruct the tesor product algebra L L C. Geeratig operators u j l q j of L form a weighted-semicircular elemets i the free product Baach -probability space, L, τ j, j Z for suitable liear fuctioals τ j o L. Ad hece, there exist s j R i C, such that s j u j become semicircular elemets, for all j Z. The costructio of our weighted-semicircular elemets is oe of the mai results of this very paper. As we discussed above, the costructio is very similar to those i [4]. From our weighted-semicircular elemets, we costruct several differet types of Baach -algebras, ad study operator-algebraic properties of such -algebras, ad cosider operator-theoretic properties of certai operators of these Baach -algebras uder spectral data, expressed by free distributios of operators.

3 Baach -algebras geerated by semicircular elemets SKETCH OF MAIN RESULTS I Sectios 3 ad 4, we established self-adjoit operators {u j } j Z i a certai Baach -algebra L L C iduced by the C -algebra geerated by a family {q j } j Z of mutually orthogoal projectios q j s. By iig a family {τ j } j Z of suitable liear fuctioals τ j s o L, we show our self-adjoit Baach-space operators u j are weighted-semicircular i the Baach -probability spaces L, τ j, for all j Z, i.e., the algebra L is sectioized by {τ j } j Z, ad i each sectioized Baach -probability space of L, our operator u j is weighted-semicircular. The costructio of our weighted-semicircular elemets, itself, is oe of the mai results of this paper. I Sectio 5, we costruct semicircular elemets {U j } j Z i L geerated by our weighted-semicircular elemets {u j } j Z of Sectio 4. It shows that wheever oe ca take a family of Z -may mutually orthogoal projectios, the correspodig semicircular elemets U j exist. I Sectio 6, based o the costructios of weighted-semicircularity ad semicircularity i Sectios 4 ad 5, we study weighted-semicircularity iduced by differet types of Baach -algebras geerated by fiite copies of L s. I particular, we are iterested i the weighted-semicircularity o Baach -algebras geerated by fiite-copies of L s uder direct product, tesor product, ad free product. Check the weighted-semicircularity 6.1 o direct products of L s, ad the weighted-semicircularity 6.3, ad the semicircularity 6.34 o tesor products of L s, ad the semicircularity 6.49 o free products of L s. I Sectio 7, we cosider a example from our mai results of Sectios 4, 5 ad 6. Especially, we study weighted-semicircular elemets ad semicircular elemets determied by the mutually orthogoal rak-1 projectios o the Hilbert space l Z.. PRELIMINARIES Readers ca check fudametal aalytic-ad-combiatorial free probability theory from [14] ad [16] ad the cited papers therei. Free probability is uderstood as the ocommutative operator-algebraic versio of classical probability theory ad statistics. The classical idepedece is replaced by the freeess. It has various applicatios ot oly i pure mathematics e.g., [13], but also i related applied topics e.g., [ 4] ad [5]. I particular, we will use combiatorial free probabilistic approach of Speicher e.g., [14]. Free momets ad free cumulats of operators will be computed without itroducig detailed cocepts. Also, readers ca check the weighted-semicircularity ad semicircularity iduced from p-adic umber fields p i [4] ad [5]. The mai results, icludig the costructios, of [4] ad [5] motivate our study. 3. FUNDAMENTAL SETTINGS I this sectio, we establish backgrouds of our study. Let A, ψ be a C -probability space, where A is a uital C -algebra i the operator algebra BH cosistig of all bouded liear operators o a Hilbert space H, ad ψ is a bouded liear fuctioal o A, satisfyig ψ1 A 1, where 1 A is the idetity operator of A.

4 504 Ilwoo Cho ad Palle E.T. Jorgese A operator a of A is said to be a free radom variable wheever it is regarded as a elemet of A, ψ. As usual, we say a is self-adjoit, if a a i A, where a is the adjoit of a. We say a self-adjoit free radom variable a A, ψ is a eve elemet, if it satisfies ψa 1 0, for all N. Defiitio 3.1. A self-adjoit free radom variable a is said to be weighted- -semicircular i A, ψ with its weight t 0 C, or i short, t 0 -semicircular, if a satisfies the free cumulat computatio, { k a, a t 0 if, k a,..., a otherwise, for all N, where k... meas the free cumulat o A i terms of ψ i the sese of [14]. If t 0 1 i 3.1, the 1-semicircular elemet a is simply said to be semicircular i A, ψ, i.e., a is semicircular i A, ψ, if a satisfies { 1 if, k a,..., a 3. 0 otherwise, for all N. By the Möbius iversio of [14], oe ca characterize the weighted-semicircularity 3.1 as follows: a self-adjoit operator a is t 0 -semicircular i A, ψ, if ad oly if ad for all N, where ψa 1 0, i.e., it is eve i A, ψ, ψa t 0 c c 1! !! is the -th Catala umber. I short, a is t 0 -semicircular i A, ψ, if ad oly if where ψa ω t 0 c ω { 1 if is eve, 0 if is odd,, 3.3 for all N. Similarly, a free radom variable a is semicircular i A, ψ, if ad oly if for all N, where ω are i the sese of 3.3. ψa ω c, 3.4

5 Baach -algebras geerated by semicircular elemets So, we will use the t 0 -semicircularity 3.1 ad its characterizatio 3.3 alteratively; ad similarly, oe ca use the semicircularity 3. ad its characterizatio 3.4, alteratively. Note that, if a is a self-adjoit free radom variable i A, ψ, the the free momets {ψa } 1, ad the free cumulats {k a,..., a} 1 provide equivalet free distributio of a i A, ψ, which are also equivalet to the spectral data of a e.g., [14] ad [16]. Ideed, the Möbius iversio of [14] satisfies ψa Π k V a,..., a, V π ad k a,..., a π NC π NC Π V θ ψa V µπ, 1, where NC is the lattice cosistig of all ocrossig partitios over {1,..., }, ad V π meas V is a block of π, ad where µ 0 k, 1 k 1 k+1 c k+1, ad µπ, 1 0, π NC where c m are m-th Catala umbers for all m N. Therefore, the free-momet formula 3.3 resp., 3.4 is equivalet to the free-cumulat iitio 3.1 resp., 3.. Now, let A, ψ be a fixed C -probability space, ad let q j A be a projectio i the sese that q j q j q j i A, 3.5 for all j Z, where Z is the set of all itegers. Moreover, assume that the projectios {q j } j Z are mutually orthogoal i A, i the sese that: q i q j δ i,j q j i A, for all i, j Z, 3.6 where δ is the Kroecker delta. Also, suppose, for coveiece, that ψq j 0 i C, for all j Z. 3.7 Now, we fix the family {q j } j Z of mutually orthogoal projectios of A, ad we deote it by, i A. {q j : j Z, q j s satisfy 3.5, 3.6 ad 3.7}, 3.8 Remark 3.. Assume that oe ca fid mutually orthogoal projectios {q 1,..., q N } i a C -algebra A 0, for some N N { }. Assume that N <, for istace, A 0 is a matricial algebra M N C, ad for all k, l j, j i {1,..., N}, etc. q j [t ij ] N N, with t jj 1, ad t kl 0,

