Matrices induced by arithmetic functions, primes and

Spec. Matrices 2015; 3:123–154

Research Article

Open Access

Ilwoo Cho* and Palle E. T. Jorgensen

Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs DOI 10.1515/spma-2015-0012 Received February 19, 2015; accepted June 17, 2015

Abstract: In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corre[︀ ]︀ sponding subalgebra AG = C* α(G) of A. We construct a suitable representation of AG , determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we consider free probability on AG . Keywords: Directed Graphs; Graph Groupoids; Groupoid Dynamical Systems MSC: 05E15, 11G15, 11R04, 11R09, 11R47, 11R56, 46L10, 46L40, 46L53, 46L54.

1 Introduction We study free-probabilistic information on certain noncommutative sub-structures of the algebra A of all arithmetic functions. In particular, we generate such noncommutative sub-structures AG by acting graph groupoids G of directed graphs G on A. Then each element of AG is understood as an operator (which is a finite or an infinite matrix), acting on a Krein space from our representations of AG (See Sections 5, 6 and 7 below). Remark that our Krein-space representations are constructed by free probability models on AG . By computing free-moments and free-cumulants, we obtain free-distributional data on AG . Based on such data, we establish sufficient freeness conditions in AG . In more technical terms, we study groupoid actions acting on arithmetic functions, and consider subalgebras induced by the actions in the algebra consisting of all arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs, called graph groupoids. Graph groupoids are understood as groupoidal version of free groups in group theory. Starting from a fixed directed graph G, we construct the corresponding graph groupoids G, the algebraic structure containing all combinatorial properties of G, and then act G on the algebra A of arithmetic functions equipped with the usual functional addition and convolution (*). We especially act G injectively. i.e., define a groupoid-monomorphism α of G on A, such that α(G) is [︀ ]︀ a subgroupoid of A under (*), which is isomorphic to G. Then we construct a subalgebra AG = C* α(G) of A, determined both by combinatorial data of G (explained by algebraic data of G) and by functional data

*Corresponding Author: Ilwoo Cho: St. Ambrose Univ., Dept. of Math. & Stat., 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, U. S. A. / Univ. of Iowa, Dept. of Math., 14 McLean Hall, Iowa City, Iowa, 52242, U. S. A., E-mail: [email protected] Palle E. T. Jorgensen: St. Ambrose Univ., Dept. of Math. & Stat., 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, U. S. A. / Univ. of Iowa, Dept. of Math., 14 McLean Hall, Iowa City, Iowa, 52242, U. S. A., E-mail: [email protected]

© 2015 Ilwoo Cho and Palle E. T. Jorgensen, licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

- 10.1515/spma-2015-0012 Downloaded from PubFactory at 08/04/2016 09:03:50PM via free access

124 | Ilwoo Cho and Palle E. T. Jorgensen inherited from A. By establishing a suitable representation of AG , the corresponding free probabilistic model on AG will be studied for a fixed prime p, motivated from the results of [1], [3], [4], [7] and [8].

1.1 Background Recently, the connections between modern number theory and operator algebra theory have been studied in various different approaches (e.g., [1], [3], [4], [7], [8], [9], [10], [11] and [12]), via free probability (e.g., [14]). In particular, number-theoretic analytic objects; arithmetic functions, corresponding Dirichlet series, and Lfunctions (e.g., [13], [15] and [16]), have been considered as operators in free-probabilistic random variables (e.g., [4], [10] and [11]). Even though we are using free probability on them to establish new tools for studying those analytic objects, the commutativity of the functions provide somewhat trivial results. However, it is true that free-probabilistic models on such commutative structures help to develop the results easily and allow to apply operator theory for studying them (e.g., [7], [9], [10], [11] and [12]). Here, we construct “noncommutative,” conditional substructures of A by acting highly noncommutative algebraic object G on it. We apply similar free-probabilistic models to study G-depending arithmetic functions of A (as elements of AG ) as operators. To do that we establish certain Krein-space representation like [9] and [10]. Independently, directed graphs and graph groupoids have been studied in operator algebra theory, connected with dynamical system theory and representation theory, under free probability (e.g., [2], [5] and [6]). Motivated by the theory, we act a graph groupoid G on A and provide a suitable representation and a freeprobabilistic model depending both on a graph G and on a fixed prime p.

1.2 More About our Krein-Space Models Here, we specifically emphasize that our Krein-spaces of this paper are isomorphic Krein-spaces in the sense of [3] of those in the sense of [9] and [10]. In [9] and [10], the authors established Krein-space representations of arithmetic functions for primes. For a fixed prime p, each arithmetic function f is realized as a Krein-space operator on the “Krein subspace Kp ” of the Krein space K2 = C2 ⊕ C2− , where C2− is the anti-space of the 2-dimensional Hilbert space C2 in the sense of Section 2.3 below. In particular, the Krein space Kp is Krein-space isomorphic to the Krein subspace C ⊕ C in K2 (See [9] and [10]), which is re-characterized by the 2-dimensional indefinite inner product space of [3]. Therefore, in this paper, without loss of generality, we define the 2-dimensional indefinite inner product space of [3] by our Krein space Kp of [9] and [10] (See Proposition 3.2).

1.3 Overview In Section 2, we provide basic definitions and concepts; the arithmetic algebra A consisting of all arithmetic functions, directed graphs and corresponding graph groupoids, Krein spaces, and combinatorial free probability; used throughout this paper. In Section 3, we briefly review some fundamental results from our previous works used or applied in the text. In particular, we concentrate on how to construct the Krein-space representations of A for fixed primes. Under such representations, arithmetic functions are understood as Krein-space operators for fixed primes. In Section 4, we introduce a suitable Hilbert-space representation for a given graph groupoid G, induced by the vertices of the graph G generating G. Such a representation allows us to understand each groupoidal element of G as (finite or infinite) matrices over C. In Section 5, we act a fixed graph groupoid G on A pure-algebraically. And such a groupoid-action of G on A is realized as a generating process of subalgebras of a Banach *-algebra, under certain Krein-space


On graph-arithmetic algebras |

125

representations in Section 6. Under convolution, as the algebra multiplication, the arithmetic algebra A is a commutative algebra, but, by acting G on A, we obtain a noncommutative subalgebra AG of A, and each element of the subalgebra is realized as a Krein-space operator. We study free-probabilistic information on new noncommutative sub-structures AG of A induced by graphs G under Krein-space representation (in the sense of Sections 5 and 6) in Section 7. By computing free moments and free cumulants, free-distributional data of elements of AG are obtained. Also, we consider freeness conditions on AG , too.

2 Preliminaries In this section, we introduce basic definitions and backgrounds of our study.

2.1 The Arithmetic Algebra A Let f : N → C be a function whose domain is N, the natural numbers. Such a function f is called an arithmetic function. Denote the set of all arithmetic functions by A. This set A is a well-defined vector space over C, under the usual functional addition and the scalar product. Define now a convolution f1 * f2 by a function, (︁ n )︁ ∑︁ f1 * f2 (n) = f1 (d)f2 , for all f1 , f2 ∈ A, (2.1.1) d d|n

where “d | n” means “d divides n,” or “n is divisible by d,” or “d is a factor of n” in N, for all n ∈ N. Then this convolution (*) of (2.1.1) is associative and distributive with the functional addition, and commutative on A. So, the vector space A of all arithmetic functions forms a commutative algebra equipped with (*). Definition 2.1. We call the algebra A, the arithmetic algebra. For a fixed arithmetic function f ∈ A, one can define the corresponding C-valued function L f , as a series depending on both the arithmetic function f , and a C-variable s, L f : C → C, by L f (s) =

∞ ∑︁ f (n) . ns

(2.1.2)

n=1

We call L f (s), the (classical) Dirichlet series of an arithmetic function f . It is well-known that (2.1.2)′ (︀ )︀ (︀ )︀ L f1 (s) L f2 (s) = L f1 *f2 (s), for all f1 , f2 ∈ A. Let L = {L f : L f are in the sense of (2.1.2),∀f ∈ A}. It is not difficult to check that the set L is an algebra over C, with the usual functional addition and multiplication. Moreover, this algebra L is algebra-isomorphic to the arithmetic algebra A. Indeed, one can define an algebra-isomorphism Φ:A→L

(2.1.3)

defined by def

Φ(f ) = L f (s) in L, for all f ∈ A, by (2.1.2)′ .


126 | Ilwoo Cho and Palle E. T. Jorgensen Definition 2.2. We call the algebra L, the L-functional algebra. Depending on convolution (*) on the arithmetic algebra A, we have the (number-theoretic) Möbius inversion: the (arithmetic) Möbius function µ is the convolution-inverse of the constant function 1 defined by 1(n) = 1, for all n ∈ N. Equivalently, if 1A is the arithmetic identity function of A, which is the (*)-identity, {︃ 1 if n = 1 1A (n) = 0 otherwise, for all n ∈ N, then µ * 1 = 1A = 1 * µ. (Note the difference between 1 and 1A !) More generally, the Möbius inversion on arithmetic functions can be expressed as follows: f * 1 = g ⇐⇒ f = g * µ, for f , g ∈ A. Let L f1 and L f2 be corresponding Dirichlet series induced by arithmetic functions f1 and f2 , respectively. Then the (usual functional) product L f1 L f2 again becomes an L-function induced by the convolution f1 * f2 . i.e., L f1 (s)L f2 (s) = L f1 *f2 (s).

(2.1.4)

The constant arithmetic function 1 induces its corresponding L-function ζ (s) = L1 (s) =

∞ ∑︁ 1 , ks k=1

called the Riemann zeta (L-)function. The Riemann zeta function plays an important role not only in functional analysis but also in analytic number theory. If we establish a Dirichlet series m(s) by the series L µ (s) induced by the Möbius functional µ, then one has that ζ (s)m(s) = 1L = m(s)ζ (s), by (2.1.3) and (2.1.4), where 1L = L i d (s) = 1.

2.2 Graphs and Graph Groupoids Throughout this paper, we say G is a (directed) graph, if it is a combinatorial quadruple (V(G), E(G), s, r), consisting of the vertex set V(G), the edge set E(G), and the functions s and r from E(G) onto V(G), where the functions s and r are the source map and the range map, indicating the initial vertices and the terminal vertices of edges, respectively. Every graph G is depicted by the points (or node), representing vertices, and the oriented curves (or oriented lines) connecting points, representing (directed) edges. If e is an edge in E(G) with s(e) = v1 and r(e) = v2 , in V(G), then we write e = v1 e, or e = ev2 , or e = v1 e v2 . Whenever a graph G is fixed, one can construct the opposite-directed graph G−1 , with



