---

P.E.T. Jørgensen 1,2, and R.F. Werner 3,4

arXiv:funct-an/9303002v1 12 Mar 1993Coherent States of theq–Canonical Commutation RelationsP.E.T. Jørgensen 1,2, and R.F.

Werner 3,4Abstract.For the q-deformed canonical commutation relationsa(f)a†(g) = (1 −q) ⟨f, g⟩1I + q a†(g)a(f) for f, g in some Hilbert spaceH we consider representations generated from a vector Ωsatisfyinga(f)Ω= ⟨f, ϕ⟩Ω, where ϕ ∈H. We show that such a representation ex-ists if and only if ∥ϕ∥≤1.

Moreover, for ∥ϕ∥< 1 these representationsare unitarily equivalent to the Fock representation (obtained for ϕ = 0).On the other hand representations obtained for different unit vectors ϕare disjoint. We show that the universal C*-algebra for the relationshas a largest proper, closed, two-sided ideal.

The quotient by this idealis a natural q-analogue of the Cuntz algebra (obtained for q = 0). Wediscuss the Conjecture that, for d < ∞, this analogue should, in fact, beequal to the Cuntz algebra itself.

In the limiting cases q = ±1 we de-termine all irreducible representations of the relations, and characterizethose which can be obtained via coherent states.AMS subject classification:46L89, 58B30, 81R10, 81R301 Dept. of Mathematics, University of Iowa, Iowa City, IA 52242, USA2 Supported in part by the NSF(USA), and NATO3 FB Physik, Universit¨at Osnabr¨uck, Postfach 4469,D-4500 Osnabr¨uck, Germany.4 Electronic mail:reinwer@dosuni1.rz.Uni-Osnabrueck.DE

1. IntroductionIn this paper we study some new aspects of a set of commutation relations, dependingon a parameter q ∈(−1, 1) studied by various authors on quite different motivations.Greenberg [15] introduced these relations as an interpolation between Bose (q = 1) andFermi (q = −1) statistics.

He was particularly interested in the observable consequencesof a hypothetical small deviation from the Pauli principle.However, due to problemswith field theoretical localizability [16] and thermodynamic stability [34], a naive particleinterpretation of systems satisfying these relations is problematic. Speicher [33] introducedthese relations as a new kind of quantum “noise”, which could be used as a driving forcein a quantum stochastic differential equation [23].

From the point of view of C*-algebratheory the relations became interesting as an example of a C*-algebra defined in termsof generators and relations. In this context it was observed that the relations reduce forq = 0 to those studied by Cuntz [9].The special case of a single generator, the so-called q-oscillator, was introduced byBiedenharn [4] and Macfarlane [27] as a means of constructing representations of quantumgroups.

In fact, the q-oscillator also appears as a subalgebra of the quantum group SνU(2)[35].The q-oscillator can be studied in full detail by representing the generator as aweighted unilateral shift (in mathematical terminology) or as a Bose creation operatormultiplied with a suitable function of the number operator (in physical terminology). Thishas been noted in a large number of papers.

We will use this representation in the presentpaper to obtain information about the non-trivial case of several generators.In this case most early work [15,6,13] focussed on showing that the scalar productin the q-analogue of Fock space is positive definite.On the other hand, from the C*-algebraic point of view the most immediate and natural problem arising from the relationswas to characterize the norm-closed operator algebra generated by any realization of therelations by bounded operators on a Hilbert space.Here the case q = 0 served as amodel: for q = 0 this algebra must be either isomorphic to the one obtained in the Fockrepresentation, called the Cuntz-Toeplitz algebra, or a quotient of the Cuntz-Toeplitzalgebra by its unique two-sided ideal (isomorphic to the compact operators), known as theCuntz algebra. For q ̸= 0 the first important step was made in [18], where we showed thatfor |q| <√2−1 ≈.41 the same results hold.

In particular, the C*-algebras generated withq in this range are exactly the same as for q = 0. The condition |q| <√2 −1 is certainlynot optimal, and all results known to us are compatible with the conjecture (which wewill refer to as “Conjecture C”, see Section 4) that the results of [18] hold for all |q| < 1.However, no decisive progress towards proving this conjecture has been made since [18].Based on an improved understanding of the Fock representation [36], Dykema and Nica[12] managed to extend the interval for q slightly, but only for the algebra generated inthe Fock representation.