6 506 Ilwoo Cho ad Palle E.T. Jorgese The oe ca costruct a C -algebra A A Z 0 C... A 0 A 0..., uder suitable product topology. The oe ca obtai the mutually orthogoal projectios, {..., q 1 1,..., q 1 N, q0 1,..., q0 N, q1 1,..., q1 N,..., } i A, with idetity: q k j q j i the direct summad A 0 of A, for all k Z, for all j 1,..., N. Similarly, if oe ca fid mutually orthogoal projectios {q 1, q, q 3,... } i a C -algebra A 0 i.e., N, the oe ca costruct a C -algebra A A 0 A 0, ad the correspodig orthogoal projectios, i A, with {..., q 1 3, q1, q1 1, q 1, q, q 3,...} q k j q j i the direct summad A 0 of A, for k 1,, for all j N. Thus, we ca assume the existece of mutually orthogoal Z -may projectios iduced by a fixed C -algebra A. Ad let be the C -subalgebra of A geerated by of 3.8, where is i the sese of 3.8. C A, 3.9 Propositio 3.3. Let be a C -subalgebra 3.9 of a give C -algebra A geerated by of 3.8. The -iso C q j -iso C Z, i A j Z Proof. The proof of the isomorphism theorem 3.10 is straightforward by the mutually-orthogoality 3.5 of the geerator set of i A. Defie ow liear fuctioals ψ j o by ψ j q i δ ij ψq j, for all i Z, 3.11 for all j Z, where ψ is a fixed liear fuctioal i the fixed C -probability space A, ψ. The liear fuctioals {ψ j } j Z of 3.11 are well-ied o by the structure theorem 3.10 of the C -subalgebra of 3.9 i A.

7 Baach -algebras geerated by semicircular elemets So, if q, the q j Zt j q j with t j C, by 3.10, ad hece, ψ j q ψ j j q j t j ψq j, j Zt by 3.11, for all j Z. Defiitio 3.4. The C -probability spaces, ψ j are called the j-th C -probability spaces of i a fixed C -probability space A, ψ, where is i the sese of 3.9, ad ψ j are i the sese of 3.11, for all j Z. Now, let us ie a Baach-space operator c ad a actig o the C -algebra by a bouded liear trasformatios satisfyig that c q j q j+1, ad a q j q j 1, 3.1 for all j Z. The c ad a are ideed well-ied bouded liear operators o. Defiitio 3.5. We call these Baach-space operators c ad a of 3.1, the creatio, respectively, the aihilatio o. Defie ow a ew Baach-space operator l actig o, by l c + a This operator l of 3.13 is said to be the radial operator o. By the iitio 3.13, oe has l j q j j Zt j q j+1 + q j 1, o. j Zt Now, ie a cyclic Baach algebra L C[{l}], 3.14 geerated the radial operator l of 3.13, equipped with the operator orm, T sup{ T q : q 1}, where is the C -orm o, where X mea the operator-orm closures of subsets X of the operator space B, cosistig of all bouded liear trasformatios o. O the Baach algebra L of 3.14, ie a uary operatio by t l t l i L, where z mea the cojugates of z, for all z C.

8 508 Ilwoo Cho ad Palle E.T. Jorgese The it is ot difficult to check that the operatio 3.15 forms a adjoit o L, i.e., the Baach algebra L of 3.14 forms a Baach -algebra with its adjoit Defiitio 3.6. We call the Baach -algebra L of 3.14, the radial Baach -- algebra o. Now, let L be the radial algebra o. Costruct ow the tesor product Baach -algebra, L L C Sice the radial algebra L of is a Baach -algebra, ad sice is a C -algebra, the topological tesor product L of them is a well-determied Baach -algebra uder the product topology. Defiitio 3.7. We call the tesor product Baach -algebra L of 3.16, the radial projectio Baach -algebra o. 4. WEIGHTED-SEMICIRCULAR ELEMENTS INDUCED BY Throughout this sectio, we use the same settigs, otatios ad cocepts of Sectio 3. We here costruct weighted-semicircular elemets iduced by the family of mutually orthogoal projectios i the sese of 3.8. Let, ψ j be j-th C -probability space of i A, ψ, where ψ j are i the sese of 3.11, for all j Z, ad let L be the radial projectio algebra 3.16 o. Remark that, if u j l q j L, for all j Z, 4.1 the u j l q j l q j, for all N, sice q j q j, for all N, for j Z. Oe ca costruct a liear fuctioal ϕ j o the radial projectio algebra L by a liear morphism satisfyig that ϕ j l q i ϕ j l q i ψ j l q i, 4. for all N, for all i, j Z. We call the Baach -probability spaces L, ϕ j, for all j Z, 4.3 the j-th Baach- -probability spaces o. Now, observe the elemets l q i i, for all N, i Z. If c ad a are the creatio, respectively, the aihilatio o i the sese of 3.1, the ca ac 1, the idetity operator o, 4.4 where 1 q j q j, for all j Z,

9 Baach -algebras geerated by semicircular elemets ad hece, ad Ideed, for ay q j i, for all j Z. More geerally, oe has 1 x x, for all x. ca q j c a q j c q j 1 q j 1+1 q j, ac q j a c q j a q j+1 q j+1 1 q j, by iductio o 4.4. By 4.4 ad 4.5, we also obtai that too. Thus, oe obtais that c a 1 a c, for all N, 4.5 c 1 a a c 1, for all 1, N, 4.6 l c + a by 4.4, 4.5, ad 4.4, for all N. Note that, for ay N, l 1 1 k0 k0 k c k a k, c k a k, 4.8 k by 4.7. So, the formula 4.8 does ot cotai 1 -terms, by 4.4 ad 4.5. Note also that, for ay N, oe has l k0 by 4.4, 4.5, ad 4.7. c k a k c a + [Rest terms], 4.9 k Propositio 4.1. Let l be the radial operator 3.13 geeratig the radial algebra L o. The l 1 does ot cotai 1 -terms i L, 4.10 l cotais 1 i L Proof. The statemet 4.10 is prove by 4.8, with help of 4.4, 4.5 ad 4.7. Ad, the statemet 4.11 is prove by 4.9 ad 4.5.