127

V(G−1 ) = V(G), and E(G−1 ) = {e−1 : e ∈ E(G)}, with the rule: e = v1 ev2 in G ⇐⇒ e−1 = v2 e−1 v1 in G−1 . i.e., a new graph G−1 is obtained by reversing directions (or orientations) of edges. We call e−1 ∈ E(G−1 ), the shadow of e ∈ E(G), and similarly, the new graph G−1 is said to be the shadow of G. It is trivial that (G−1 )−1 = G. Let G1 and G2 be graphs. The union G1 ∪ G2 is defined by a new graph G with V(G) = V(G1 ) ∪ V(G2 ), and E(G) = E(G1 ) ∪ E(G2 ), which preserves the directions of G1 and G2 . The disjoint union G1 ⊔ G2 is similarly determined with empty intersections, V(G1 ) ∩ V(G2 ) = ∅ = E(G1 ) ∩ E(G2 ). However, in general, the union of graphs allows nonempty intersection. ̂︀ of a given graph G is defined by the union G ∪ G−1 . i.e., The shadowed graph G ̂︀ = V(G) ∪ V(G−1 ) = V(G) = V(G−1 ), V(G) and ̂︀ = E(G) ∪ E(G−1 ) = E(G) ⊔ E(G−1 ). E(G) ̂︀ be the shadowed graph of a given graph G. The set FP(G), ̂︀ consisting of all finite paths on G, ̂︀ is Let G ̂︀ All finite paths on G ̂︀ take the forms of products of edges in E(G). ̂︀ In other words, called the finite path set of G. ̂︀ is a subset of the set E(G) ̂︀ ′ , consisting of all words in E(G). ̂︀ In general, FP(G) ̂︀ becomes the finite path set FP(G) ′ ̂︀ ̂︀ ̂︀ a proper subset of E(G) . For instance, if e1 = v1 e1 v2 , and e2 = v3 e2 v4 in E(G), with v2 ≠ v3 in V(G), then e1 e2 ̂︀ ′ of all words in E(G)) ̂︀ is undefined in FP(G), ̂︀ as a finite path on G. ̂︀ (in the set E(G) ̂︀ for k ∈ N. Then one can extend the maps s and r on FP(G) ̂︀ as follows; Now, let w = e1 . . . e k ∈ FP(G), s(w) = s(e1 ), and r(w) = r(e k ). ̂︀ we also write If s(w) = v1 , and r(w) = v2 in V(G), w = v1 w, or w = wv2 , or w = v1 w v2 , ̂︀ for all w ∈ FP(G). ̂︀ by Define a set F+ (G)


128 | Ilwoo Cho and Palle E. T. Jorgensen ̂︀ def ̂︀ ∪ FP(G), ̂︀ F+ (G) = {∅} ∪ V(G) and define a binary operation (·) by def

w1 · w2 =

{︃

w1 w2 ∅

̂︀ if r(w1 ) = s(w2 ) in V(G) otherwise,

(2.2.1)

̂︀ where ∅ is the empty word, representing the “undefinedness of w1 · w2 , as finite paths for all w1 , w2 ∈ F+ (G), ̂︀ or vertices of G.” ̂︀ is called the admissibility. If w1 · w2 ≠ ∅ in F+ (G), ̂︀ then w1 and w2 are The operation (·) of (2.2.1) on F+ (G) said to be admissible; if w1 · w2 = ∅, then they are said to be not admissible. ̂︀ with v1 , v2 ∈ V(G), ̂︀ then v1 w, wv2 , v1 wv2 , and w, itself, are automatically If w = v1 wv2 , for w ∈ F+ (G), ̂︀ identified. i.e., v1 and w are admissible, w and v2 are admissible. So, one can axiomatize that: if v ∈ V(G), then v = vv = vvv, with s(v) = v = r(v). ̂︀ then Therefore, if v ∈ V(G), v k = v = v−1 = v−k = (v k )−1 , for all k ∈ N. The above axiomatization is indeed meaningful, because ̂︀ in F+ (G). ̂︀ V(G) = V(G−1 ) = V(G) ̂︀ does not contain ∅. For example, if a graph G is the one-vertex-k-loopRemark that, in some cases, F+ (G) ⃒ ⃒ + ̂︁ edge graph O k , then F (O k ) does not contain ∅. However, in general, if ⃒V(G)⃒ > 1, then ∅ is always contained ̂︀ So, if there is no confusion, we always assume the empty element ∅ is contained in F+ (G). ̂︀ in F+ (G). + ̂︀ + ̂︀ The algebraic pair F (G) = (F (G), ·), equipped with the admissibility, is called the free semigroupoid of ̂︀ G. ̂︀ define a natural reduction (RR) by For a fixed free semigroupoid F+ (G), w−1 w = v2 and ww−1 = v1 ,

(RR)

̂︀ \ {∅}, with v1 , v2 ∈ V(G). ̂︀ whenever w = v1 wv2 ∈ F+ (G) ̂︀ Then this reduction (RR) acts as a relation on the free semigroupoid F+ (G). ̂︀ / (RR), equipped with the inherited admissibility (·) from F+ (G), ̂︀ Definition 2.3. The quotient set G = F+ (G) is called the graph groupoid of G. ̂︀ = V(G) (e.g., [2] and The graph groupoid G of G is indeed a categorial groupoid with its (multi-)units V(G) ̂︀ [6]). The subset of G, consisting of all “reduced” finite paths, is denoted by FP r (G). i.e., ̂︀ ⊔ FP r (G), ̂︀ G = {∅} ⊔ V(G) set-theoretically, where ⊔ means the disjoint union. Notice that every graph groupoid G of a graph G is in fact a collection of all “reduced” words in the edge ̂︀ of the shadowed graph G ̂︀ under (RR). set E(G)

2.3 Krein Spaces and Krein-Space Operators Let H be a Hilbert space equipped with its (positive-definite) inner product H , i.e., it is the complete inner product space under the metric topology generated by the metric d H ,



d H (ξ1 , ξ2 ) = ‖ξ1 − ξ2 ‖H =

√︀

129

< ξ1 − ξ2 , ξ1 − ξ2 > H ,

for all ξ1 , ξ2 ∈ H, where ‖.‖H is the norm generated by H . For a given Hilbert space H, the anti-space H − of H is defined by the pair (H, − H ) of the same vector space H equipped with the negative-definite inner product − H , inducing the norm ‖.‖H− , ‖ ξ ‖H− =

√︀

|− < ξ , ξ >|, for all ξ ∈ H− ,

where |.| means the modulus on C, and the corresponding metric d H− . It is clear by definition that, as normed space (and hence, as metric spaces), the Hilbert space H and its anti-space H− are homeomorphic (topologically), but by the positive-definiteness of H in the sense that: < ξ , ξ > H = ‖ξ ‖2H ≥ 0, for all ξ ∈ H, the form − H is negative-definite in the sense that: − < ξ , ξ > H = − ‖ξ ‖2H ≤ 0, for all ξ ∈ H− . i.e., H and H− are identical from each other set-theoretically, and they are homeomorphic topologically, however, they are positive-definite respectively negative-definite as (indefinite) inner product spaces. Define the (topological algebraic) direct product K = H1 ⊕ H2− of a Hilbert space H1 and the anti-space − H2 of a Hilbert space H2 . We define an indefinite inner product [, ]K on K by def

[ξ1+ + ξ1− , ξ2+ + ξ2− ]K =

(︀

)︀ (︀ )︀ < ξ1+ , ξ2+ > H1 + − < ξ1− , ξ2− > H2 ,

(2.3.1)

for all ξ j+ ∈ H1 , and ξ j− ∈ H2− , where H j means the positive-definite inner products on the Hilbert spaces H j , for j = 1, 2. Then, indeed, the inner product [, ]K on K is “indefinite” in the sense that: [ξ , ξ ]K ∈ R, for ξ ∈ K. Definition 2.4. Let K be an indefinite inner product H1 ⊕ H2− , where H j are Hilbert spaces, for j = 1, 2, and H2− is the anti-space of H2 . Then, under the product metric topology induced by the metric d H1 ⊕ d H2− , generated by the metrics d H1 and d H2− , the space K is a topological indefinite inner product space equipped with the indefinite inner product [, ]K of (2.3.1). It is called the Krein space induced by Hilbert spaces H1 and H2 . For example, if H j = Cn j are finite-dimensional Hilbert spaces with their dimensions n j ∈ N, for j = 1, 2, then the Krein space K = H1 ⊕ H2− is well-defined, and this Krein space K is sometimes called the Pontryagin space. Bounded (or Continuous) linear transformations on Krein spaces are called Krein-space operators (simply operators if there is no confusion with Hilbert-space operators). Operator-theoretic properties of Krein-space operators are similarly defined as those of Hilbert-space operators. For example, a Krein-space operator T on a Krein space K is self-adjoint, if T * = T, where T * is the (Krein-space) adjoint of T, satisfying [Tx, y]K = [x, T * y]K in C, for all x, y ∈ K. For more details about Krein spaces and Krein-space operators, see [9], [10] and cited papers therein.

2.4 Free Probability In this section, we briefly introduce free probability (e.g., [12] and [17]). Free probability is one of a main branch of operator algebra theory, establishing noncommutative probability theory on noncommutative (and hence, on commutative) algebras (e.g., pure algebraic algebras, topological algebras, topological *-algebras, etc).


130 | Ilwoo Cho and Palle E. T. Jorgensen It has the original analytic approach in the sense of Voiculescu (See [12]), and alternative combinatorial approach in the sense of Speicher (See [17]). We use Speicher’s combinatorial free probability. Let A be an arbitrary algebra over the complex numbers C, and let ψ : A → C be a linear functional on A. Then the pair (A, ψ) is called a free probability space (over C). All operators a ∈ (A, ψ) are called free random variables. Remark that free probability spaces are dependent upon the choice of linear functionals. Let a1 , ..., a s be free random variables in a (A, ψ), for s ∈ N. The free moments of a1 , ..., a s are determined by the quantities ψ(a i1 ...a i n ), for all (i1 , ..., i n ) ∈ {1, ..., s}n , for all n ∈ N. And the free cumulants k n (a i1 , ..., a i n ) of a1 , ..., a s is determined by the Möbius inversion; )︂ ∑︁ ∑︁ (︂ (︀ )︀ k n (a i1 , ...,a i n ) = ψ π (a i1 , ...,a i n )µ(π, 1n ) = Π ψ V (a i1 , ...,a i n )µ 0|V| , 1|V| , π∈NC(n)

π∈NC(n)

(2.4.1)

V∈π

for all (i1 , ..., i n ) ∈ {1, ..., s}n , for all n ∈ N, where ψ π (...) means the partition-depending moments, and ψ V (...) means the block-depending moment; for example, if π0 = {(1, 5, 7), (2, 3, 4), (6)} in NC(7), with three blocks )︁ (2, 3, 4), and (6), then (︁ (1, 5, 7), ψ π0 a ri11 , ...,a ri77 = ψ(1,5,7) (a ri11 , ..., a ri77 ) ψ(2,3,4) (a ri11 , ..., a ri77 ) ψ(6) (a ri11 , ..., a ri77 ) = ψ(a ri11 a ri55 a ri77 ) ψ(a ri22 a ri33 a ri44 ) ψ(a ri66 ). Here, the set NC(n) means the noncrossing partition set over {1, ..., n}, which is a lattice with the inclusion ≤, such that def

θ ≤ π ⇐⇒ ∀ V ∈ θ, ∃ B ∈ π, s.t., V ⊆ B, where V ∈ θ or B ∈ π means that V is a block of θ, respectively, B is a block of π, and ⊆ means the usual set inclusion, having its minimal element 0n = {(1), (2), ..., (n)}, and its maximal element 1n = {(1, ..., n)}. Especially, a partition-depending free moment ψ π (a, ..., a) is determined by (︁ )︁ ψ π (a, ..., a) = Π ψ a|V| , V∈π

where |V | means the cardinality of V . ∞ Also, µ is the Möbius functional from NC × NC into C, where NC = ∪ NC(n). i.e., it satisfies that n=1

µ(π, θ) = 0, for all π > θ in NC(n),

(2.4.2)

and µ(0n , 1n ) = (−1)n−1 c n−1 , and

∑︀

µ(π, 1n ) = 0,

π∈NC(n)

for all n ∈ N, where (︃ ck =

1 k+1

2k k

)︃ =

1 (2k)! k+1 k!k!

means the k-th Catalan numbers, for all k ∈ N. Notice that since each NC(n) is a well-defined lattice, if π < θ are given in NC(n), one can decide the “interval” [π, θ] = {δ ∈ NC(n) : π ≤ δ ≤ θ},