More importantly, they established, for the Fock representationonly, the existence of the homomorphism between the algebras for q = 0 and for general−1 < q < 1, which according to Conjecture C should be an isomorphism. We will brieflydescribe and apply their results in Section 4.2

The main aim of this paper is to study the q-analogue of a structure which is well-known in the limiting cases q = 0, ±1, namely the generalization of the Fock state tothe so-called coherent states. In the case of a single relation such states appear in [26],although, due to a different choice of generators, their work makes sense only in the Fockrepresentation, and gives states different from ours.

We will determine all coherent states,and discuss under what circumstances they generate the same representation, or are mu-tually singular. Using coherent states, we show that the universal C*-algebra generated bythe relations has a unique largest closed two-sided ideal.

(If Conjecture C holds this idealis also the only proper ideal, and isomorphic to the compact operators). The quotient ofthe algebra by this ideal is then simple, and the natural analogoue of the Cuntz algebra forq ̸= 0.

Finally, we consider the limiting case q = −1, and compute all irreducible represen-tations of the relations with Clifford algebra methods. It turns out that in this degeneratecase the coherent states exhaust only a small subclass of irreducible representations.We emphasize that when we talk about representations in the sequel we always mean*-representations of some involutive algebra by bounded operators on a Hilbert space.

Thuseven if the relations may have interesting unbounded realizations we do not consider them.2. q-relations and coherent statesThe following Proposition introduces the “q-relations” which are the object of our study.1 Proposition.

Let H be a Hilbert space, and let q ∈IR, |q| < 1.Then there is aC*-algebra EH(q) generated by elements a†(f) for f ∈H, such that f 7→a†(f) is linear,anda(f)a†(g) = (1 −q) ⟨f, g⟩1I + q a†(g)a(f),(1)where a(f) := a†(f)∗. For q = 1, and orthogonal unit vectors e1, .

. .

, en ∈H the boundnXi=1a†(ei)a(ei) ≤1I(2)holds.Moreover, EH(q) is uniquely determined by the following universal property: whenever eEis a C*-algebra containing elements ea†(f) satisfying the above conditions, there is a uniqueunital homomorphism ϕ : EH(q) →eE such that ϕ(a†(f)) = ea†(f).The proof of this result is given in [18]. Note that in comparison with [18,6] we havechanged the normalization of the operators a†(f).

This modification makes no essentialdifference for |q| < 1. However, it removes the singularity of the relations for q →1,and simplifies all algebraic expressions.

Moreover, it was shown in [28] that with thisnormalization the algebras EC(q) form a continuous field of C*-algebras [10]. We may3

consider the relations (1) for all q ∈IR∪{∞}, where for q = ∞we set a†(g)a(f) = ⟨f, g⟩1I.The study of the case |q| ≥1 is then reduced to the case |q| ≤1 by the symmetryq 7→q−1a†(f) 7→a(f)(3)for some antiunitary operator f 7→f.The crucial feature of the relations (1) is that they allow us to order any polynomialin the generators in such a way that in every monomial all operators a(f) are to the rightof every a†(f). This normal ordered, or “Wick ordered” form of a polynomial is unique[2,3,20], hence we can define a linear functional ω on the polynomial algebra over the rela-tions by choosing an arbitrary multilinear expression for ω(a†(f1) · · ·a†(fn)a(g1) · · ·a(gm).Since such monomials generate EH(q) this is also a way to parametrize all states on theC*-algebra EH(q).

The following Theorem introduces the coherent states on EH(q) usingsuch a parametrization.2 Theorem. Let |q| ≤1, and ϕ ∈H with ∥ϕ∥≤1.

Then there is a unique state ωϕ onEH(q) such thatωϕ(a†(f)X) = ⟨ϕ, f⟩ω(X)(4)for all f ∈H, and all X ∈EH(q). The state ωϕ is pure.

For ∥ϕ∥> 1, there is no statesatisfying (4).We will call ωϕ the coherent state associated with ϕ. This terminology originatedin quantum optics, where these states are used to describe states of the electromagneticradiation field [22,17].