10 510 Ilwoo Cho ad Palle E.T. Jorgese Sice oe ca get that u j l q j l q j, for all N, ad j Z, by 3.11 ad Similarly, we have ϕ j u 1 j ψj l 1 q j 0, 4.1 ϕ j u j ψj l q j ψ j q j + [Rest terms]q j by 4.9 ψ j q j ψ q j, 4.13 by 3.11, for all N, i.e., by 4.10 ad 4.11, we obtai the followig free-distributioal data o the j-th probability space L, ϕ j, for j Z. Theorem 4.. Fix j Z, ad let u j l q j be the correspodig geeratig operator 4.1 of the j-th probability space L, ϕ j. The ϕ j u j ω + 1 ψ q j c, 4.14 for all N, where ω are i the sese of 3.3, ad c k are the k-th Catala umbers, for all k N. Proof. By 4.1 ad 4.13, oe ca get that: ϕ j u 1 j 0, ad ϕ j u + 1 j ψ q j ψ q + 1 j 1 + 1ψ q j ψ q j c, for all N. Therefore, the formula 4.14 holds. Motivated by the free-distributioal data 4.14 of the geeratig operator u j of L, we ie the followig morphism E j, : L L by a surjective liear trasformatio satisfyig E j, u ψ q j 1 i δ j,i [ ] + 1 u j, 4.15

11 Baach -algebras geerated by semicircular elemets for all N, i, j Z, where δ is the Kroecker delta, ad [ ] mea the miimal itegers greater tha or equal to, for example, [ 3 [ 4. ] ] The liear trasformatios E j, of 4.15 are well-ied liear trasformatios o L, because of the costructio 3.16 of L, ad by the structure theorem 3.10 of, i.e., all elemets T of L has its form, t k,j u k k1 j Z j with k N, t k,j C. So, uder liearity, the morphisms E j, of 4.15 are ideed well-ied o the radial projectio algebra L, for all j Z. Defie ow a ew liear fuctioal τ j o L by τ j ϕ j E j, o L, for all j Z, 4.16 where ϕ j is the liear fuctioal 4. o L. By the liearity of ϕ j ad E j,, the above morphisms τ j are ideed well-ied liear fuctioals o L, for all j Z. Defiitio 4.3. The well-ied Baach -probability spaces L j deote L, τ j 4.17 are called the j-th filtered Baach- -probability spaces of L, for all j Z. O the j-th filtered probability space L j of 4.17, oe ca obtai that τ j u j ϕj Ej, u j by 4.14, i.e., we ca get that for all N, for j Z. ψ q j 1 ϕ j [ ] + 1 u j ψ q j 1 [ ] + 1 ϕ j u j ψ q j 1 [ ] + 1 ω + 1 ψ q j c τ j u j { ψ q j c if is eve, 0 if is odd, 4.18 Theorem 4.4. Let L j L, τ j be the j-th filtered probability space of L, for j Z. The τ j u k δ j,k ω ψ q j c 4.19 for all N, for all k Z, where ω are i the sese of 3.3.

12 51 Ilwoo Cho ad Palle E.T. Jorgese Proof. If k j i Z, the the free-distributioal data 4.19 holds true by 4.18, for all N. If k j i Z, the, by the very iitio 4.15 of the j-th filterizatio E j,, ad also by the iitio 4. of ϕ j, τ j u k 0. Therefore, the above formula 4.19 holds true, for all k Z. The followig theorem is the direct cosequece of the above free distributioal data Theorem 4.5. Let L j be the j-th filtered probability space L, τ j of L, for j Z, ad let u j l q j be the j-th geeratig operator of L i L j. The u j is ψq j -semicircular i L j. Proof. First of all, the geeratig operators u k are self-adjoit i L. Ideed, u k l q k l q k l q k u k, by 3.15 ad by the self-adjoitess of projectios q k, for all k Z. Let us fix j Z, ad let u j l q j be a geeratig free radom variable of the j-th filtered probability space L j. The it is self-adjoit i L j. Also, by 4.19, we have that τ j u j ω ψ q j c ω ψqj c, where ω are i the sese of 3.3, for all N. Therefore, by the free-mometal characterizatio 3.3 of weighted-semicircularity, the elemet u j is ψq j -semicircular i the j-th filtered probability space L j of L. The above theorem shows that, for ay k Z, the k-th geeratig operator u k l q k of the k-th filtered probability space L k of is ψq k -semicircular. Observe also that if k j... is the free cumulat i terms of the liear fuctioal τ j o L i the sese of [14], the ku j j, u j,..., u j }{{} -times { k j u j, u j ψq j if, 0 otherwise, 4.0 for all N, by 4.19, ad by the Möbius iversio of [14]. So, by 3.1, the j-th geeratig operators u j are ψq j -semicircular i the j-th filtered probability spaces L j, for all j Z.

13 Baach -algebras geerated by semicircular elemets SEMICIRCULAR ELEMENTS INDUCED BY As i Sectio 4, we will keep workig o the j-th filtered probability space L j, for j Z. The mai results of Sectio 4 show that, for ay fixed j Z, the j-th geeratig operator u j l q j of L is ψq j -semicircular i L j, satisfyig that equivaletly, τ j u j ω ψq j c, k j u j,..., u j { ψq j if, 0 otherwise, 5.1 for all N, where k j... meas the free cumulat o L i terms of the liear fuctioal τ j of Corollary 5.1. Let u j l q j be the j-th geeratig operator of the radial projectio algebra L, for j Z. If the projectio q j of satisfies ψq j 1, the u j is semicircular i the j-th filtered probability space L j. However, geerally ψq j may ot be idetical to 1, for j Z. So, we cosider the followig semicircularity iduced by our weighted-semicircularity i L j. Defie operators U j of L j by U j 1 ψq j u j L j, 5. for all j Z, where u j are the ψq j -semicircular elemets i L j. The these operators U j ca be semicircular i L j uder a certai additioal coditio. Theorem 5.. Let U j 1 ψq j u j be a free radom variable 5. of the j-th filtered probability space L j, for j Z. If ψq j R i C, the U j is semicircular i L j. Proof. Fix j Z, ad assume that ψq j R i C. The 1 Uj ψq j u j U j, by the self-adjoitess of u j i L. So, the operator U j is self-adjoit i L. Cosider ow that: if k. j.. is the free cumulat o L i terms of τ j, the 1 ku j j, U j,..., U j k j }{{} ψq j u 1 j,..., ψq j u j -times 1 k j ψq j u j, u j,..., u j }{{} -times by the bimodule-map property of free cumulats e.g., [14] 1 ψq j k j u j, u j if, 0 otherwise

14 514 Ilwoo Cho ad Palle E.T. Jorgese by the ψq j -semicircularity of u j i L j { 1 ψq j ψq j 1 if, 0 otherwise, 5.3 for all N. Therefore, if ψq j R i C, the the self-adjoit operator U j satisfies the free-distributioal data 5.3. Thus, by 5.3 ad 3., this operator U j is semicircular i L j. The above theorem shows that, from the ψq j -semicircular elemets u j l q j i L j, oe ca costruct semicircular elemets U j 1 ψq u j j i L j, wheever ψq j R, for all j Z. 6. WEIGHTED-SEMICIRCULARITY ON BANACH -ALGEBRAS GENERATED BY COPIES OF L Let A, ψ be a fixed C -probability space, ad, the C -subalgebra of A geerated by the family {q j } j Z of Z -may projectios satisfyig the coditios 3.4, 3.6 ad 3.7. Ad let L be the radial projectio algebra 3.16, iducig its j-th filtered probability spaces L j L, τ j i the sese of 4.17, for all j Z. The the free radom variables u j l q j L 6.1 are ψq j -semicircular i L j, for all j Z. If ψq j R i C, for j Z, the the free radom variables U j 1 ψq j u j L 6. are semicircular i L j. I the rest of this sectio, we assume ψq j R i C, for all j Z, for coveiece ON THE BANACH -PROBABILITY SPACES L N j 1,..., j N Let W j 1,..., j N be a N-tuple of itegers j 1,..., j N i Z, for N N. Remark that the chose iteger-etries j 1,..., j N of W are ot ecessarily distict i Z, ad ie the direct product Baach -algebra L N by L N L... L, 6.3 }{{} N-times where meas the direct sum of Baach -algebras uder product topology. The o a ew Baach -algebra L N of 6.3, oe ca ie a coditioal expectatio E W : L N C N