131

and it is always lattice-isomorphic to [π, θ] = NC(1)k1 × NC(2)k2 × ... × NC(n)k n , for some k1 , ..., k n ∈ N, where NC(l)k t means “l blocks of π generates k t blocks of θ,” for k j ∈ {0, 1, ..., n}, for all n ∈ N. By the multiplicativity of µ on NC(n), for all n ∈ N, if an interval [π, θ] in NC(n) satisfies the above set-product relation, then we have n

µ(π, θ) = Π µ(0j , 1j )k j . j=1

(For details, see [17]). By the very definition of free cumulants, one can get the following equivalent Möbius inversion; ∑︁ (︀ )︀ (︀ )︀ ψ a i1 a i2 ...a i n = k π a i1 , ...,a i n , (2.4.3) π∈NC(n)

where k π (a i1 , ..., a i n ) means the partition-depending free cumulant, for all (a i1 , ..., a i n ) ∈ {a1 , ..., a s }n , for n ∈ N, where a1 , ..., a s ∈ (A, ψ), for s ∈ N. Under the same example; π0 = {(1, 5, 7), (2, 3, 4), (6)} in NC(7); we have (︀ )︀ (︀ )︀ (︀ )︀ k π0 (a i1 , ..., a i7 ) = k(1,5,7) a i1 , ...,a i7 k(2,3,4) a i1 , ...,a i7 k(6) a i1 , ...,a i7 (︀ )︀ (︀ )︀ = k3 a i1 , a i5 , a i7 k3 a i2 , a i3 , a i4 k1 (a i6 ). In fact, the free moments of free random variables and the free cumulants of them provide equivalent free distributional data. For example, if a free random variable a in (A, ψ) is a self-adjoint operator in the von Neumann algebra A in the sense that: a* = a, then both free moments {ψ(a n )}∞ n=1 and free cumulants { k n (a, ..., a)}∞ give its spectral distributional data. n=1 However, their uses are different case-by-case. For instance, to study the free distribution of fixed free random variables, the computation and investigation of free moments is better, and to study the freeness of distinct free random variables in the structures, the computation and observation of free cumulants is better (See [17]). Definition 2.5. We say two subalgebras A1 and A2 of A are free in (A, ψ), if all “mixed” free cumulants of A1 and A2 vanish. Similarly, two subsets X1 and X2 of A are free in (A, ψ), if two subalgebras A1 and A2 , generated by X1 and X2 respectively, are free in (A, ψ). Two free random variables x1 and x2 are free in (A, ψ), if {x1 } and {x2 } are free in (A, ψ). Suppose A1 and A2 are free subalgebras in (A, ψ). Then the subalgebra A generated both by these free subalgebras A1 and A2 is denoted by A

denote

=

A1 *C A2 .

Inductively, assume that A is generated by its family {A i }i∈Λ of subalgebras, and suppose the subalgebras A i are free from each other in (A, ψ), for i ∈ Λ. Then we call A, the free product algebra of {A i }i∈Λ (with respect to ψ), i.e., A = *C A i i∈Λ

is the free product algebra of {A i }i∈Λ (with respect to ψ). In the above text, we concentrated on the cases where (A, ψ) is a “pure-algebraic” free probability space. Of course, one can take A as a topological algebra, for instance, A can be a Banach algebra. In such a case, ψ


132 | Ilwoo Cho and Palle E. T. Jorgensen is usually taken as a “bounded (or continuous)” linear functional (under topology). Similarly, A can be taken as a *-algebra, where (*) means here the adjoint on A, satisfying that: a** = a, for all a ∈ A, (a1 + a2 )* = a*1 + a*2 , (a1 a2 )* = a*2 a*1 , for all a1 , a2 ∈ A. Then we put an additional condition on ψ, called the (*)-relation on ψ: ψ(a* ) = ψ(a), for all a ∈ A, where z means the conjugate of z, for all z ∈ C. Finally, the algebra A can be taken as a topological *-algebra, for example, a C* -algebra or a von Neumann algebra. Then usually we take a linear functional ψ satisfying both the boundedness and the (*)-relation on it. In the following, to distinguish the differences, we will use the following terms; (i) if A is a Banach algebra and if ψ is bounded, then (A, ψ) is said to be a Banach probability space, (ii) if A is a *-algebra and if ψ satisfies the (*)-relation, then (A, ψ) is called a *-probability space, (iii) if A is a C* -algebra and if ψ is bounded with (*)-relation, then (A, ψ) is a C* -probability space, (iv) if A is a von Neumann algebra and if ψ is bounded with (*)-relation, then (A, ψ) is a W * -probability space.

3 Free Probability and Representations on A Determined by Primes Let A be the arithmetic algebra consisting of all arithmetic functions, and let p be an arbitrarily fixed prime. Then the point-evaluation g p at p is a well-defined linear functional on A, i.e., def

g p (f ) = f (p), for all f ∈ A.

(3.1)

i.e., the pair (A, g p ) forms a free probability space in the sense of Section 2.4. Definition 3.1. A free probability space (A, g p ) of the arithmetic algebra A and a linear functional g p of (3.1) is said to be the arithmetic p-prime probability space. By the commutativity of A, the freeness on (A, g p ) is not so interesting in operator algebra point of view, however, it provides a new model to study number-theoretic objects by using operator-algebraic and operatortheoretic tools, filterized by a fixed prime p. Proposition 3.1. (See [3] and [8]) Let (A, g p ) be the arithmetic p-prime probability space. g p (f1 * f2 ) = f1 (1)f2 (p) + f1 (p)f2 (1), for all f1 , f2 ∈ (A, g p ), (︂ gp

n

* fj

j=1

)︂ =

n ∑︁ j=1

(︂ f j (p)

Π

l≠j∈{1,...,n}

)︂ f l (1) , forallf1 , . . . , f n ∈ (A, g p ), for all n ∈ N.

(3.2)

(3.3)

(︁ )︁ g p f (n) = nf (1)n−1 f (p) = nf (1)n−1 g p (f ), for all f ∈ (A, g p ), where f (n) means the convolution f * . . . * f of − copies of f , foralln ∈ N. Clearly, the computation (3.4) is proven by (3.3), and the computation (3.3) is obtained by (3.2) inductively. We can check that our linear functional g p acts like a derivation on A.



133

As we have discussed in [3], if we fix an arithmetic p-prime probability space (A, g p ), then all arithmetic functions (as free random variables) f are classified by two quantities f (1) and f (p). It shows that, the arithmetic algebra A is classified by the equivalence relation Rp ; )︀ (︀ )︀ def (︀ f1 Rp f2 ⇐⇒ f1 (1), f1 (p) = f2 (1), f2 (p) ,

(3.4)

for all f1 , f2 ∈ (A, g p ). i.e., we have equivalence classes [f ]Rp , [f ]Rp = {h ∈ (A, g p ) : h Rp f }, where Rp is in the sense of (3.5). Construct now a quotient algebra Ap = A/Rp = {[f ]Rp : f ∈ A}.

(3.5)

It is indeed an algebra because the addition [f1 + f2 ]Rp = [f1 ]Rp + [f2 ]Rp , and the convolution [f1 * f2 ]Rp = [f1 ]Rp * [f2 ]Rp are well-defined. Notice here that, by (3.5), the convolution is well-defined “for the fixed p,” as above. Definition 3.2. The quotient algebra Ap is called the (arithmetic) p-prime Banach algebra. Furthermore, by [3], the quotient algebra Ap is bijective (or equipotent) to the 2-dimensional space C2 , set-theoretically. So, as a 2-dimensional set, Ap is complete under the quotient norm because it is finitedimensional. (Moreover, by the finite-dimensionality, all norms are equivalent.) Define now a 2-dimensional space Kp by the set C2 , having the usual 2-dimensional vector addition, the usual scalar product, equipped with an indefinite inner product [, ]p defined by [(t1 , t2 ), (s1 , s2 )]p = t1 s2 + t2 s1 ,

(3.6)

for all (t1 , t2 ), (s1 , s2 ) ∈ C2 . Then, indeed, it is an “indefinite” inner product on C2 , since )︀ (︀ [(t1 , t2 ), (t1 , t2 )]p = 2 Re t1 t2 ∈ R. i.e., the space Kp = (C2 , [, ]p ) is an indefinite inner product space. Moreover, one has that: Proposition 3.2. (See [9]) The space Kp = (C2 , [, ]p ) with the indefinite inner product [, ]p of (3.6) is a Krein space in the Section 2.3. More precisely, Kp is a Krein subspace of the Krein space K2 , K2 = (C2 , 2 ) ⊕ (C2 , − 2 ), where 2 is the usual (positive-definite) inner product on C2 . Sketch of the Proof For more details about the proof, see Sections 5 and 6 of [9]. Let K2 be the Krein space as above. Define a Krein subspace K0 of K2 by K0 = ∆2 ⊕ ∆−2 , where ∆2 = {(t, t) ∈ C2 : t ∈ C},


134 | Ilwoo Cho and Palle E. T. Jorgensen and ∆−2 = {(t, −t) ∈ C2− : t ∈ C}. Then ∆2 and ∆−2 are 1-dimensional subspaces of C2 and C2− , respectively. So, Krein

Kp = K0 in K2 . It is not difficult to check that our p-prime Banach algebra Ap acts on the Krein space Kp (See [9] and [10]). Notation and Assumption In the rest of this paper, we denote each element [f ]Rp of the p-prime Banach algebra Ap simply by f , if there is no confusion. Also, we understand all elements (t, s) of Kp by (h(1), h(p)), for some h ∈ Ap . By [9] and [10], we act Ap on Kp by an algebra-action α p , where (︃ )︃ f (1) 0 α p (f ) = on Kp , (3.7) f (p) f (1) (︃ )︃ (︃ )︃ (︃ )︃ (︀ )︀ f (1) 0 h(1) f (1)h(1) for all f ∈ Ap . It shows that α p (f ) (h(1), h(p)) = = = f (p) f (1) h(p) f (p)h(1) + f (1)h(p) (︃ )︃ (f * h) (1) , i.e., (f * h)(p) (︀ )︀ (︀ )︀ α p (f ) h(1), h(p) = f * h(1), f * h(p) , (3.8) (︀ )︀ for all f ∈ Ap and h(1), h(p) ∈ Kp . Moreover, we have that: (︀ )︀ (︀ )︀ α p (f1 ) α p (f2 ) = α p (f1 * f2 ) on Kp , for all f1 , f2 ∈ Ap , (3.9) (︀

α p (f )

)︀*

(︁ )︁ = α p f * , for all f ∈ Ap ,

(3.10)

where f * = [f * ]Rp , such that f * (n) = f (n), for all n ∈ N, where z means the conjugate of z, for all z ∈ C. Based on the above discussion, one can understand all free random variables of the arithmetic p-prime probability space (A, g p ) as Krein-space operators acting on Kp , for fixed primes p. Definition 3.3. The pair (Kp , α p ) of the Krein space Kp and the algebra-action α p is called the p-prime Kreinspace representation of the p-prime Banach algebra Ap . And we denote the subalgebra α p (Ap ) in the operator Banach *-algebra B(Kp ) by Ap , i.e., Ap = α p (Ap ) in B(Kp ).

(3.11)

Let Ap be the Banach *-subalgebra of B(Kp ) in the sense of (3.11). Define now a linear functional φ p : Ap → C (︃(︃ (︀

)︀

φ p α p (f ) = φ p

f (1) f (p)

0 f (1)

)︃)︃

def

= π2,1

by (︃(︃

f (1) f (p)

0 f (1)

)︃)︃ = f (p) = g p (f ),

where π ij : M n (C) → C, for i, j ∈ {1, ..., n}, for n ∈ N means “taking (i, j)-entry of a matrix.” We will use this linear functional in Section 7 later.