The special state ω0 is called the Fock state. If ∥ϕ∥= 1, we will callωϕ a peripheral coherent state.

For any ϕ we will denote by πϕ the GNS-representationassociated with ωϕ, and call it the coherent representation associated with ϕ. For thespecial case q = 0, coherent states in this sense have been studied in [7].The proof of Theorem 2 is based on an analysis of the case of a single relation.

Wesummarize the relevant results in the following Lemma. The assumption that a is boundedis essential for this result, i.e.

there are also unbounded operators satisfying the relation,and the conclusion fails for these.3 Lemma. Let |q| < 1, and let a ≡a(e1) with ∥e1∥= 1 be a bounded operator on aHilbert space R satisfying the relation aa∗= (1 −q)1I + qa∗a.

(1) Then a is reduced by a unique decomposition R ∼= (R0 ⊗R′) ⊕R1, such that(a) if R1 ̸= {0}, a|`R1 is unitary.4

(b) if R′ ̸= {0}, a|`R0 ⊗R′ acts as a = a0 ⊗1I, where a0 is given explicitly as theweighted shifta∗0|n⟩= (1 −qn+1)1/2 |n + 1⟩,(5)where |n⟩for n = 0, 1, . .

. is an orthonormal basis of R0.

(2)∥a∥= 1for q > 0, or a unitary√1 −qfor q < 0, and a not unitary. (6)(3) There are functions β+(q) < ∞and β−(q) > 0 such thatβ−(q)1I ≤an(a∗)n ≤β+(q)1I,(7)uniformly for n ∈IN.

In particular, the spectral radius of a is equal to 1. (4) Let a∗ξ = λξ for ξ ̸= 0.

Then ξ ∈R1, |λ| = 1, and aξ = λξ.Proof : For (1) see [18]; for (2) see [6,18]. (3) For the unitary part a|`R1 we only need β−(q) ≤1 ≤β+(q), which will be true for theβ± constructed below.

Hence it suffices to take a = a0. Thenan0(a∗0)n|k⟩= λk+1 · · · λk+n|k⟩,where λk = (1 −qk).

We will take β± as the supremum (resp. infimum) over all productsQk∈M λk for M ⊂IN.

Explicitly,β+(q) = 1q ≥0Q∞k=11 −q2k+1q ≤0β−(q) = Q∞k=11 −qkq ≥0Q∞k=11 −q2kq ≤0(8)Since these products (related to Theta functions, and to “q-factorials” [1]) are absolutelyconvergent, β±(q) is finite and non-zero for all q, |q| < 1. For computing the spectralradius we let n →∞in the inequalityβ−(q)1/2n ≤∥an∥1/n ≤β+(q)1/2n.

(4) Given the decomposition it suffices to show that a∗0ξ = λξ implies ξ = 0. This followsimmediately from the weighted shift structure (5) of a∗0, by solving the recursion for thecoefficients ξn in ξ = Pn ξn|n⟩.Consider the GNS-representation πϕ associated with the coherent state ωϕ.

This hasa cyclic vector Ωϕ, which is a joint eigenvector of the generators, i.e.a(f)Ωϕ = ⟨f, ϕ⟩Ωϕ. (9)Conversely, any unit vector satisfying (9) will give the coherent state via ωϕ(X) = ⟨Ωϕ, XΩϕ⟩.Therefore, in order to show that ωϕ is positive, it is sufficient to exhibit such a vector ina representation which is known to be positive.

Now the Fock representation π0 has been5

proven to be positive [6,13,36,20]. Hence it suffices to find such vectors in the Fock rep-resentation.

The basic construction for such vectors can be carried out in the case of asingle generator. For Boson commutation relations the operator transforming the vacuuminto a coherent state is well-known to be expza∗.

For the q-relations a similar role isplayed by the “q-exponential” function Expq, defined by the functional equation [21]Dq Expq(z) ≡Expq(z) −Expq(q z)z −qz= Expq(z). (10)The q-exponential satisfies no simple addition formula, and therefore the operator connect-ing different coherent state can only be expressed as a quotient of two such exponentials.Rather than defining first the q-exponential, and then studying its invertibility, we define,in the following Lemma, all these quotients at the same time.