15 Baach -algebras geerated by semicircular elemets by the liear trasformatio satisfyig that for all E W T 1,..., T N τ j1 T 1,..., τ jn T N, 6.4 T 1,..., T N deote N T l L N where C N is the N-dimesioal algebra which is regarded as the diagoal subalgebra i the matricial algebra M N C, ad τ jk are the liear fuctioals 4.16 o L, iducig the correspodig j k -th filtered probability space L j k L, τ jk, for k 1,..., N, where j 1,..., j N are the etries of the fixed N-tuple W. The the morphism E W of 6.4 is ideed a well-ied coditioal expectatio from L N oto C N e.g., [14]. Propositio 6.1. The liear morphism E W : L N C N of 6.4, for a fixed N-tuple W j 1,..., j N of itegers j 1,..., j N, is a coditioal expectatio. Proof. By the very iitio 6.4 of the liear morphism E W, it is a well-determied bouded liear trasformatio from L N oto C N, because τ jl are bouded, for all l 1,..., N. Clearly, if oe takes T t 1 I,..., t N I L N, for l1 I 1 L 1 L 1 i L, for t 1,..., t N C, where 1 L is the idetity elemet l 0 of the radial algebra L, satisfyig 1 L l l, for all N, the E W T t 1,..., t N T, by uderstadig T N C I -iso C N i L N. Therefore, uder liearity, we have k1, E W T T, wheever T C N i L N. 6.5 Now, let S 1 t 1,..., t N, S t 1,..., t N C N i L N, with t k, t k C, for all k 1,..., N, ad let T T 1,..., T N L N. The i the algebra C N E W S 1 T S E W t 1 T 1 t 1,..., t 1 T N t N τ j1 t 1 T 1 t 1,..., τ jn t N T N t N t 1 τ j1 T 1 t 1,...., t N τ jn T N t N t 1,..., t N τ j1 T 1,..., τ jn T N t 1,..., t N S 1 E W T S.

16 516 Ilwoo Cho ad Palle E.T. Jorgese So, So, for ay S 1, S C N, ad T L N, oe obtais that E W S 1 T S S 1 E W T S i C N. 6.6 Fially, observe that, if T T 1,..., T N L N, the T T1,..., TN i L N. E W T τ j1 T 1,..., τ jn T N where z are the cojugates of z, for all z C i the algebra C N τ j1 T 1,..., τ jn T N E W T. Thus, for ay T L N, oe has that τ j1 T 1,..., τ jn T N, E W T E W T i C N. 6.7 Therefore, by 6.5, 6.6 ad 6.7, the bouded surjective liear trasformatio is a coditioal expectatio from L N oto C N. Let tr N be the usual ormalized trace o the matricial algebra M N C, i.e., tr N [t ij ] N N 1 N N t jj, j1 for all N N-matrices with their i, j-etries t ij i C. Sice the N-dimesioal algebra C N is the diagoal subalgebra of M N C cosistig of all N N-diagoal matrices, oe ca aturally restrict the ormalized trace tr N o M N C to the liear fuctioal tr N C N, also deoted simply by tr N, o C N, i.e., tr N t 1,..., t N 1 N N j1 t j, 6.8 for all t 1,..., t N C N. Now, ie a liear fuctioal τ W o the direct product Baach -algebra L N by τ W tr N E W o L N, 6.9 where tr N is the restricted trace 6.8 o C N, ad E W is the coditioal expectatio 6.4 from L N oto C N.

17 Baach -algebras geerated by semicircular elemets The the liear morphism η W of 6.9 is ideed a well-ied liear fuctioal o L N, moreover, it is a trace i the sese that: So, the pair L N τ W T 1 T τ W T T 1, for all T 1, T L N., τ W forms a well-ied tracial Baach -probability space. Defiitio 6.. We deote the Baach -probability space L N W, where η W is the liear fuctioal 6.9 o L N, i.e., for all N-tuples W i Z. L N W L N, τ W L N, τ W simply by, 6.10 Let us take a operator u W i L N, as a free radom variable of the Baach -probability space L N W i the sese of 6.10, where u W 1 N U j 1, U j,..., U jn L N, 6.11 ad U jk 1 ψq jk u j k 1 ψq jk l q j k L j k are our semicircular elemets i the j k -th filtered probability space L j k L, τ jk, for all k 1,..., N. Recall ad remark that i the rest of this paper, we automatically assumed ψq j R i C, for all j Z. So, the direct summads U jk of the operator u W of 6.11 i L W are well-determied semicircular elemets i L j k, for k 1,..., N W. Theorem 6.3. Let u W be a free radom variable 6.11 i L N W of 6.10, for a fixed N-tuple W j 1,..., j N of itegers. The u W is 1 N -semicircular i L N W, i.e., for ay W Z N, there exists u W L W, such that 1 u W is -semicircular i L W W. 6.1 W Proof. Let u W be i the sese of 6.11 i the Baach -probability space L N W, τ W of 6.10, for a fixed W j 1,..., j N Z N. Clearly, the operator L N u W is self-adjoit i L N, because every direct summad U j k is self-adjoit by its semicircularity, for all k 1,..., N.

18 518 Ilwoo Cho ad Palle E.T. Jorgese By the iitio 6.9 of τ W, oe ca get that 1 τ W u W τ W N U j 1, U j,..., U jn 1 N τ W 1 N tr N 1 N tr N 1 N tr N U j1, U j,..., U j N EW U j1, U j,..., U j N τj1 U j1,..., τjn U jn ω c,..., ω c by the semicircularity of U jk i L, τ jk, for k 1,..., N ω N tr N ω 1 N N c,..., c N k1 by the iitio 6.8 of the ormalized trace tr N o C N c ω N N N c ω 1 N c ω c N, 6.13 where ω are i the sese of 3.3, for all N. Therefore, by 6.13 ad 3.3, oe ca coclude that the free radom variable T W 1 is N -semicircular i L N W. Just for sure, we cosider the followig extra parts of the proof. By the free-momet computatio 6.13, we obtai that: if k W... is the free cumulat o L N W with respect to the liear fuctioal τ W, the k W u W, u W τ W u W τw u W τ W u W 1 N, uder the Möbius iversio of [14], i.e., oe obtais that Now, for ay N, k W 1 u W,..., u W k W u W, u W 1 N π NC 1 τ W V π u V W µ π, 1 1 by the Möbius iversio of [14] 0,

19 Baach -algebras geerated by semicircular elemets because every ocrossig partitio π over {1,..., 1} cotais at least oe odd block V 0, satisfyig the block-depedig free momet τ W u V0 W 0, by the eveess from the formula 6.13, i.e., k W u W,..., u W 0, wheever is odd i N Observe that, for ay N \ {1}, k W u W,..., u W η W u V µθ, 1 θ NC e V θ W where ad hece by 6.13 NC e {θ NC : V θ V is eve}, θ NC e θ NC e θ NC e θ NC e 1 N V θ 1 N V θ 1 c V µθ, 1 N V 1 c N V V µθ, 1 V θ Σ V θ V V θ c V µθ, 1 1 c V µθ, 1 N V θ µθ, 1 c +1 θ NC e 1 c +1 µπ, 1 0, N π NC by [14], i.e., k W u W,..., u W 0, wheever > 1 i N Thus, by 6.14, 6.15 ad 6.16, ideed, we obtai that k W u W,..., u W { 1 N if, 0 otherwise, 6.17 for all N. Therefore, by 6.17 ad 3.1, the free radom variable u W is ideed 1 N -semicircular i L N W.