(3.12)


135

4 Vertex-Representations Induced by Graph Groupoids Let G be a directed graph with its graph groupoid G. In the rest of this paper, we restrict our interests to the ⃒ ⃒ ⃒ ⃒ cases where given graphs G are finite in the sense that ⃒V(G)⃒ < ∞, and ⃒E(G)⃒ < ∞. Assumption In the rest of this paper, all given graphs are finite and connected. The algebra def

MG = C[G]

(4.1)

generated by G is called the graph-groupoidal algebra of G. i.e., the algebra MG is the polynomial ring over C, consisting of all elements ∑︀

t w w, with t w ∈ C,

w∈G

where

∑︀

means “finite sum.” For example, if e1

e

G0 = v1 • • →3 • , e2

v2

e4

then 5 T = 3e1 e2 e1 e3 + (2i)e−1 3 − v2 + e4

(4.1′ )

√

is an element of MG , where i = −1 in C. One can understand MG as a *-algebra having its adjoint; w* = w−1 , the shadow of w, for all w ∈ G, more generally, )︂*

(︂ ∑︀ w∈G

tw w

=

∑︀ w∈G

t w w* =

∑︀

t w w−1 .

w∈G

For instance, if T ∈ MG0 is given as in (4.1)′ , then −1 −1 −1 −1 5 T * = 3e−1 3 e 1 e 2 e 1 +(−2i) e 3 − v 2 + (e 4 ) in MG0 .

Now, for the vertex set V(G), let’s index vertices, i.e., identify V(G) = {v1 , v2 , ..., v|V(G)| }.

(4.2)

Under the above indexing process on V(G), one can have the corresponding indexes on the edge set E(G), ⃒ {︀ E(G) = e ij:l ⃒e ij:l is a directed edge connecting v i to v j , for v i , v j ∈ V(G), with l = 1, ..., k ij in N, }︀ where k ij means the cardinaly of edges connecting v i to v j . (4.2′ ) Define now a Hilbert space H G by the l2 -space l2 (V(G)) over the indexed vertex set V(G) of (4.2). i.e., (︁ )︁ (︀ )︀ def (4.3) H G = l2 V(G) = l2 {v1 , ...,v|V(G)| } . In fact, the construction of H G is free from the choice of indexes (4.2) on V(G), because different indexings on V(G) provide graph-isomorphic graphs with G, which are identified with G, by (4.2)′ . Thus it suffices to fix an arbitrarily chosen indexes on V(G) as in (4.2), providing corresponding indexes on E(G) of (4.2)′ . The Hilbert space H G of (4.3) has its orthonormal basis BH G ,


136 | Ilwoo Cho and Palle E. T. Jorgensen BH G = {ξ v ∈ H G : v ∈ V(G)}, satisfying < ξ v i , ξ v j > G = δ i,j , for all v i , v j ∈ V(G), where δ means the Kronecker delta, where G means the canonical l2 -space inner product on H G = l2 (V(G)). Definition 4.1. We call H G of (4.3), the vertex Hilbert space of G. Let H G be the vertex Hilbert space of G, with ⃒ ⃒ N G = ⃒V(G)⃒ inN∞ = N ∪ {∞}.

(4.4)

Then the graph groupoid G of G acts on H G via a groupoid action α G , satisfying the following conditions (4.5), (4.6) and (4.7); ̂︀ = V(G), for k ∈ {1, ..., N G }, if v k ∈ V(G)

(4.5)

then α G (v k ) = [t ij ]N G ×N G , an (N G × N G )-matrix on H G , with {︃

t kk = 1 0

t ij =

ifi = k = j otherwise,

for all i, j ∈ {1, ..., N G }, where N G is the cardinality of V(G) in the sense of (4.4); if e sr:l ∈ E(G), for s ≠ r ∈ {1, ..., N G }

(4.6)

and l ∈ {1, ..., k sr } in the sense of (4.2)′ , then α G (e sr:l ) = [t ij ]N G ×N G , an (N G × N G )-matrix on H G , with {︃ t ij =

ωl 0

if i = s and j = r otherwise,

where ω l is the l-th root of unity of the C-polynomial equation z k ij = 1; if e ss:l ∈ E(G), for s ∈ {1, ..., N G } and l ∈ {1, ..., k ss } as in (4.2)′ (equivalently, if e ss:l is a loop-edge connecting a vertex v s to itself), then α G (e ss:l ) = [t ij ]N G ×N G , an (N G × N G )-matrix on H G , with {︃ t ij =

ω l+1 0

if i = s = j otherwise,

where “ω l+1 ≠ 1” is the (l + 1)-st root of unity of the C-polynomial equation z k ss +1 = 1.


(4.7)


137

̂︀ and e ij:l ∈ E(G) ̂︀ in G, then Of course, by (4.5), (4.6) and (4.7), if v j ∈ V(G) α G (v j )* = α G (v j ),

(4.8)

and (︀ )︀* [︀ ]︀ α G (e ij:l )* = [t ij ]N G ×N G = t ji N

G ×N G

(︁ )︁ = α G e−1 ij:l ,

for all i, j ∈ {1, ..., N G } and l ∈ {1, ..., k ij }. So, one can extend α G to all reduced finite path as follows; if w is a reduced finite path ee′ · · · e′′ in G, then )︁ (︁ )︁ (︁ )︁ (︀ )︀ (︁ α G (w) = α G ee′ · · · e′′ = α G (e) α G (e′ ) · · · α G (e′′ ) (4.9) on H G . Also, we axiomatize that α G (∅) = O N G , the zero matrix on H G .

(4.10)

i.e., the action α G of the graph groupoid G, satisfying (4.5), (4.6) and (4.7), on the vertex Hilbert space H G is well-determined, by (4.8), (4.9) and (4.10). Proposition 4.1. The morphism α G from G to (Hilbert-space) operators in H G is a well-determined groupoidaction. Proof. Let w ∈ G. Then α G (w) are well-defined matrices on H G , by (4.5) through (4.10). Moreover, if w1 , w2 ∈ G, then α G (w1 w2 ) = α G (w1 )α G (w2 ),

(4.11)

̂︀ by (4.9) and by the fact that all elements of G are the (RR)-reduced words in E(G). Indeed, if w1 and w2 are admissible, equivalently, if w1 w2 ≠ ∅ in G, then the formula (4.11) holds, by (4.8) and (4.9). Otherwise, if w1 and w2 are not admissible, equivalently, if w1 w2 = ∅ in G, then the formula (4.11) also hods, because α G (w1 w2 ) = α G (∅) = O N G = α G (w1 ) α G (w2 ), by (4.5), (4.6), (4.7) and (4.10). By the above proposition, we conclude that the pair (H G , α G ) of the vertex Hilbert space H G and the graphgroupoid-action α G of G on H G is a well-defined groupoid representation of G. Definition 4.2. The groupoid representation (H G , α G ) of the graph groupoid G of a graph G is called the vertex-representation of G. Under linearity, the above vertex-representation (H G , α G ) of G is extended to the algebra-representation of the graph-groupoidal algebra MG of G. i.e., one can have the extended morphism, also denoted by α G , from MG to the operator algebra B(H G ), consisting of all bounded (or continuous) linear operators (unitarily equivalent to (finite or infinite) matrices) on H G ; (︃ )︃ ∑︁ def ∑︁ αG tw w = t w α G (w). (4.12) w∈G

w∈G

Then the action α G of (4.12) is an algebra-action of MG acting on the vertex Hilbert space H G . Definition 4.3. The algebra representation (H G , α G ), where α G now is in the sense of (4.12), of the graphgroupoidal algebra MG is called the vertex-representation of MG .


138 | Ilwoo Cho and Palle E. T. Jorgensen The above observation shows that the graph-groupoidal algebra is embedded in the operator algebra B(H G ). Definition 4.4. Let (H G , α G ) be the vertex-representation of the graph-groupoidal algebra MG of a graph G. The closed subalgebra M G , def

M G = α G (MG ) in B(H G )

(4.13)

is called the graph-groupoidal C* -algebra of G, where X means the operator-norm-topology closure of subsets X of B(H G ). Under the vertex-representation (H G , α G ) of the graph groupoid G of a given graph G, one can establish the corresponding graph-groupoidal C* -algebra M G as a C* -subalgebra of B(H G ). Now, let M G be the graph-groupoidal C* -algebra in the sense of (4.13). Define now a C* -subalgebra D G of M G by [︁ ]︁ ̂︀ ⊆ M G in B(H G ). D G = C α G (V(G)) It is trivial to realize that all elements of this C* -subalgebra D G are diagonal matrices (or operators), by (4.5). Thus, we call D G the diagonal subalgebra of M G . Then, one can define a conditional expectation EG : MG → DG (︃ EG

)︃ ∑︁

t w α G (w)

def

=

w∈G

by

∑︁

t v α G (v).

(4.14)

̂︀ v∈V(G)

For example, if T is in the sense of (4.1)′ in MG0 , then (︀ )︀ 5 E G0 (T) = E G0 3e1 e2 e1 e3 + (2i)e−1 3 − v 2 + e 4 = −v 2 , in D G0 . Indeed, the morphism E G of (4.14) is a well-defined conditional expectation because (i) E G is onto D G , and hence, it is a bounded linear transformation, (ii) it satisfies E G (d) = d, for all d ∈ D G , (iii) it also satisfies that E G (d1 md2 ) = d1 E G (m) d2 , for all d1 , d2 ∈ D G , and m ∈ M G , and (iv) for any m ∈ M G , (︀ )︀ E G m* = E G (m)* , by (4.5), (4.6), (4.7), (4.8) and (4.9). Based on the conditional expectation E G of (4.14), we define a linear functional tr G on M G by tr G = tr|V(G)| ∘ E G ,

(4.15)

where tr k means the usual matricial trace on M k (C), for all k ∈ N. So, by (4.15), one has (︃ )︃ (︂ )︂ (︂ (︂ )︂)︂ ∑︀ ∑︀ ∑︀ ∑︀ tr G t w α G (w) = tr|V(G)| E G t w α G (w) = tr|V(G)| t v α G (v) = tv , w∈G

w∈G

̂︀ v∈V(G)

̂︀ v∈V(G)

We will use this linear functional tr G of (4.15) later in Section 7.



139

5 Graph Groupoids Acting on Arithmetic Functions Let G be a fixed finite, connected directed graph with its graph groupoid G, generating the graph-groupoidal C* -algebra M G in the sense of (4.13) under the vertex-representation (H G , α G ) of M G . We act G (and hence, M G ) on the p-prime arithmetic algebra Ap of all arithmetic functions up to the equivalence relation Rp . Recall that Rp and Ap are in the sense of (3.5) and (3.11), respectively. Notation and Assumption In the rest of this paper, as we assumed in Section 3, all elements [f ]Rp of Ap , the equivalence classes f ∈ A under Rp , are simply denoted by f of Ap , for f ∈ A. First, define an injective map ̂︀ → Ap , u G : E(G)

(5.1)

u G (e) = f e in Ap ⇐⇒ u G (e−1 ) = f e* inAp ,

( 5.1′ )

with additional rules (5.1)′ and (5.1)′′ below:

̂︀ where f * means the adjoint of f , which is the arithmetic function satisfying for all e ∈ E(G), f * (n) = f (n), for all n ∈ N, and ⎧ ⎪ if i ≠ j ∈ {1, ...,N G } ⎨ ω l f ij u G (e ij:l ) = ⎪ ⎩ω f if i = j ∈ {1, ...,N G }, l+1 ij

(5.1′′ )

where f ij is fixed for all l, and ω l and ω l+1 are in the sense of (4.6) and (4.7), respectively, for i, j ∈ {1, ..., N G } ⃒ ⃒ and l ∈ {1, ..., k ij }, where N G = ⃒V(G)⃒ is in the sense of (4.4). ̂︀ is a countable discrete set and Ap is an unWe can always define such an injective map because E(G) countable continuous set. (︁ )︁ ̂︀ of an injective map u G of (5.1) satisfying (5.1)′ and (5.1)′′ in Ap by EG . Let’s denote the image u G E(G) Define then a (pure-algebraic) subspace VG of Ap by def

VG = spanC EG inAp .