The connection with theq-exponential is Vα0(z) = Eqαz/(q −1).4 Lemma. (1) Let |q| < 1, and α, β ∈IR.

Then the functional equationVαβ(qz) = 1 −αz1 −βz Vαβ(z);Vαβ(0) = 1(11)has a unique analytic solution near z = 0, which is analytic for |αz| < 1. For |αz| < 1,and |βz| < 1, and γ ∈IR we have VαβVβγ = Vαγ.

(2) Let a be a bounded operator on a Hilbert space R with aa∗= (1 −q)1I + qa∗a. Then,for Ωβ ∈R, and |α| < 1 we have the implicationa −βΩβ = 0⇒a −αVαβ(a∗)Ωβ = 0,(12)where the function Vαβ is evaluated on a∗in the analytic functional calculus.Proof : Let Vαβ(z) = Pk ckzk.

Then equation (12) together with the iterated relationa(a∗)k = qk(a∗)ka + (1 −qk)(a∗)k−1(13)gives a functional equation for the coefficients ck:ck+1 = α −qkβ1 −qk+1 ck;c0 = 1..(14)By an elementary computation this is the same recursion which holds for the coefficientsof Vαβ defined through the functional equation. By standard theorems on power series itsradius of convergence is |α|−1.

The chain relation VαβVβγ = Vαγ follows directly from thefunctional equation.Proof of Theorem 2: Let ω be a state satisfying equation (4). Then we can compute iton any polynomial in the generators by Wick ordering the polynomial, and then applyingsuccessively equation (4) and its adjoint ω(Xa(g)) = ⟨g, ϕ⟩ω(X).

Since polynomials aredense in EH(q), ω = ωϕ is uniquely determined. It is also clear that ωϕ must be a pure state,6

since it is the only state on which the positive elements (a†(f) −⟨ϕ, f⟩1I)(a(f) −⟨f, ϕ⟩1I)have zero expectation for all f ∈H.If there is a state ωϕ we have, for ∥f∥= 1: |⟨ϕ, f⟩|2 = ωϕa†(f)Na(f)N1/N ≤1, sincethe spectral radius of a(f) is 1 by Lemma 3(2). With f = ϕ/ ∥ϕ∥this implies ∥ϕ∥≤1.It remains to be shown that ωϕ(X∗X) is positive for ∥ϕ∥≤1, and X ∈EH(q).

Sincepolynomials in the generators are norm dense in EH(q) by the universal property, it sufficesto show this for polynomials X. For such X, ωϕ(X∗X) is obviously a continuous functionof ϕ.

Hence it suffices to show positivity for ∥ϕ∥< 1.Let ∥ϕ∥< 1.We know from [6] that ω0, the Fock state, is positive.By Lemma3(2), a†(ϕ) has spectral radius < 1. Hence we can apply V10 from Lemma 4 to a†(ϕ)in the analytic functional calculus.Let V ≡V10(a†(ϕ)) = V∥ϕ∥,0(a†(ϕ/ ∥ϕ∥)).Thensince V0,∥ϕ∥(a†(ϕ/ ∥ϕ∥)) = V −1 we have that Ωϕ = V Ω0 is non-zero.

By Lemma 4 wehave a(ϕ)Ωϕ = ⟨ϕ, ϕ⟩Ωϕ. On the other hand, when ψ ⊥ϕ, we have a(ψ)a†(ϕ)nΩ0 =qna†(ϕ)na(ψ)Ω0 = 0.

With the series expansion for V we find a(ψ)Ωϕ = 0. Combiningthis with the result for ϕ = ψ we get a(ψ)Ωϕ = ⟨ψ, ϕ⟩Ωϕ, and ωϕ(X∗X) = ⟨Ωϕ, X∗XΩϕ⟩=∥XΩϕ∥2 ≥0.The proof gives more information than just the positivity of ωϕ: by composing theoperators V10(a†(ϕ)) and V10(a†(ψ))−1 we get the following consequence:5 Corollary.