20 50 Ilwoo Cho ad Palle E.T. Jorgese The above theorem shows that, for ay fiite sequeces W of itegers, if we ie a operator u W 1 U j i L W, W j W 1 the it is -semicircular i the Baach -probability space L W W W, where U j 1 ψq l q j j are the semicircular elemets of L, τ j, for all j W i Z. 6.. ON THE BANACH -PROBABILITY SPACES L N j 1,..., j N Like i Sectio 6.1, let W j 1,..., j N Z N be a arbitrarily fixed N-tuple of itegers j 1,..., j N, which are ot ecessarily distict from each other i Z, ad let L be the radial projectio algebra of the C -algebra of the fixed C -probability space A, ψ. Let τ j be the liear fuctioals 4.16 o L, iducig the correspodig j-th filtered probability space L j L, τ j of. Also, let U j 1 ψq j u j 1 ψq j l q j be the semicircular elemet i L j, for all j Z. Now, ie the tesor product Baach -algebra L N by L N L C L C... C L }{{} N-times 6.18 uder product topology. Defie a liear trasformatio F W from the tesor product Baach -algebra L N of 6.18 oto the fiite-dimesioal algebra C N by the liear morphism, deote F W N τ jk o L N, 6.19 k1 satisfyig that N F W T k N τ jk T k I, k1 k1 for all N T k L N, where I is the idetity elemet of L itroduced i Sectio 6.1. k1 Propositio 6.4. Let L N be i the sese of 6.18 for a fixed N-tuple W with N W. The the liear morphism F W of 6.19 is a coditioal expectatio from L N oto C N. Proof. By the boudedess of the liear fuctioals τ jk of 4.16 o L, for k 1,..., N, the liear trasformatio F W is a bouded liear trasformatio from to C N. Also, by the very iitio 6.19, it is surjective. L N

21 Baach -algebras geerated by semicircular elemets Now, take T N k1 t k I i L N So, uder liearity, oe has, for t 1,..., t N C, i.e., T C N L N. The F W T N τ jk t k I N t k i C N. k1 k1 F W S S i C N, for all S L N. 6.0 Now, let T 1 N t k I with t k C, ad T N t k I with t k C be i C N, k1 k1 ad S N S k L N k1 with S k L. The F W T 1 ST F W N k1 t k S k t k N t k τ jk S k t k k1 by 6.19 N N N t k I τs k k1 T 1 F W ST. k1 k1 t ki Thus, uder liearity, for all T 1, T C N, ad S L N, oe obtais that Take T N k1 T k L N F W T 1 ST T 1 F W ST i C N The T N Tk k1 i L N. So, by 6.19, F W T F W T. 6. Therefore, the surjective bouded liear trasformatio F W is a coditioal expectatio from L N oto C N, by 6.0, 6.1 ad 6.. O the N-dimesioal tesor product algebra C N, ie a liear fuctioal ϕ N by a liear morphism satisfyig that: N ϕ N t k k1 N t k, for all k1 N t k C N. 6.3 k1 Defie ow a liear fuctioal ϕ o W o L N by ϕ o W ϕ N F W o L N, 6.4 where ϕ N is i the sese of 6.3 ad F W is the coditioal expectatio 6.19.

22 5 Ilwoo Cho ad Palle E.T. Jorgese By the iitio 6.4 of ϕ o W, it is a well-determied liear fuctioal o L N satisfyig N N ϕ o W T k τ jk T k. 6.5 k1 k1 Defie a elemet I W,jk of L N by I W,jk 1 L qĵk, 6.6 with qĵk q jk k-th , for all j k W i Z, for k 1,..., N. From the operators I W,jk of 6.6, ie a operator I W of L N I W by N I W,jk 6.7 k1 Note that, if the chose projectios q j1,..., q jn are mutually orthogoal o, equivaletly, if W j 1,..., j N is a N-tuple of mutually distict itegers i Z N, the oe ca get that I W N k1 N k1 1L qĵk 1 L qĵk N 1L qĵk k1 N k1 1L qĵk IW, 6.8 for all N, by 6.6 ad 6.7. So, if I W is i the sese of 6.7 i L N, the oe ca get that k1 N N ϕ o W IW ϕ o W I W τ jk 1 L q jk ψ q jk, k1 ad ϕ o W N I W,jk ϕ o W I W,jk τ jk 1 L q jk ψ q jk, 6.9 k1 i C, by 6.5 ad 6.8, for all N, where ϕ o W is i the sese of 6.4. For coveiece, we deote the C-quatity ϕ o W I W of 6.9, simply by ψ W, below. Note that, sice N <, the quatity ψ W coverges i C.

23 Baach -algebras geerated by semicircular elemets Now, ie ew liear fuctioals ϕ W,k s o L N by ϕ W,k ψq j k ψ W ϕ o W o L N, 6.30 for all k 1,..., N, where ϕ o W is the liear fuctioal 6.4 o L N satisfyig 6.5, ad where ψ W is the C-quatity 6.9. The the liear fuctioals ϕ W,k of 6.30 are well-ied o L N, ad hece, we obtai the tesor product Baach -probability spaces L N k, W, L N k, W L N, ϕ W,k, 6.31 where ϕ W,k are i the sese of 6.30, for all k 1,..., N. O our tesor product Baach -probability space L N k, W of 6.31, oe ca obtai the followig semicircular elemets i L N. Theorem 6.5. Let W j 1,..., j N be a N-tuple of mutually distict itegers j 1,..., j N i Z, for N N, ad fix k 0 {1,..., N}. For a fixed k 0, let L N k 0, W be the correspodig Baach -probability space 6.31 equipped with its liear fuctioal ϕ k0,w of Defie the free radom variables U W,k0, for some k 0 {1,..., N}, by U W,k0 N S jk k1 L N k 0, W L N, ϕ k 0,W, 6.3 where ψ W C is i the sese of 6.9, ad S jk { u jk0 if k k 0, I W,jk otherwise, for all k 1,..., N, where u jk0 l q jk0 is our ψqjk0 -semicircular elemet of the j k0 -th filtered probability space L j k0 of, ad I W,jk are i the sese of 6.6, for k k 0 i {1,..., N}. The U W,k0 of 6.3 is a ψq jk0 -semicircular elemet i L N W. Proof. By the very iitio 6.3 of U W,k0, it is self-adjoit i L N. Ideed, the tesor factors I W,jk ad u jk0 are self-adjoit i L, ad hece, the operator U W,k0 is self-adjoit i L N.