(5.2)

Then each element of this vector space VG is expressed by ∑︀

te fe

(over a finite sum).

̂︀ e∈E(G)

Now, let A0 be the minimal subalgebra of the arithmetic algebra A containing the subspace VG of (5.2). i.e., A0 = C* [VG ], where C* [X] mean subalgebras of Ap generated by subsets X of Ap under vector-multiplication (*). Define now a binary operation (*G ) on A0 by the operation satisfying that: {︃ f e1 * f e2 if e1 e2 ≠ ∅ def denote f e1 *G f e2 = f e1 e2 = 0A0 = 0A otherwise,

(5.3)

̂︀ where (*) on the far-right-hand side of (5.3) means the usual convolution on Ap . More for all e1 , e2 ∈ E(G), precisely, ⎛ ⎞ ⎛ ⎞ ∑︁ ∑︁ ∑︁ ⎝ t e f e ⎠ *G ⎝ sw hw ⎠ = (5.3′ ) ( t e s w ) ( f e *G h w ) , ̂︀ e∈E(G)

̂︀ w∈E(G)

̂︀ 2 (e,w)∈E(G)


140 | Ilwoo Cho and Palle E. T. Jorgensen where the summands f e *G h w on the right-hand side of (5.3)′ satisfy (5.3). ̂︀ Also, by the notation we defined in (5.3), if w = e1 ... e n is a reduced finite path in G with e1 , ..., e n ∈ E(G), then n

n

j=1

j=1

f w = *G f e j = * f e j inA0 .

(5.3′′ )

Clearly, if w = ∅ in G, then f w = f∅ = 0A in A0 . Now, inside A0 , we set a subset AG , satisfying (5.3) and (5.3)′ , i.e., establish the subalgebra AG of Ap (as a subalgebra of A0 ) by def

A G = C * G [V G ] = C * G [E G ]

(5.4)

in A0 ⊆ Ap , where C*G [X] means the subalgebra of A generated by subsets X of A, equipped with its algebra multiplication (*G ) of (5.3), satisfying (5.3)′ and (5.3)′′ . Remark 5.1. Notice that the arithmetic algebra A is a commutative algebra under the convolution (*), while the subalgebra AG of (5.4) is a noncommutative algebra under the conditional convolution (*G ), determined ̂︀ (and hence, that on the graph groupoid G) of a given graph G. by the admissibility on the shadowed graph G ̂︀ Indeed, if e1 , e2 ∈ E(G), and assume w1 = e1 e2 ≠ ∅ and w2 = e2 e1 = ∅ in G. For instance, if the shadowed ̂︀ of G contains graph G e

e

1 2 · · · −→ • −→ • −→ • −→ · · ·,

then e1 and e2 are admissible, but e2 and e1 are not admissible in G. So, even though f w1 = f e1 *G f e2 = f e1 * f e2 is a nonzero arithmetic function in AG , but f w2 = f e2 *G f e1 = f∅ = 0A ≠ f e2 * f e1 in AG . So, indeed the G-arithmetic algebra AG is highly noncommutative in general. Remark again that, in the p-prime arithmetic algebra Ap , f e1 * f e2 = f e2 * f e1 . However, in the subalgebra AG , f e1 *G f e2 ≠ f e2 *G f e1 . Definition 5.1. We call the subalgebra AG of the p-prime arithmetic algebra Ap in the sense of (5.4), the G-arithmetic (sub)algebra (of Ap ). And we call the operation (*G ) on AG , the G-convolution. By the very construction of the G-arithmetic algebra AG , if f ∈ AG , then it is expressed by f=

∑︀

t w f w in AG ,

w∈G

by (5.3)′ , where the sum on the right-hand side is a finite sum, and where f w are in the sense of (5.3)′′ . ̂︀ then one can understand it by e−1 e, for some e ∈ E(G), ̂︀ whenever e = ev in G. So, Note that, if v ∈ V(G), −1 by (5.1), if v = e e is a vertex in G, then f v = f e−1 e = f e−1 *G f e = f e−1 * f e = f e* * f e in AG , by (5.3) and (5.3)′ .


On graph-arithmetic algebras

| 141

̂︀ with e1 = e1 v and e2 = e2 v, then we have If e1 ≠ e2 in E(G) f v = f e−1 e1 = f e−1 e2 , 1

2

by (5.1)′ and (5.1)′′ , since |ω l |2 = 1 = |ω l+1 |2 . By construction of our graph-arithmetic algebras in the arithmetic algebra Ap , one can realize that the graph groupoid G is acting on Ap by an injective map u G of (5.1) satisfying (5.1)′ and (5.1)′′ . Define now a morphism U G from G into the functions on AG by a morphism satisfying that: def

U G (w) (f w′ ) = u G (w) *G f w′ = f w *G f w′ = f ww′ , for all w, w′ ∈ G. One may extend U G under linearity from MG into the functions on AG by (︃ )︃ ∑︁ def ∑︁ t w U G (w)(f w′ ), UG t w w ( f w′ ) = w∈G

(5.5)

(5.5′ )

w∈G

for all w′ ∈ G, where the summands U G (w)(f w′ ) of the right-hand side of (5.5)′ are in the sense of (5.5). Theorem 5.1. Let G be a graph with its graph groupoid G, and let AG be the G-arithmetic algebra in the sense of (5.4). The triple (G, AG , U G ) is a well-determined groupoid-dynamical system of G acting on A (and hence, on AG ) via the groupoid-action U G of G in the sense of (5.5). Proof. It suffices to show that the morphism U G of (5.5) is a well-defined groupoid-action of G acting on the G-arithmetic algebra AG . Clearly, each image U G (w) are well-defined convolution operators on AG under linearity (5.5)′ . Moreover, U G (w1 w2 ) (f w ) = u G (w1 w2 )*G f w = f w1 w2 *G f w = f w1 *G f w2 *G f w (︀ )︀ (︀ )︀ = f w1 *G u G (w2 ) *G f w = f w1 *G U G (w2 )(f w ) (︀ )︀ (︀ )︀ = U G (w1 ) U G (w2 )(f w ) = U G (w1 ) ∘ U G (w2 ) (f w ), for all w1 , w2 , w ∈ G. i.e., on AG , the morphism U G of (5.5) satisfies that U G (w1 w2 ) = U G (w1 ) ∘ U G (w2 ), for all w1 , w2 ∈ G. Moreover, if w ∈ G, then (︀ )︀* U G (w)* (f w′ ) = u G (w) *G f w′ = u G (w−1 ) *G f w′ = f w−1 * f w′ by (5.1)′ = f w−1 * f w = U G (w−1 ), for all w, w′ ∈ G, i.e., we have U G (w)* = U G (w−1 ), for all w ∈ G. Therefore, U G is a well-defined groupoid-action of G acting on AG . Equivalently, the triple (G, AG , U G ) is a well-defined groupoid-dynamical system. The above theorem shows that graph groupoids act suitably on the arithmetic algebra A.

6 Krein-Space Representations of Graph-Arithmetic Algebras Remark again that we will work on Krein-space representations of our graph-arithmetic algebras to maintain the free-distributional data of arithmetic functions (containing the full number-theoretic data of arithmetic functions) obtained from a fixed prime. Again, we will use same notations used before. Fix a prime p throughout this section, and let (A, g p ) be the arithmetic p-prime probability space inducing the corresponding p-prime Banach algebra Ap acting on the Krein space Kp of Section 3. Let G be a fixed directed graph with its graph groupoid G, and let AG be the


142 | Ilwoo Cho and Palle E. T. Jorgensen G-arithmetic algebra, inducing the groupoid-dynamical system (G, AG , U G ), where U G is in the sense of (5.5) and (5.5)′ . Recall again that all graphs G are finite and connected. In this section, we establish a suitable Krein-space representation for AG , motivated by both Section 3 and Section 4. In particular, we want our representations contain both analytic-and-number-theoretic data of arithmetic functions in AG and combinatorial-and-algebraic data of G. Define now a topological tensor product space Kp,G by def

Kp,G = Kp ⊗ H G ,

(6.1)

where Kp is the Krein-space in the sense of Section 3, where the p-prime Banach algebra Ap is acting, and H G is the vertex Hilbert space of G in the sense of Section 4, equipped with its “indefinite” inner product [, ]p,G , (︀ )︀ [η1 ⊗ ξ1 , η2 ⊗ ξ2 ]p,H = [η1 , η2 ]p (< ξ1 , ξ2 > G ) ,

(6.1′ )

where [, ]p is the indefinite inner product on the Krein space Kp in the sense of (3.6) and G is the positivedefinite inner product on the Hilbert space H G in the sense of (4.3). Then, the tensor product space Kp,G of (6.1) is a Krein space with its indefinite inner product (6.1)′ , by [9] and [10]. In general, if K = H1 ⊕ H2− is a Krein space with its Hilbert-space part H1 and its anti-space part H2− , and if H is a Hilbert space, then K ⊗ H = (H1 ⊕ H2− ) ⊗ H = (H1 ⊗ H) ⊕ (H2− ⊗ H)

(6.2)

becomes a Krein space, since H2− ⊗ H is the anti-space of the Hilbert space H2 ⊗ H. Indeed, it is equipped with its negative-definite inner product, (︀ )︀ (︀ )︀ −[, ]2,H = − H2 (H ) = − H2 ⊗ (H ) , induced by the positive-definite inner product, (︀ )︀ (︀ )︀ [, ]2,H = H2 ⊗ (H ) = H2 (H ) , i.e., the anti-space (H2 ⊗ H)− of H2 ⊗ H is identified with the negative-definite inner product space H2− ⊗ H. Since Kp is a well-defined Krein space by [9] and [10], the tensor product space Kp,G becomes a welldefined Krein space for the vertex Hilbert space H G , by (6.2). Definition 6.1. The Krein space Kp,G = Kp ⊗ H G of (6.1) equipped with its indefinite inner product [, ]p,G of (6.1)′ is called the p-prime G-Krein space. Notation and Assumption As we assumed in previous sections, we denote elements [f ]Rp of AG simply by (︀ )︀ f , for f ∈ A. So, the term “f ∈ AG ” means “f = U G [f ]Rp .” Define now an algebra-action α p,G of the quotient algebra Ap,G acting on the Krein space Kp,G of (6.1) by the linear morphism satisfying that: def

α p,G (f w ) = α p (f w ) ⊗ α G (w), for all f w ∈ AG , i.e., more precisely, (︃ α p,G

)︃ ∑︁ w∈G

tw fw

=

∑︁

(︀ )︀ t w α p (f w ) ⊗ α G (w) ,

w∈G


(6.3)


| 143

where α p is the algebra-action of the quotient algebra Ap in the sense of (3.7) acting on the Krein space Kp , and α G is the algebra-action of the graph groupoidal C* -algebra M G in the sense of (4.5), (4.6), (4.7) and (4.9) acting on the vertex Hilbert space H G . Definition 6.2. Define an algebraic *-subalgebra Ap,G of the tensor product *-algebra Ap ⊗ M G by def

Ap,G = α p,G (AG ) ,

(6.4)

where ⊗ means a pure-algebraic tensor product of *-algebras. Remark that α p (f w ) and α G (w) are well-defined Krein-space operators on Kp and Hilbert-space operators on H G , respectively, and hence, the images α p,w (f w ) are well-defined Krein-space operators on Kp,G , for all w ∈ G. Remark also that each element of Ap,G has its form, ∑︁ T= t w f w with t w ∈ C, (6.5) w∈G