For ∥ϕ∥, ∥ψ∥< 1, the states ωϕ and ωψ are connected by an invertibleelement vϕψ ∈EH(q) via ωϕ(X) = ωψ(v∗ϕψXvϕψ).The operators vϕψ are Radon-Nikodym derivatives in the sense of [31]. Since they aredefined by norm convergent series, they end up in the C*-algebra EH(q), and not merelyin some bigger von Neumann algebra.We close this section with a brief discussion of the coherent states for certain variationsof the q-relations found in the literature.

Most of the literature is concerned with the Fockrepresentation of the relations with a single generator, and the relations are frequentlywritten in a form explicitly involving the number operator N of the Fock representation.This operator is defined by exp(itN)a†(f) exp(−itN) = a†exp(it)f, and NΩ0 = 0,where Ω0 is the Fock vacuum. We will continue to denote by a†(f) the generators with theconventions fixed in Proposition 1.

Then the generators found elsewhere are b†(f) withb†(f) = β qαN a†(f) = β a†(f)qα(N+1)b(g)b†(f) = |β|2(1 −q) ⟨f, g⟩q2α(N+1) + q2α+1 b†(f)b(g). (15)The normalization used in this paper agrees with [28], implicitly with [35], and with oneof the versions introduced by [27] (written with a different parameter eq = q−1/2).

In most7

of the papers in the bibliography we have the convention β = (1 −q)−1/2, α = 0. Theexistence of a vector Ψ ̸= 0 in Fock space withb(f)Ψ = ⟨f, ϕ⟩Ψ(16)is then equivalent to ∥ϕ∥≤(1 −q)−1/2, and the joint eigenvectors of the b(f) are preciselythose of the a(f).On the other hand, when α > 0, the series for Ψ satisfying (16) diverges for allf ̸= 0, and no joint eigenvectors can be found.

The interesting cases are for α < 0. Thecoefficients of the power series then decrease more rapidly, and the series defines an entirefunction.

Hence no constraint is placed on ∥ϕ∥, and the notion of peripheral coherent statesmakes no sense. This is related to the fact that the relations then explicitly involve theoperator N, and hence make sense only in the Fock representation.

The relations appearedfor the first time (for a single generator, a so called q-oscillator) in [4] with α = −1/4,|β|2 = q1/2(1 −q)−1/2, and in [27] with the same constants, but using eq = q−1/2. Ofpotential interest is also the case α = −1/4, |β|2 = q(1 −q)−1 in which the relations canbe expressed by an ordinary commutator, i.e.

[a(f), a†(g)] = ⟨f, g⟩q−N.3. Peripheral coherent statesA remarkable fact about the peripheral coherent states, i.e.

the coherent states ωϕ with∥ϕ∥= 1, is the following: if H′ ⊂H, there is a canonical embedding EH′(q) ֒→EH(q). Withrespect to this embedding a peripheral coherent state on EH′(q) has a unique extension toEH(q), which is also a peripheral coherent state.

This follows readily from the first item ofthe following Proposition.7 Proposition. Let ϕ ∈H, with ∥ϕ∥= 1.

Then(1) ωϕ is the uniquely characterized by the condition ωϕ(a†(ϕ) −1I)(a(ϕ) −1I)= 0. (2) For dim H > 1 the kernel of the GNS-representation πϕ contains every closed two-sidedideal of EH(q).Proof : We set a = a(ϕ), for short.

(1) Let Ωdenote the GNS-vector of a state ω with ω(a∗−1I)(a −1I)= 0. Then we haveaΩ= Ω, and since on the subspace generated by the (a∗)nΩ, a is unitary, we also havea∗Ω= Ω.

Then for any vector ψ ∈H, we geta(ψ)Ω= a(ψ)(a∗)nΩ= (1 −qn)⟨ψ, ϕ⟩(a∗)n−1Ω+ qn(a∗)na(ψ)Ω.Since ∥(a∗)n∥is uniformly bounded we can take the limit n →∞on the right hand side,and obtain a(ψ)Ω= ⟨ψ, ϕ⟩Ω. Hence Ωimplements ωϕ.8

(2) Let J ⊂EH(q) be a closed two-sided ideal, and consider the algebra eE = EH(q)/Jwith quotient mapping η : EH(q) →eE. Since dim H > 1 we know from Proposition 4 in[18] that η(a) ∈eE cannot be unitary, and consequently that the spectrum of η(a) containsthe spectrum of a0, the generator in the Fock representation.