24 54 Ilwoo Cho ad Palle E.T. Jorgese ϕ k0,w Observe ow that by 6.7 U W,k0 ϕk0,w ϕ k0,w by 6.5 ad 6.30 I W,j 1... I W,j k0 1 u j k0 k 0-th I W,j1... I W,jk0 1 u j k0 k 0-th ψq j k0 τ j1 I W,j1... τ jk0 u jk0... τ jn I W,jN ψ W ψq j1... ω ψq jk0 c... ψqjn k0 1 NΠ ψq jk0 Π ψq j k ψq j k k1 kk 0+1 ψq j k0 ψ W ψ W I W,j k I W,j N I W,jk I W,j N ω ψq jk0 c by 6.9 ω ψqjk0 c 6.33 for all N. Therefore, by 6.33 ad 3.3, the free radom variable U W,k0 is ψq jk0 -semicircular i L N k 0, W. The above theorem shows that, wheever W is a fiite sequece of mutually distict W -may itegers of Z, the the operators U W,k of 6.3 i L W are ψq jk -semicircular i L W k, W, for all k 1,..., W. By the weighted-semicircularity 6.33 of the operator U W,k of 6.17, oe ca obtai the followig corollary. Corollary 6.6. Let W j 1,..., j N be the N-tuple of mutually distict itegers j 1,..., j N i Z, ad let L N k, W be the Baach -probability space i the sese of 6.31, for k 1,..., N. Defie a free radom variable T W,k by where I W,jk T W,k I W,j1... I W,jk 1 U jk I W,jk+1... I W,jN, 6.34 k-th are i the sese of 6.6, ad U jk 1 ψq jk l q j k is our semicircular elemet i L j k, for all k 1,..., N. The the operator T W,k is semicircular i L N k, W, for k 1,..., N.

25 Baach -algebras geerated by semicircular elemets Proof. Similar to 6.33, oe ca get that L N ϕ k,w T W,k ω c, for all N Thus, by 6.35 ad 3.4, the free radom variables T W,k are semicircular i k, W, for all k 1,..., N ON THE BANACH -PROBABILITY SPACES L N j 1,..., j N Let be the C -subalgebra of a fixed C -probability space A, ψ geerated by the family {q j } j Z of mutually orthogoal projectios satisfyig ψq j R i C, for all j Z, ad let L be the radial projectio algebra of, havig the correspodig j-th filtered probability spaces L j L, τ j, for all j Z. As i Sectios 6.1 ad 6., let W j 1,..., j N be the N-tuple of itegers j 1,..., j N i Z, for N N. As i Sectio 6.1 ad differet from the mai results of Sectio 6., the itegers j 1,..., j N are ot ecessarily distict from each other i Z. For a fixed N-tuple W, ie the free product Baach -probability space L N W by the Baach -probability space, i the sese of [14] ad [16], where L N W deote L N, τ W N L j k k1 N k1 L, τ jk is the free product liear fuctioal o τ W N k1 τ jk N k1 L, N k1 τ j k L N N L L L... L. k1 }{{} N-times 6.36 Note that the above free product Baach -probability space L N W of 6.36 is highly dictated by the choice of liear fuctioals {τ jk } N k1 e.g., see [14] ad [16]. Now, ie a set N by the family of all fiite sequeces i {1,..., N}, i.e., with its partitio blocks N l1 N l, 6.37 N l {i 1,..., i l : i j {1,..., N} for all j 1,..., l}, for all l N.

26 56 Ilwoo Cho ad Palle E.T. Jorgese ad For each l N, ie the subsets AN l of N l by AN l w N l w is alteratig i the sese that: if w i 1,..., i l, the i 1 i, i i 3,..., i l 1 i l Suppose N 3, ad let w 1 1,,, 3, 1, 1 N 6 i N, w 1, 3, 1,, 1, N 6 i N, where N ad N 6 are i the sese of The w 1 / AN 6, but w AN 6, i N, where AN 6 is i the sese of Defie ow the subset AN of N by AN l1 AN l, i N Now, let us take w AN i AN of 6.39, ad choose a operator T w, T w Π l w T jl L N W, 6.40 as a free reduced word i {T j1,..., T jn }, where T jk are i the free blocks L j k i L N W, for all k 1,..., N. Remark 6.7. By [14] ad [16], if L N W is idetified with L N, τ W i the sese of 6.36, the free product Baach -algebra L N of L j 1,..., L j N has its Baach-space expressio, L N Baach-Sp C L o w, 6.41 l1 w AN l where L o w Baach-Sp L o j i1 L o j i1... L o j il, wheever w i 1,..., i l AN l i AN, where L o j ik Baach-Sp L C, for all k 1,..., l. Here, Baach-Sp meas beig Baach-space isomorphic ad is the direct product, is the tesor product, ad is the complemet of Baach spaces uder topology, i.e., if a operator T is cotaied i L o w, ad if w AN, the T is uderstood as the free reduced words i L N.

27 Baach -algebras geerated by semicircular elemets By 6.36, 6.38 ad 6.39, oe obtais the followig result. Propositio 6.8. If T w is i the sese of 6.40 i L N W, the τ W T w l w τ jl T jl, 6.4 wheever w AN. Proof. The free-momet formula 6.4 is prove by 6.36, 6.38 ad 6.39 See [14] ad [16]. ad if By the free-momet formula 6.4, oe ca get that, if w i 1,..., i AN i AN, M r1,...,r w Π l w U r l j l l1 U r l j il L N W, 6.43 for all r 1,..., r N, where U j 1 ψq j l q j are our semicircular elemets of L j, for all j Z, the oe has by 6.4 τ W Mw r1,...,r τ jl U r l l w l1 j l l1 τ jil U r l j il, ω rl c r l 6.44 by the semicircularity of U jl, for all l 1,...,, where c k are the k-th Catala umbers for all k N. Theorem 6.9. Let w j i1,..., j i AN i AN, for some N, ad let Mw r1,...,r be a free radom variable 6.43 i L N W of The τ W Mw r1,...,r ω r1,...,r l1 c r l, 6.45 with for r 1,..., r N. ω r1,...,r { 1 if all r 1,..., r are eve, 0 otherwise, Proof. The free-momet formula 6.45 is prove by Similar to the free-momet formulas 6.45 for a free radom variable Mw r1,...,r 6.43 i L N W, oe ca get the followig result. of