∑︀

where the sum is the finite sum. The image α p,G (T) of T in the sense of (6.5) is a well-defined Krein-space operator on Kp,G by (6.3), as an element of Ap,G of (6.4). Let’s denote the operator (Banach *-)algebra consisting of all bounded (or continuous) linear Krein-space operators on Kp,G by B(Kp,G ). Then the above observation can be summarized by the following theorem. Theorem 6.1. The pair (Kp,G , α p,G ) forms a well-defined Krein-space representation of the quotient algebra Ap,G . Proof. Each image α p,G (T) of T ∈ Ap,G is a well-defined Krein-space operator on Kp,G , by (6.5). For w1 , w2 ∈ G, we also have that (︀ )︀ α p,G (f w1 *G f w2 ) = α p,G (f w1 w2 ) = (α p (f w1 w2 )) ⊗ α G (w1 w2 ) ⎧(︀ )︀ (︀ )︀ ⎪ ifw1 w2 ≠ ∅ ⎨ α p (f w1 * f w2 ) ⊗ α G (w1 )α G (w2 ) = (︀ )︀ (︀ )︀ ⎪ ⎩ α p (0A ) ⊗ α G (∅) ifw1 w2 = ∅ ⎧(︀ )︀ (︀ )︀ ⎪ if w1 w2 ≠ ∅ ⎨ α p (f w1 )α p (f w2 ) ⊗ α G (w1 )α G (w2 ) = ⎪ ⎩ OKp ⊗ O H G = OKp,G if w1 w2 = ∅ where O X means the zero operators (or the zero matrices) on vector spaces X ⎧ (︀ )︀ (︀ )︀ ⎪ if w1 w2 ≠ ∅ ⎨ α p (f w1 ) ⊗ α G (w1 ) α p (f w2 ) ⊗ α G (w2 = ⎪ ⎩ O if w1 w2 = ∅ Kp,G ⎧ ⎪ ⎨ α p,G (f w1 )α p,G (f w2 ) if w1 w2 ≠ ∅ = ⎪ ⎩ O if w1 w2 = ∅, Kp,G

(6.6)

(6.6′ )

for all w1 , w2 ∈ G. Observe now that in the formula (6.6), if w1 w2 = ∅ in G, then one can get that: (︀ )︀ (︀ )︀ (︀ )︀ (︀ )︀ α p,G (f w1 ) α p,G (f w2 ) = α p (f w1 ) ⊗ α G (w1 ) α p (f w2 ) ⊗ α G (w2 ) (︀ )︀ (︀ )︀ (︀ )︀ (︀ )︀ = α p (f w1 )α p (f w2 ) ⊗ α G (w1 )α G (w2 ) = α p (f w1 )α p (f w2 ) ⊗ α G (w1 w2 ) (︀ )︀ = α p (f w1 )α p (f w2 ) ⊗ O H G = OKp,G . (6.7) Therefore, the formula (6.6)′ is without loss of generality can be re-written simply by (︀ )︀ (︀ )︀ α p,G (f w1 *G f w2 ) = α p,G (f w1 ) α p (f w2 ) ,

(6.8)


144 | Ilwoo Cho and Palle E. T. Jorgensen for all w1 , w2 ∈ G, by (6.7). Thus we obtain that, if T1 and T2 are in Ap,G , then α p,G (T1 *G T2 ) = α p,G (T1 )α p,G (T2 ),

(6.9)

under linearity, by (6.8). Also, we have that: (︀ )︀* )︀* (︀ α p,G (f w ) = α p (f w ) ⊗ α G (w) = α p (f w )* ⊗ α G (w)* = α p (f w* ) ⊗ α G (w−1 ) = α p (f w−1 ) ⊗ α G (w−1 ) where w−1 is the shadow of w in G (︀ )︀ = α p,G (f w−1 ) = α p,G f w* , i.e., α p,G (f w )* = α p,G (f w* ), for all w ∈ G.

(6.10)

Therefore, by (6.10), if T ∈ Ap,G , then α p,G (T)* = α p,G (T * ),

(6.11)

under linearity. Therefore, by (6.9) and (6.11), the morphism α p,G is a well-defined algebra-action acting on the Krein space Kp,G . Equivalently, the pair (Kp,G , α p,G ) is a well-determined Krein-space representation of Ap,G of (6.4). The above theorem provides a suitable representation for Ap,G in Krein space. Definition 6.3. For the topology on the Krein-space operator algebra B(Kp,G ), construct the closure Ap,G of the algebra Ap,G of (6.4) in B(Kp,G ), and denote it by Ap,G . i.e., def

Ap,G = Ap,G inB(Kp,G ).

(6.12)

Then this Banach *-algebra is called the p-prime G-algebra acting on Kp,G . Now, let Ap,G be the p-prime G-algebra of (6.12) acting on the Krein space Kp,G of (6.1). Then every element T of Ap,G is expressed by ∑︀

T=

t w α p,G (f w ) =

w∈G

∑︀

(︀ )︀ t w α p (f w ) ⊗ α G (w) ,

w∈G

(finite or infinite (under limits) sum). ̂︀ for i, j ∈ {1, ..., N G }, l ∈ {1, ..., k ij }, then For instance, if e ij:l is an edge in E(G), ⎛

(︃ (︀

)︀

α p,G e ij:l =

f e ij:l (1) f e ij :l (p)

O

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ )︃ ⎜ ⎜ 0 ⎜ ⊗⎜ ⎜ f e ij:l (1) ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

O

··· ···

.. . 0 0

.. . 0 ω

.. . 0 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

··· ···

(i, j)−entry

···

O

0 .. .

0 .. .

0 .. .

⎞

···

O

(6.13)

N G ×N G

on Kp,G , where



{︃ ω=

ωl ω l+1

| 145

if i ≠ j if i = j,

in the sense of (4.6) and (4.7).

7 Free Probability on Ap,G Now, let p be a fixed prime and G, a fixed directed graph with its graph groupoid G, and let Ap,G be the p-prime G-algebra of (6.12) acting on the Krein space Kp,G = Kp ⊗ H G . Recall the linear functional φ p of (3.11) on the Banach *-subalgebra Ap = α p (Ap ) , and the conditional expectation E G of (4.14) from the graph-groupoidal C* -algebra M G onto its diagonal subalgebra D G , and the linear functional tr G of (4.15) on M G , induced by E G . Now, define a linear functional φ p,G : Ap,G → C

by

φ p,G = φ p ⊗ tr G ,

(7.1)

satisfying (︀ )︀ (︀ )︀ def (︀ (︀ )︀)︀ (︀ )︀ (︀ (︀ )︀)︀ φ p,G α p,G (f w ) = φ p,G α p (f w ) ⊗ α G (w) = (φ p (f w )) tr G α G (w) = f w (p) tr G α G (w) {︃ ̂︀ f w (p) = g p (f w ) if w ∈ V(G) = 0 otherwise,

(7.2)

for all w ∈ G, where g p is in the sense of (3.1) (up to quotient; See (6.13)). We write the above relation (7.2) simply by (︀ )︀ φ p,G α p,G (f w ) = V(w)f w (p),

(7.2′ )

where def

{︃

V(w) =

1 0

̂︀ = V(G) if w ∈ V(G) otherwies,

for all w ∈ G. So, more precisely, one has (︃ φ p,G

)︃ ∑︁

t w α p,G (f w )

=

w∈G

∑︁

t v f v (p),

(7.3)

̂︀ v∈V(G)

by (7.2) and (7.2)′ . Proposition 7.1. Let T =

∑︀

t w α p,G (f w ) be an element of the p-prime G-algebra Ap,G acting on the Krein

w∈G

space Kp,G , with t w ∈ C, and let φ p,G be the linear functional on Ap,G of (7.1). Then α p,G (T) =

∑︁ ̂︀ v∈V(G)

t v f v (p) =

∑︁

t v g p (f v ) ,

(7.4)

̂︀ v∈V(G)

where g p is in the sense of (3.1) (up to quotient). Proof. The proof of (7.4) is done by (7.2)′ and (7.3).


146 | Ilwoo Cho and Palle E. T. Jorgensen So, the formula (7.4) can be used as a definition of the linear functional φ p,G of (7.1) on Ap,G . In the rest of this section, we will restrict our interests to the generating operators f w of the p-prime Galgebra Ap,G on the Krein space Kp,G , for w ∈ G. Definitely, by (7.2)′ , one has {︃ φ p,G (f w ) = V(w) f w (p) =

f w (p) 0

̂︀ if w ∈ V(G) otherwise,

for all w ∈ G. Definition 7.1. The free probability space (Ap,G , φ p,G ) is called the p-prime G-probability space (acting on the Krein space Kp,G ). Let (w1 , ..., w n ) ∈ {w, w−1 }n , for n ∈ N, w ∈ G \ {∅}. Then, by (7.4), one has φ p,G (f w1 *G ... *G f w n ) = φ p,G (f w1 w2 ...w n ) = V(w1 ...w n )f w1 ...w n (p).

(7.5)

By (7.5), we directly obtain the following lemma. ̂︀ Lemma 7.1. Let (w1 , ..., w n ) ∈ {w, w−1 }n , for n ∈ N, w ∈ FP r (G). If n is odd in N, then φ p,G (f w1 *G ... *G f w n ) = 0.

(7.6)

If n is even in N, and if (w1 , ..., w n ) is an alternating finite sequence, i.e.,

(7.7)

{︃ (w1 , ..., w n ) =

(w, w−1 , w, w−1 , ...,w, w−1 ) (w−1 , w, w−1 , w, ...,w−1 , w),

or

then

φ p,G (f w1 *G ... *G f w n ) =

⎧ ⎪ ⎨ f ww−1 (p)

respectively

⎪ ⎩ f −1 (p), ww where ww−1 is the initial vertex of w and w−1 w is the terminal vertex of w in G. But the converse of (7.7) does not hold in general. Proof. The statement (7.6) is proven directly from (7.1) and (7.2)′ . Indeed, if n is odd in N, then for any n-tuple (w1 , ..., w n ) in {w, w−1 }, the groupoid element w1 w2 ...w n cannot be a vertex in G. So, by (7.2)′ , φ p,G (f w1 ...w n ) = 0. Assume now that n is even in N, moreover the corresponding n-tuple (w1 , ..., w n ) is an alternating n-tuple (w, w−1 , ..., w, w−1 ). Then w1 w2 ...w n = ww−1 ww−1 ...ww−1 = ww−1 , where ww−1 is the initial vertex of w in G. Therefore, φ p,G (f w1 *G ... *G f w n ) = φ p,G (f ww−1 ) = V(ww−1 )f ww−1 (p) = f ww−1 (p), by (7.4) and (7.5). Similarly, if the n-tuple (w1 , ..., w n ) is an alternating n-tuple (w−1 , w, ..., w−1 , w), then φ p,G (f w1 *G ... *G f w n ) = φ p,G (f w−1 w ) = V(w−1 w)f w−1 w (p) = f w−1 w (p),



| 147

where w−1 w is the terminal vertex of w in G. ̂︀ with v ∈ V(G) ̂︀ in G. Then, even though a Now, let w = vwv be a loop reduced finite path in FP r (G), −1 n sequence (w1 , ..., w n ) is not alternating in {w, w } , for n ∈ 2N, if it contains same numbers of w’s and w−1 ’s, then φ p,G (f w1 *G ... *G f w n ) = φ p,G (f v ). −1