This is the unit disk, andhence the spectrum of η(a) includes 1. It follows (by compactness of the state space ofa C*-algebra) that there is a representation eπ : eE →B(R) in which 1 is an eigenvalue ofeπ(η(a)).

But then by part (1) we have⟨ξ, π(η(X))ξ⟩= ωϕ(X),(∗)where ξ is the corresponding normalized eigenvector. The kernel of πϕ is the set of Y ∈EH(q) such that ωϕ(X∗Y Z) = 0 for all X, Z ∈EH(q).

By equation (∗) it is now plain thatJ = ker η ⊂ker π ◦η ⊂ker πϕ.The second part of this Proposition suggests the following terminology:8 Definition. Let H be a Hilbert space with dim H > 1.

Then the q-Cuntz algebraOH(q) over H is the quotient of EH(q) by its unique largest ideal. Equivalently, OH(q) =πϕEH(q)for any peripheral coherent representation.Of course, for q = 0 the q-Cuntz algebra is just the usual Cuntz algebra Odim H.For dim H < ∞Conjecture C says that OH(q) ∼= OH(0), and this is proven [18] for|q| <√2 −1.

We will further extend this interval in Section 4, using the results of [12].When dim H = ∞, one can show that the Fock representation of EH(q) is simple [24].Hence in that case, OH(q) is isomorphic to the Fock representation of EH(q).From Corollary 6 we know that the non-peripheral coherent representations are allequivalent.For the peripheral coherent representations we know that the C*-algebrasπϕ(EH(q)) are all equal. However, the von Neumann algebras πϕ(EH(q))′′ are not: in thefollowing Proposition we show that all peripheral coherent representations are disjoint.9 Proposition.

Let ϕ, ψ ∈H, with ∥ϕ∥= ∥ψ∥= 1, and let π : EH(q) →B(R) be anyrepresentation. (1) The strong operator limitP(ϕ) = limn→∞1nnXk=1πa†(ϕ)k(15)exists, and is a self-adjoint projection.

(2) For χ ∈H:πa(χ)P(ϕ) = ⟨χ, ϕ⟩P(ϕ). (3) Let P(0) denote the orthogonal projection onto the spaceN =Ω∈R ∀ϕ ∈H : πa(ϕ)Ω= 09

of Fock vectors. Then, for ϕ, ψ unit vectors in H, or zero, and for X ∈EH(q):P(ϕ) π(X) P(ψ) =ωϕ(X)P(ϕ)ϕ = ψ0ϕ ̸= ψ.Proof : In the proof we will suppress the representation π for notational convenience.

Theexistence of the limit (1) follows from the Mean Ergodic Theorem (e.g. Corollary VIII,5.4in [11], and the fact that the powers a†(ϕ)k are uniformly norm bounded by Lemma 3.

(3).Let χ ∈H. Thena(χ) 1nnXk=1a†(ϕ)n = 1nnXk=1n(1 −qk)⟨χ, ϕ⟩a†(ϕ)k−1 + qka†(ϕ)ka(χ)o= ⟨χ, ϕ⟩1nnXk=1a†(ϕ)n + Restwith the estimate∥Rest∥≤1n1I −a†(ϕ)n + 1n11 −|q|β+(q),and β+(q) from equation (8).

Taking the strong limit n →∞we find (2). In particular, wehave a(ϕ)P(ϕ) = P(ϕ), which implies P(ϕ)∗P(ϕ) = weak−lim(1/n) Pnk=1 a(ϕ)kP(ϕ) =P(ϕ), and hence that P(ϕ) is an orthogonal projection.To prove (3), let X be a polynomial in the generators, which we may assume to be Wickordered.

Then after finitely many applications of (2) we find that P(ϕ)XP(ψ) is equalto some factors times P(ϕ)P(ψ). If ϕ = ψ(possibly ϕ = ψ = 0), the factors add up toωϕ(X), and the result follows because P(ϕ) is a projection.It remains to show that P(ϕ)P(ψ) = 0, when ϕ ̸= ψ, and ϕ ̸= 0.Since P(ϕ) isalso the weak limit of (1/n) Pnk=1 a(ϕ)k this follows from (2) and the observation thatlimn→∞(1/n) Pnk=1⟨ϕ, ψ⟩k vanishes, unless ϕ = ψ.We can use the universal representation for π.