28 58 Ilwoo Cho ad Palle E.T. Jorgese Corollary Let w j 1,..., j AN i AN, for some N. For a fixed w AN, ad fixed quatities r 1,..., r N, ie a operator mw r1,...,r L N by m r1,...,r w l1 u r k jil L N W, 6.46 where u j l q j are our ψq j -semicircular elemets of L, τ j, for all j Z. The τ W m r1,...,r w ω r1,...,r l1 for all k N, where ω r1,...,r is i the sese of ψq jil r l c r l, 6.47 Proof. Let m r1,...,r w L N W be i the sese of 6.46, for w AN, ad, r 1,..., r N. Observe that τ W m r1,...,r w τ W l1 u r l j il τ W l1 u r l j il by uderstadig m r1,...,r w as a free reduced word i L o w of 6.41 l1 τ jil u r l j il l1 ω rl ψq jil r l c r l, by the ψq il -semicircularity of q jil. Thus, we obtai the free-distributioal data The above free-momet formulas 6.45 ad 6.47 provide ways to compute free momets of certai free radom variables of L N W geerated by our semicircular elemets U j, ad weighted-semicircular elemets u j of the j-th filtered probability spaces L, τ j, for j Z. Also, by 6.45 ad 6.47, we obtai the followig trivial, but iterestig free-distributioal data o L N W. Theorem Let L N W L N, τ W be the Baach -probability space 6.36 for a fixed N-tuple W j 1,..., j N i Z. Let u jk l q jk L j k i L N W, for all k 1,..., N, 6.48 are ψq jk -semi- where L j 1,..., L j N are the free blocks of L N W. The u j k circular i L N W, for all k 1,..., N. Now, let U jk 1 ψq jk u j k L j k i L N W, for all k 1,..., N The they are semicircular i L N W.

29 Baach -algebras geerated by semicircular elemets Proof. Let u jk be i the sese of 6.48 i L N W, for k {1,..., N}. The it forms a free reduced word with its legth-1 i the free block L j k of L N W, iducig free reduced words u j k with their legth-1 i L j k i L N W, for all N. So, oe obtais that τ W u j k τjk u jk ω ψqjk c, by 6.47, for all N, for all k 1,..., N. Therefore, the operators u jk of 6.48 are ψq jk -semicircular i L N W, for all k 1,..., N. Similarly, oe obtais that, if U jk are i the sese of 6.49 i L N W, the τ W U j k τjk U jk ω c, by 6.45, for all N, for all k 1,..., N. So, the operators U jk of 6.49 are semicircular i L N W, for all k 1,..., N. The above theorem shows that our weighted-semicircularity, ad semicircularity i free blocks L j, the j-th filtered probability spaces, for j W, are preserved by those i the free product Baach -probability space L W W. 7. EXAMPLE: FROM ORTHOGONAL RANK-1 PROJECTIONS ON l Z Let H l Z be the caoical l -Hilbert space with its orthoormal basis B H {ξ j } j Z satisfyig where ξ j ξ j1, ξ j δ j1,j, for all j 1, j Z, , 0,..., 0, 1, 0,..., 0,..., for all j Z, j-th ad where, is the usual l -ier product o H,..., t 1, t 0, t 1,...,..., s 1, s 0, s 1,... k t k s k. The o the operator algebra BH, oe ca aturally ie the rak-1 projectios q j by q j, ξ j ξ j, for all j Z. 7. i.e., for ay h H, oe has q j h h, ξ j ξ j, for all j Z, ad hece, if h Zt ξ H, with t C, the by 7., for all j Z. q j h t ξ, ξ j ξ j t j ξ j, Z

30 530 Ilwoo Cho ad Palle E.T. Jorgese Thus, oe ca get the family {q j } j Z of mutually orthogoal projectios q j s of 7., ad the correspodig C -subalgebra of the C -algebra BH, which is -isomorphic to C Z. Now, fix a arbitrary vector h 0 of H, C 7.3 h 0 Zt ξ H, with t C. For the fixed vector h 0 of H, ie ow a liear fuctioal ψ o BH by ψt T h 0, h 0, for all T BH. 7.4 By the sesqui-liearity of the ier product,, the morphism ψ of 7.4 is ideed a liear fuctioal o BH, i.e., BH, ψ forms a well-ied C -probability space. Observe that the liear fuctioal ψ of 7.4 o BH satisfies that by 7.1 ad 7., i.e., ψq j q j h 0, h 0 t j ξ j, h 0 t j ξ j, h 0 t j, ψq j t j, for all j Z, 7.5 where t j is the j-th coefficiet of a fixed vector h 0 of 7.3. Clearly, oe ca restrict the liear fuctioal ψ o BH to that o the C -subalgebra of BH, i the sese of 7.3, also deoted by ψ. The, for ay T j Zs j q j, with s j C, oe has ψ T j Zs j t j i C, by 7.5. So, oe ca determie the family {ψ j } j Z of liear fuctioals providig sectioized free probability o, where they are liear fuctioal satisfyig that ψ j s q ψ s j q j s j ψq j s j t j, 7.6 Z by 7.5, for all j Z, i.e., we obtai the followig sectioized C -probability spaces, for the liear fuctioals ψ j of 7.6. {, ψ j : j Z},

31 Baach -algebras geerated by semicircular elemets Now, costruct the radial projectio algebra L, L L C C[{l}] B C 7.7 of, by iig the radial operator l o by the liear trasformatio satisfyig l q j c q j + a q j q j+1 + q j 1, for all j Z, where c ad a are the creatio, respectively, the aihilatio o. O the Baach -algebra L of 7.7, ie the sectioized liear fuctioals ϕ j by liear morphisms satisfyig that ϕ j l q i ϕ j l q i ψ j l q i, 7.8 for all i, j Z, for all N, where ψ j are i the sese of 7.6. Note that, by 7.6, oe has that ψ j l q i ψ j k0 k0 c k k a k q i ψ k j qi+k k ψ k j q i+k k0 k0 k δj,i+k t j, with help of 7.5, for all N, for j Z. Defie ow a ew liear fuctioal τ j o the radial projectio algebra L of 7.7 by a liear morphism satisfyig τ j l q j ψ q j 1 ϕ j δ j,i [ ] + 1 l q j for all N, for all j Z, where ϕ j are i the sese of 7.8. The, by 4.18, if u j l q j L, τ j, the we obtai that τ j u j ω ψq j c ω t j c, 7.9, 7.10 by 7.5 ad 7.6, for all N, where ω 1, if is eve, ad ω 0, if is odd i N, for all j Z. Recall agai that t j are the j-th etry of the fixed vector h 0 of 7.3 i H l Z.