For example, if we have (w, w, w , w−1 ), which is not alternating, then φ p,G (f w *G f w * f w−1 *G f w−1 ) = φ p,G (f www−1 w−1 ) = φ p (f v ). Therefore, the converse of (7.7) does not hold in general. More precisely, one can generalize the statement (7.7) as follows. ̂︀ in G. Lemma 7.2. Let w be a reduced finite path in FP r (G) If w is a “non-loop” reduced finite path, and if (w1 , ..., w2n ) is a (2n)-tuple in {w, w−1 }2n , for n ∈ N, then ⎧ −1 −1 ⎪ ⎨ f ww−1 (p) if (w1 , ..., w2n ) = (w, w , ..., w, w ) −1 −1 φ p,G (f w1 *G ... *G f w2n ) = (7.8) f w−1 w (p) if (w1 , ..., w2n ) = (w , w, ..., w , w) ⎪ ⎩ 0 otherwise. And the converse also holds true. If w is a “loop” reduced finite path, and if (w1 , ..., w2n ) is a (2n)-tuple in {w, w−1 }2n , for n ∈ N, then

φ p,G (f w1 *G ... *G f w n ) =

⎧ ⎪ ⎨ f w−1 w (p) = f ww−1 (p)

if #(w1 ,...,w2n ) (w) = #(w1 ,...,w2n ) (w−1 )

⎪ ⎩ 0

otherwise,

(7.9)

where #(w1 ,...,w2n ) (w) = the number of w’s in (w1 , ..., w2n ) and #(w1 ,...,w2n ) (w−1 ) = the number of w−1 ’s in (w1 , ..., w2n ). Also, the converse does hold, too. Proof. Let n ∈ N, and 2n, a corresponding even number in N, and let w be a nonempty reduced finite path in G. Let (w1 , ..., w2n ) be (2n)-tuples in {w, w−1 }2n . Assume first that w is a non-loop finite path in G. Then the only ways to make the element w1 w2 ...w2n be vertices in G are when the (2n)-tuple (w1 , ..., w2n ) is alternating, i.e., {︃ (w, w−1 , w, w−1 , ...,w, w−1 ) or (w1 , ..., w2n ) = (w−1 , w, w−1 , w, ...,w−1 , w), because if w is non-loop, then w k = ∅ = (w k )−1 in G, whenever k ≥ 2 in N. Assume now that w is a non-loop finite path in G, and suppose an n-tuple (w1 , ..., w n ) of {w, w−1 } does not satisfy the alternating property. Equivalently, there exists at least one i, satisfying 1 ≤ i < n, such that w i = w i+1 in {w, w−1 }. Then w1 w2 ... w n = w1 ... (w i w i+1 ) ... w n = ∅,


148 | Ilwoo Cho and Palle E. T. Jorgensen and hence, φ p,G (f w1 *G ... *G f w2n ) = 0. So, we obtain the statement (7.8). ̂︀ in G. So, by (7.7), if the (2n)-tuple Now, suppose w is a loop finite path in G. Then w−1 w = ww−1 in V(G) (w1 , ..., w2n ) is alternating we have φ p,G (f w1 *G ... *G f w2n ) = f w−1 w (p) = f ww−1 (p). Furthermore, since w is a loop, if #(w1 ,...,w2n ) (w) = #(w1 ,...,w2n ) (w−1 )

(7.10)

in (w1 , ..., w2n ), where they are as in (7.9), then w1 w2 ...w2n = w−1 w = ww−1 in G. In fact, the alternating cases are contained in this case (7.10). Now, for a loop finite path w, suppose #(w1 ,...,w2n ) (w) ≠ #(w1 ,...,w2n ) (w−1 ), in (w1 , ..., w2n ). Say, #(w1 ,...,w2n ) (w) = #(w1 ,...,w2n ) (w−1 ) + m, with some m ∈ N. Then ⎛ φ p,G (f w1 *G ... *G f w2n )

⎞

(︀ )︀ = φ p,G ⎝f w *G ... *G f w ⎠ = φ p,G (f w m ) = φ p (f w m )tr G (w m ) = φ p (f w m ) (0) = 0. ⏟ ⏞ m−times

So, the statement (7.9) holds. −1 Take a subset X = {w1 , ..., w N }, for N ∈ N, in G \ {∅}. Also, let X −1 = {w−1 1 , ..., w N } in G. Construct

X = X ∪ X −1 in G \ {∅}.

(7.11)

Lemma 7.3. Let X be a subset of G in the sense of (7.11), and let W = (w i1 , ..., w i k ) ∈ Xk , for k ∈ N. If k is odd in N, then (︃ )︃ φ p,G

k

*G f w i j

= 0.

(7.12)

j=1

Let k is even in N, and let W o be the product w i1 w i2 ...w i k of W in G. Then (︃ )︃ φ p,G

k

*G f w i j

= V(W o )f W o (p).

(7.13)

j=1

Proof. The proof of (7.12) is clear, by (7.6). The proof of (7.13) is directly from (7.3) and (7.4). The results (7.12) and (7.13) are generalizations of (7.6), (7.7), (7.8) and (7.9). Now, we refine the result (7.13) of the above lemma. Let X be as in (7.11), and let W be the k-tuple of the above lemma inducing the corresponding product W o in G. In the formula (7.13), the relation “V(W o ) ≠ 0” is ̂︀ So, if there are m-distinct nonempty reduced finite paths w i , ..., equivalent to “W is admissible in V(G).” j1 w i jm in W which are admissible in W , and if each w i j in W has the same number of shadows w−1 , for all l= i jl l 1, ..., m, and if all of them satisfy the conditions of (7.8) and (7.9), and all other non-reduced finite paths in W are a unique vertex which is either an initial or a terminal vertex of w i j ’s, for l = 1, ..., m, then V(W o ) ≠ 0. l



| 149

Proposition 7.2. Let X be a subset of G in the sense of (7.11), and let W = (w i1 , ..., w i k ) ∈ Xk with its product W o in G, for k ∈ N. Also, assume that all entries w i1 , ..., w i k of W are nonempty reduced finite paths of G in X. If k ∈ N, then (︂ )︂ {︃ ̂︀ ∩ X f w0 (p) if w i1 = ... = w i k = w0 in V(G) φ p,G *G f w = . (7.14) 0 otherwise. w∈W If k is odd in N, and if W consists of reduced finite paths in X, then (︂ )︂ φ p,G *G f w = 0.

(7.15)

w∈W

If (i) k is even in N, (ii) W is partitioned by W = (W1 , ..., W m ), which is order-preserving in W, with ∑︀m ⃒ ⃒ ⃒W j ⃒ = |W | = k, and (iii) each block W j satisfies the conditions for (7.8) and (7.9), for a fixed {w i , w−1 }, j=1

j

ij

̂︀ for j = 1, ..., m, then and (iv) the products W jo of W j are identical from each other in V(G), (︂ φ p,G

)︂ *G f w

= f W o (p).

(7.16)

w∈W

The converse also holds true. Proof. Let W, W o be given as above in X = X ∪ X −1 of (7.11), for k ∈ N. Assume first that W consists of a single element w0 ∈ X, and suppose w0 is a vertex in G, i.e., ⎞ ⎛ ̂︀ W = ⎝w0 , w0 , w0 , ...,w0 ⎠ , with w0 ∈ X ∩ V(G). ⏞ ⏟ k−times

o

̂︀ So, Then W = w0 w0 ...w0 = w0 in V(G). (︂ )︂ φ p,G *G f w = φ p,G (f w0 w0 ...w0 ) = φ p,G (f w0 ) = f w0 (p), w∈W

by (7.5). So, the statement (7.14) holds. Suppose now that W = (w i1 , ..., w i k ) in Xk , and assume that k is odd in N, where all entries w i j of W are ̂︀ Then there exists a blocks W1 , ..., W m of W , such that reduced finite paths in FP r (G). W = (W1 , ..., W m ), for some m ∈ N,

(7.17)

where W j consist only of elements of {w i j , w−1 in W . Since k is i j } ⊂ X, where the blocks are order-preserving ⃒ ⃒ assumed to be odd, there exists at least one block W j0 of W in (7.17) such that ⃒W j0 ⃒ is odd in N. It shows that the corresponding product W jo0 of W j0 cannot be a vertex, and hence, the product W o of W cannot be a vertex in G, equivalently, V(W o ) = 0. Therefore, by (7.13), if k is odd in N, then φ p,G (f W o ) = 0. So, the statement (7.15) holds. Now, assume all conditions (i), (ii), (iii) and (iv) of (7.16) hold for an arbitrarily fixed k-tuple W of Xk with ̂︀ So, by (7.5), (7.8), (7.9) and its product W o in G. In such a case, the product W o becomes a vertex in V(G). (7.13), one has that φ p,G (f W o ) = f W o (p). Assume now that one of the conditions (i) or (ii) or (iii) or (iv) of (7.16) does not hold. Then, by (7.6), (7.7), (7.8), (7.9) and (7.13), φ p,G (f W o ) = 0.


150 | Ilwoo Cho and Palle E. T. Jorgensen Thus, the converse holds true, too. i.e., the conditions; (i), (ii), (iii) and (iv); hold, if and only if φ p,G (f W o ) = f W o (p). So, the statement (7.16) holds. In particular, the statement (7.16) shows that if a k-tuple W consists of all reduced finite paths in X of (7.11) with its corresponding product W o in G, then φ p,G (f W o ) ≠ 0, if and only if W satisfies the conditions (i), (ii), (iii) and (iv) of (7.16) altogether. Therefore, one obtains the following main result of this section. Theorem 7.1. Let W be a k-tuple of X of (7.11) with its corresponding product W o in G. Suppose W = (W1 , ..., W m ) in Xk , for some m ∈ N, where W1 , ..., W m are ordered-preserving maximal blocks in W , such that each W j consisting only of entries o in {w j , w−1 j } in X. If all W j ’s satisfy (7.8), or (7.9), or (7.14), and if the corresponding products W j of W j are all identical to a single vertex in G, then (︂ )︂ φ p,G *G f w = f W o (p) = f W jo (p), w∈W

for all j = 1, ..., m. Proof. The proof is done by (7.14), (7.15) and (7.16), inductively. The above theorem provides a general tool to compute joint free-moments on the p-prime G-algebra Ap,G on the Krein space Kp,G . Example 7.1. For example, let −1 −1 X = {v1 , v2 , w1 , w−1 1 , w2 , w2 , w3 , w3 },

̂︀ and w1 is a loop-edge, and w2 are non-loop edges. as in (7.11), where v1 , v2 ∈ V(G), Assume now that W contains both v1 and v2 . Then, by (7.14) and (7.16), φ p,G (f W o ) = 0. Suppose now −1 −1 W = (w1 , w1 , w1 , w−1 1 , w 1 , w 2 , w 2 , v 2 , v 2 ).

Then one can have the ordered blocks W1 , W2 , W3 of W , with −1 −1 W1 = (w1 , w1 , w1 , w−1 1 , w 1 ), W 2 = (w 2 , w 2 ), W 3 = (v 2 , v 2 ),

such that W = (W1 , W2 , W3 ). In this case, W1 has #W1 (w1 ) = 3 ≠ 2 = #W1 (w−1 1 ). Therefore, φ p,G (f W o ) = 0. Now, let



151

−1 −1 −1 W = (w1 , w1 , w−1 1 , w 1 , w 1 , w 1 , w 2 , v 1 , v 1 , v 1 , w 2 ).

Then W has ordered blocks −1 −1 −1 W1 = (w1 , w1 , w−1 1 , w 1 , w 1 , w 1 ), W 2 = (w 2 ), W3 = (v1 , v1 , v1 ), and W4 = (w2 ),

such that W = (W1 , W2 , W3 , W4 ). Since W2 and W4 do not satisfy (7.16), one has φ p,G (f W o ) = 0. If −1 −1 −1 W = (w1 , w−1 1 , w 1 , w 1 , w 2 , w 2 , w 2 , w 2 , v 1 , v 1 , v 1 ),

then one has −1 −1 −1 W1 = (w1 , w−1 1 , w 1 , w 1 ), W 2 = (w 2 , w 2 , w 2 , w 2 ), and W3 = (v1 , v1 , v1 ),

such that W = (W1 , W2 , W3 ). In W1 , we have #W1 (w1 ) = 2 = #W2 (w−1 1 ), for the loop-finite path w1 , and the ordered block W2 is an alternating 4-tuple of the non-loop finite path w2 , and W3 consists only of an identical vertex v1 . −1 −1 So, if w−1 1 w 1 = w 1 w 1 = w 2 w 2 = v 1 in G, then φ p,G (f W o ) = f W o (p) = f v1 (p).