Then the projections P(ϕ) are inter-preted as projections in the universal enveloping algebra EH(q)∗∗. By (4) their centralsupports in EH(q)∗∗are mutually disjoint.

Hence, the projections P(ϕ) with ∥ϕ∥= 1, andthe single projection P(0) (for all the non-peripheral coherent representations) preciselylabel the quasi-equivalence classes of coherent representations.From (3) one readily concludes that any representation space R can be split into adirect sum R = Rϕ ⊕R⊥ϕ , where Rϕ is the cyclic subspace containing ¶(ϕ)R. Then therepresentation restricted to the first summand is a direct multiple of πϕ with multiplicitydim P(ϕ)R. The decomposition into a Fock and a non-Fock sector (of which Lemma 3is a special case) is obtained for ϕ = 0. It is especially useful because the orthogonalcomplement R⊥0 has an interesting description [19,20]: it consists of all vectors with an10

“infinite iteration history” with respect to the operators a†(f), i.e. it is the intersectionover n ∈IN of the closed subspaces generated by all vectors of the form a†(f1) · · ·a†(fn)Φwith f1, .

. ., fn ∈R, and Φ ∈R.

This decomposition can be viewed as an analogue of the“Wold decompositon” of a contraction operator in Hilbert space [29].4. Conjecture C and the Fock representationWe begin by making precise the Conjecture C mentioned in the introduction.

We presentit here, not because we are completely convinced of its truth, but because we believe thatit presents an excellent target for future research.10 Conjecture C. Let −1 < q < 1, and let H be a Hilbert space with dim H = d < ∞.Let EH(q), and EH(0) denote the universal algebras introduced in Proposition 1, and denotethe respective generators by a†(f) ∈EH(q), and v†(f) ∈EH(0). Letρ =Pdi=1 a†(ei)a(ei) 12 ∈EH(q)for some (or any) orthonormal basis e1, .

. ., ed ∈H.Then there is a C*-isomorphism η : EH(0) →EH(q) such thata†(f) = ρ ηv†(f).Moreover, 0 is an isolated point in the spectrum of ρ, and the eigenprojection correspondingto this eigenvalue is η1I −Pdi=1 v†(ei)v(ei).Note that this Conjecture can only be formulated for finite d, since the sum definingρ cannot converge in norm (even though it converges strongly in every representation).For this reason the universal C*-algebras for the case of infinitely many generators haveto be treated separately.

For q = 0, d = ∞it is well known that EH(0) ∼= OH(0) is simple,whereas it has an ideal isomorphic to the compact operators for d < ∞.Analogousphenomena occur for q ̸= 0, at least in the Fock representation [24]. There are a numberof interesting equivalent reformulations of the Conjecture.

The following one is the form inwhich this Conjecture was proven [18] for all finite d, and the restricted range |q| <√2−1.11 Proposition. Let −1 < q < 1, and let H be a Hilbert space with dim H = d < ∞,and let e1, .

. ., ed ∈H be an orthonormal basis.

Then Conjecture C is equivalent to theconjunction of the following two statements:(A) In the C*-algebra Md(EH(q)) of d × d-matrices with entries in EH(q), the matrixXij = a(ei)a†(ej) is strictly positive. (B) Let R be a Hilbert space, and let vi, i = 1, .

. ., d be bounded operators on R satisfyingthe relations viv∗j = δij1I.Then there is a unique positive semidefinite boundedoperator ρ on R such that a†(ei) = ρv∗i satisfies relations (1), and such that 1I−P v∗i viprojects onto the kernel of ρ.

Moreover, this unique ρ necessarily lies in the C*-algebragenerated by the operators vi.11

Proof : Assume (A). Consider in the universal representation π : EH(q) →B(R) theoperatorsA† : H ⊗R →RA† : f ⊗ψ 7→πa†(f)ψA : R →H ⊗RA : ψ 7→Pdi=1 ei ⊗a(ei)ψ.Then ρ2 = A†A, and X = AA†.