32 53 Ilwoo Cho ad Palle E.T. Jorgese Observatio 7.1. Let u j l q j be a free radom variable of the Baach -probability space L, τ j, where τ j are i the sese of 7.9, for all j Z. If the j-th etry t j of the vector h 0 of 7.3 is ozero, the u j is t 4 j -semicircular i L, τ j, for all j Z, by 7.10 ad 3.3. By the Möbius iversio of [14], oe obtais the equivalet free-distributioal data of 7.10 as follows: if t j 0 i h 0, the for all N, for j Z. k j u j,..., u j { t 4 j if, 0 otherwise, Therefore, by Observatio 7.1, we immediately get the followig result Observatio 7.. Let u j l q j i L, ad let U j t j u j L, for some j Z, where the j-th coefficiets t j are the ozero i the fixed vector h 0 of 7.3 i H. The U j is semicircular i L, τ j, for such j Z, by Sectio 5. The above Observatios 7.1 ad 7., we obtai the weighted-semicircularity ad correspodig semicircularity o L, for suitable j Z. From below, let us assume t j 0, as the j-th etry of a fixed Hilbert-space vector h 0 of 7.3, ad let u j l q j, ad U j 1 ψq j u j 1 t u j i L, j for such j Z. For the fixed vector h 0 H l Z of 7.3, ie a subset Supph 0 of Z by Supph 0 {j Z : t j 0 i h 0 }. 7.1 We call this set Supph 0 of 7.1, the support of h 0. Observatio 7.3. Let h 0 Z t ξ H be i the sese of 7.3, ad let Supph 0 be the support 7.1 of h 0 i Z. If j Supph 0 i Z, the oe has a t 4 j -semicircular elemet u j, ad a semicircular elemet U j i the j-th filtered probability space L j. Now, take a fiite sequece W j 1,..., j N of itegers j 1,..., j N i Supph 0, for some N N. The, by Observatio 7.3, oe ca take correspodig weighted-semicircular elemets u j1,..., u jn, ad semicircular elemets U j1,..., U jn which are ot ecessarily distict from each other i L. Costruct the direct product Baach -algebra L N geerated by W N-copies of L. O L N, ie a liear fuctioal τ W by a liear morphism satisfyig that for all N k1 T k L N, as i 6.9. N τ W T k 1 k1 N N τ jk T k, 7.13 k1

33 Baach -algebras geerated by semicircular elemets The we obtai a well-ied Baach -probability space, where τ W is i the sese of L N W L N, τ W, 7.14 Observatio 7.4. For a fixed fiite sequece W i Supph 0, ie a free radom variable T W of the Baach -probability space L W W of 7.14 by T W 1 U j L W W, 7.15 W j W where U j are our semicircular elemets of L j, for j W. The the free radom 1 variable T W of 7.15 is -semicircular i L W W W, by 6.1. Now, for a fixed fiite sequece W j 1,..., j N of mutually distict itegers j 1,..., j N i Supph 0, ie the tesor product Baach -algebra L N geerated by N-copies of L. O this Baach -algebra L N, ie liear fuctioals {ϕ W,k} N k1 by the liear morphisms satisfyig l1 N N ϕ W,k T k t 1 j k k1 t τ jl T l, 7.16 j l for all k 1,..., N, as i The oe ca get the family { L N deote W, k L N, ϕ W,k } N k of Baach -probability spaces, where ϕ W,k are i the sese of Observatio 7.5. For a fixed fiite sequece W j 1,..., j N of mutually distict itegers i Supph 0, oe ca costruct the family 7.17 of Baach -probability spaces L W W, k, for k 1,..., N. For a fixed k {1,..., N}, ie a free radom variable T W,k of L W W, k by T W,k I W,j1... I W,jk1 u jk I W,jk+1... I W,jN, 7.18 k-th where u jk is our t j -semicircular elemet of L, τ jk, ad I W,jk 1 L qĵk L, for k 1,..., N. The the operator T W,k of 7.18 is t 4 j k -semicircular i L W W, k by 6.33, for all k 1,..., N.

34 534 Ilwoo Cho ad Palle E.T. Jorgese Observatio 7.6. Uder the same coditios of Observatio 7.5, if we take a free radom variable S W,k I W,j1... I W,jk 1 U jk I W,jk+1... I W,jN, k-th i L N W, k, for k {1,..., N}, where U j k is our semicircular elemet of L, τ jk, the S W,k is semicircular i L N W, k, for k 1,..., N. Now, let W j 1,..., j N be a N-tuple of itegers i Supph 0, ad let L j k L, τ jk, for k 1,..., N, be the correspodig j k -filtered probability spaces of. Costruct the free product Baach -probability space, L N W L N, τ W N L, N τ jk k1 k1 Observatio 7.7. Let W be a fiite sequece of itegers i Supph 0, ad let L W W be i the sese of The the free reduced words u k ad U k with their legth-1 i the free blocks L j k are t 4 k-semicircular, respectively, semicircular i L W W, for all k 1,..., W, by 6.48 ad REFERENCES [1] M. Braa, K. Kirkpatrick, uatum groups ad geeralized circular elemets, Pacific J. Math , [] I. Cho, O dyamical systems iduced by p-adic umber fields, Opuscula Math , [3] I. Cho, p-adic free stochastic itegrals for p-adic weighted-semicircular motios determied by primes p, Libertas Math. New Series , [4] I. Cho, Free semicircular families i free product Baach -algebras iduced by p-adic umber fields, Complex Aal. Oper. Theory , [5] I. Cho, P.E.T. Jorgese, Semicircular elemets iduced by p-adic umber fields, Opuscula Math , [6] A. Deya, R. Schott, O the rough-paths approach to ocommutative stochastic calculus, J. Fuct. Aal , [7] T. Gillespie, Prime umber theorems for Raki-Selberg L-fuctios over umber fields, Sci. Chia Math , [8] I. Gomez, M. Caetagio, A quatum versio of spectral decompositio theorem of dyamical systems, uatum Chaos Hierarchy: Ergodic, Mixig ad Exact, Chaos Solitos Fractals ,

35 Baach -algebras geerated by semicircular elemets [9] N.A. Ivaov, T. Omlad, C -simplicity of free products with amalgamatio ad radical classes of groups, J. Fuct. Aal , [10] Y. Liu, Y. Yu, O a extesio of Stevic-Sharma operator from the geeral space to weighted-type spaces o the uit ball, Complex Aal. Oper. Theory , [11] M.A. Navarro, S. Villegas, Semi-stable radial solutios of p-laplace equatios i R N, Noliear Aal , [1] A. Nica, Free probability aspect of irreducible meadric systems, ad some related observatios about meaders, Ifiite Dimes. Aal. uatum Probab. Relat. Top , , pp. [13] F. Radulescu, Radom matrices, amalgamated free products ad subfactors of the C -algebra of a free group of osigular idex, Ivet. Math , [14] R. Speicher, Combiatorial theory of the free product with amalgamatio ad operator-valued free probability theory, Amer. Math. Soc. Mem [15] V.S. Vladimirov, I.V. Volovich, E.I. Zeleov, p-adic aalysis ad mathematical physics, Ser. Soviet & East Europea Math., vol. 1, World Scietific, [16] D. Voiculescu, K. Dykemma, A. Nica, Free radom variables, CRM Moograph Series, vol. 1, Amer. Math. Soc., Providece, 199. [17] Y. Yi, Z. Bai, J. Hu, O the semicircular law of large-dimesioal radom quaterio matrices, J. Theor. Probab , Ilwoo Cho choilwoo@sau.edu Sait Ambrose Uiversity Departmet of Mathematics ad Statistics 41 Ambrose Hall, 518 W. Locust St. Daveport, Iowa, 5803, USA Palle E.T. Jorgese palle-jorgese@uiowa.edu The Uiversity of Iowa Departmet of Mathematics Iowa City, IA , USA Received: Jue 1, 017. Accepted: August 1, 017.