By Theorem 7.6, we obtain the following interesting free-distributional data, too. (...) Theorem 7.2. Let W = (w i1 , ..., w i k ) be an arbitrary k-tuple of X in the sense of (7.11), for k ∈ N, and let k p,G k means the free cumulant on the p-prime G-algebra Ap,G in terms of the linear functional φ p,G as in Section 2.4, for all k ∈ N. If W has its ordered blocks W1 , ..., W m satisfying the hypothesis of Theorem 7.6 inducing W = (W1 , ..., W m ) and the product w o ∈ G, then there exists µ W o ∈ C, such that (︀ )︀ k p,G f w i1 , ...,f w ik = µ W o f W o (p), k

(7.18)

where µW o =

∑︀

µ(π, 1k ),

θ∈NC W

where


152 | Ilwoo Cho and Palle E. T. Jorgensen def

NC W = {π ∈ NC(k) : π ≥ {W1 , ..., W m }}. Proof. By the Möbius inversion of Section 2.4, one has that (︀ )︀ ∑︀ k p,G (f w i1 , ..., f w ik ) = φ p,G:π f w i1 , ...,f w ik µ(π, 1k ) k π∈NC(k)

(︂ =

∑︀

(︂ Π φ p,G

V∈π

π∈NC(k)

(︂

)︂)︂ *G f w

w∈V

(︂

∑︀

µ(π, 1k ) =

Π

π∈NC(k), π≥W

V⊇W j ∈{W1 ,...,W m }

φ p,G

)︂)︂ *G f w

w∈V

µ(π, 1k )

by (7.7), (7.8), (7.9), (7.14), (7.15) and (7.16) =

∑︀ π∈NC W

(︀ )︀ φ p,G:π f w i1 , ...,f w ik µ(π, 1k )

where NC W is in the sense of (7.18) (︃ =

∑︀

(︀

)︀ φ p,G (f W o ) µ(π, 1k ) = φ p,G (f W o )

π∈NC W

)︃ ∑︀

µ(π, 1k )

= µ W o f W o (p)

π∈NC W

by (7.5), where µW o =

∑︀

µ(π, 1k ).

π∈NC W

Clearly, if NC W is empty in NC(k), then (︀ )︀ k p,G f w i1 , ...,f w ik = 0, n for W = (w i1 , ..., w i k ), for k ∈ N. The above free-cumulant computation (7.18) provides an equivalent free-distributional data with the freemoment computations (7.14), (7.15) and (7.16). Now, let w1 and w2 are distinct nonempty elements in G such that; −1 −1 −1 ̂︀ w−1 1 w 1 ≠ w 2 w 2 and w 1 w 1 ≠ w 2 w 2 in V( G).

(7.19)

i.e., the initial and terminal vertices of w1 are distinct from the initial and terminal vertices of w2 in G. If two nonempty elements w1 and w2 of G satisfy (7.19), we say that they are vertex-distinct in G. We extend this concept by axiomatizing that: all nonempty elements of G and the empty element ∅ of G are vertex-distinct in G. ±1 Theorem 7.3. Let w1 ≠ w2 ∈ G, and let X = {w±1 1 , w 2 } in the sense of (7.11) in G. Then f w1 and f w2 are free in (Ap,G , φ p,G ), if and only if w1 and w2 are vertex-distinct in the sense of (7.19). i.e.,

f w1 and f w2 are free ⇔ w1 and w2 are vertex − distinct inG.

(7.20)

Proof. (⇐) Suppose w1 and w2 are vertex-distinct in G, in the sense of (7.19). Then, by (7.14), (7.15) and (7.16), in particular, by (7.16), for any “mixed” k-tuples W of Xk , for all k ∈ N \ {1}, φ p,G (f W o ) = 0.

(7.21)

Thus, for any mixed k-tuples W = (w i1 , ..., w i k ) of X, we have (︀ )︀ k p,G f w i1 , ...,f w ik = 0, k by (7.18) and (7.21). So, by Section 2.4, the free random variables f w1 and f w2 are free in (Ap,G , φ p,G ). (⇒) Suppose that f w1 and f w2 are free in (Ap,G , φ p,G ), for w1 ≠ w2 in G, and assume that w1 and w2 are not vertex-distinct in G. So, one may assume that



153

̂︀ ⊂ G. w1 = w and w2 = w−1 in FP r (G) Then, clearly, w1 ≠ w2 in G. Assume further that f w−1 w (p) ≠ 0, and f w−1 w (p) ≠ 0. By hypothesis, we have ±1 −1 ̂︀ X = {w±1 1 , w 2 } = { w, w } in FP r ( G).

Now, take W = (w, w−1 ), a “mixed” (2)-tuple of X. Then, by (7.16), k2p,G (f w , f w−1 ) = f ww−1 (p) ≠ 0, which contradicts our assumption that f w1 = f w and f w2 = f w−1 are free in (Ap,G , φ p,G ). Therefore, if f w1 and f w2 are free in (Ap,G , φ p,G ), then w1 and w2 are vertex-distinct in the sense of (7.19) in G. The above freeness characterization (7.20) shows that the freeness on (Ap,G , φ p,G ) is combinatorially characterized by the vertex-distinctness (7.19). Let G be the graph groupoid of a fixed graph G. Define subsets Gvv21 of G by def

Gvv21 = {w ∈ G : w = v1 wv2 },

(7.22)

̂︀ for all v1 , v2 ∈ V(G). By the freeness characterization (7.20), we obtain that: ′

̂︀ Let Gxv , Gvx , Gx′ , Corollary 7.1. Let X be in the sense of (7.11), and let v, v′ , x, x′ be distinct vertices in V(G). v ′

Gvx′ be subsets of G in the sense of (7.22). Now, let (︁ ′ )︁ (︀ )︀ ′ Xv,x = X ∩ Gxv ∪ Gvx and Xv′ ,x′ = X ∩ Gxv′ ∪ Gvx′ in X. Then the families {f w : w ∈ Xv,x } and {f w′ : w′ ∈ Xv′ ,x′ }

are free in (Ap,G , φ p,G ). The converse also holds true. Proof. It Xv,x and Xv′ ,x′ are given as above in X, then they are vertex-distinct in the sense that, for all pairs (w1 , w2 ) in Xv,x × Xv′ ,x′ , the elements w1 and w2 are vertex-distinct in G, and hence, the free random variables f w1 and f w2 are free in (Ap,G , φ p,G ), by (7.20). The converse also holds again by the vertex-distinctness of (7.20). The freeness characterization (7.20) of the above theorem and the generalized freeness condition of the above corollary, one can verify the following special case. Corollary 7.2. Let e i1 j1 :l1 and e i2 j2 :l2 be edges in E(G) under the indexing (4.2)′ ; e ij:l means the l-th edge connecting its initial vertex v i to its terminal vertex v j for the vertex indexing (4.2), where l = 1, ..., k ij , where k ij means the cardinality of edges connecting v i to v j in E(G). Then the corresponding free random variables f k = f e ik jk :lk are free in (Ap,G , φ p,G ), if and only if j1 ≠ i1 , j1 ≠ j2 , j2 ≠ i1 and j2 ≠ j1 . (Remark that it is not necessary that i1 ≠ j1 or i2 ≠ j2 .) The above corollary is equivalent to the following result.


154 | Ilwoo Cho and Palle E. T. Jorgensen Corollary 7.3. Let e1 = e i1 j1 :l1 and e2 = e i2 j2 :l2 be edges in E(G) under the indexing (4.2)′ as in the above corollary. Then the corresponding free random variables f e1 and f e2 are free in (Ap,G , φ p,G ), if and only if (︀

α G (e1 )r1

)︀ (︀

)︀ (︀ )︀ (︀ )︀ α G (e2 )r2 = O H G = α G (e2 )r2 α G (e1 )r1 ,

for r1 , r2 ∈ {1, *}, on the vertex Hilbert space H G , where α G is in the sense of (4.5), (4.6) and (4.7). Motivated by Corollaries 7.10 and 7.11, we obtain the following equivalent freeness characterization of (7.20). Theorem 7.4. Let w1 , w2 be nonempty elements of G, and let f w1 and f w2 be corresponding free random variables of (Ap,G , φ p,G ). Then f w1 and f w2 are free in (Ap,G , φ p,G ), if and only if (︀

α G (w1 )r1

)︀ (︀

)︀ (︀ )︀ (︀ )︀ α G (w2 )r2 = O H G = α G (w2 )r2 α G (w1 )r1 ,

(7.23)

for r1 , r2 ∈ {1, *}, on H G . Proof. By (7.20), f w1 and f w2 are free in (Ap,G , φ p,G ), if and only if w1 and w2 are vertex-distinct in G, if and only if (7.23) holds on the vertex Hilbert space H G .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

I. Cho, Operators Induced by Prime Numbers, Methods Appl. Math. Sci. 19, no. 4, (2013) 313 - 340. I. Cho, Graph Groupoids and Partial Isometries, ISBN: 978-3-8383-1397-9, (2009) Lambert Academic Press I. Cho, Classification on Arithmetic Functions and Corresponding Free-Moment L-Functions, Bulletin Korea Math. Soc., (2015) To Appear. I. Cho, p-Adic Banach-Space Operators and Adelic Banach-Space Operators, Opuscula Math., 34, no. 1, (2014) 29 - 65. I. Cho, Fractals on Graphs, ISBN: 978-3-639-19447-0, (2009) Verlag with Dr. Muller I. Cho, Operations on Graphs, Groupoids, and Operator Algebras, ISBN: 978-8383-5271-8, (2010) Lambert Academic Press. I. Cho, C -Valued Functions Induced by Graphs, Compl. Anal. Oper. Theo., DOI:10.1007/s11785-014-0368-0, (2014). I. Cho, and P. E. T. Jorgensen, An Application of Free Probability to Arithmetic Functions, Compl. Anal. Oper. Theo., DOI: 10.1007/s11785-014-0378-y, (2014) I. Cho and P. E. T. Jorgensen, Krein-Space Representation of Arithmetic Functions Determined by Primes, Alg. Rep. Theo, DOI: 10.1007/s11785-014-9473-z, (2014) I. Cho, and P. E. T. Jorgensen, Krein-Space Operators Induced by Dirichlet Characters, Contemp. Math.: Commutative and Noncommutative Harmonic Analysis and Applications, (2014) 3 - 33. I. Cho, and P. E. T. Jorgensen, Actions of Arithmetic Functions on Matrices and Corresponding Representations, Ann. Funct. Anal., (2014) To Appear. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol 1, ISBN: 978-981-02-0880-6, (1994) World Scientific. D. Bump, Automorphic Forms and Representations, Cambridge Studies in Adv. Math., 55, ISBN: 0-521-65818-7, (1996) Cambridge Univ. Press. D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, Vol. 1, ISBN: 0-8218-1140-1, (2002) Published by Amer. Math. Soc. J. P. S. Kung, M. R. Murty, and G-C Rota, On the Ré dei Zeta Function, J. Number Theo., 12, (1980) 421 - 436. P. Flajolet and R. Sedgewick, Analytic Combinatorics, ISBN: 978-0-521-89806-5, (2009) Cambridge Univ. Press. R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Amer. Math. Soc. Mem., vol 132, no. 627, (1998).


Matrices induced by arithmetic functions, primes and

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