The polar decomposition of A† takes the form A† = ρV † =V †X1/2. Since X > 0, the components vi of V † : f ⊗ψ 7→= Pi⟨ei, f⟩viψ are in the C*-algebra π(EH(q)), and V V † = 1IH⊗R.

The latter relation translates into viv∗j = δij1I. Withv†(f) := Pi⟨ei, f⟩v∗i , these are the q-relations for v with q = 0.

Hence by the universalproperty there is a homomorphism η : EH(0) →EH(q) with the required property. SinceEH(0) has only one proper two-sided ideal, and this ideal is clearly not annihilated by η(consider the Fock representation of EH(q)), η is injective.

It remains to be shown that ηis onto. This readily follows from condition (B).Conversely, assume that Conjecture C holds.Then in the universal representation ofEH(q) the isomorphism η provides a polar decomposition of the operator A†.

Since thepolar isometry in this case is an isometry, we must have that AA† has no kernel.Ifthe spectrum of AA† in Md(EH(q)) had an accumulation point at zero, zero would alsohave to be an eigenvalue by compactness of the state space, and the universality of therepresentation. Hence the spectrum must be bounded away from zero (A).

Suppose thatvi and ρ are as in (B). Then by the universality of EH(q) there is a unique *-representationΦ : EH(q) →B(H) such that Φ(a†(ei)) = ρv∗i .

The polar decomposition of A† in thisrepresentation is given by ρ and vi. On the other hand, by the isomorphism with EH(0) weknow that ρ = Φ(Pi a†(ei)a(ei)) is in the C*-algebra generated by the v∗i = Φ(v†(ei)).Some consequences of Conjecture C would be the following: (1) The Fock represen-tation of EH(q) is faithful for all q.

(2) EH(q) has only one proper ideal, isomorphic to thecompact operators. (3) the resulting quotient is isomorphic to the Cuntz algebra OH(0).Statement (3) may be extended to a version of Conjecture C on the level of the q-Cuntzalgebras OH(q).

Since OH(q) can be obtained as a quotient of any other representationof EH(q), we can utilize specific information about the Fock representation in approachingthis problem.Dykema and Nica [12], building on results of Zagier [36], were able to verify parts ofConjecture C in the Fock representationπ0. For example, they verified statement (A) ofProposition 11 for that representation, by unitary implementation of a homomorphismη0 : π0(EH(0)) →π0(EH(q))(16)12

satisfying the properties required in Conjecture C, except surjectivity. Their results implya lower boundπ0(X) ≥1 −q1 −|q| ε(|q|) 1I > 0ε(s) =∞Yk=11 −sk1 + sk =∞Xk=−∞(−1)k sk2(17)for X ∈Md(EH(q)), Xij = a(ei)a†(ej).

(We remind the reader of the difference innormalization between [12], and this paper). Moreover, they showed surjectivity of η0 forq2 < ε(|q|)i.e.|q| <≈0.44.

(18)We can immediately translate these results into a partial verification of Conjecture C onthe Cuntz algebra level:12 Theorem. Let −1 < q < 1, and let H be a Hilbert space with dim H = d < ∞.Let OH(q), and OH(0) ≡Od denote the q-Cuntz algebra, and the Cuntz algebra, as inDefinition 8, and denote by π1 : EH(q) →OH(q) (or EH(0) →OH(0)) the respectivequotient maps.

Let ρ ∈EH(q) be as in Conjecture C.(1) Then there is a (not necessarily surjective) C*-homomorphism bη : OH(0) →OH(q)such that π1(a†(f)) = π1(ρ) bηπ1v†(f). (2)π1(ρ2) ≥1 −q1 −|q|ε(|q|)1I.

(3) Let ωϕ be a peripheral coherent state on OH(q). Then ωϕ◦bη is the peripheral coherentstate on OH(0) associated with ϕ.

(4) When q2 < ε(|q|), bη is onto, and hence an isomorphism.Proof : The eigenprojection onto the kernel of π0(ρ) is P0 ≡η0(1I −π0(P v∗i vi)) ∈π0(EH(q)).Consider a peripheral coherent representation πϕ of π0(EH(q)).Since pe-ripheral coherent states are pure, this representation is irreducible. On the other hand,πϕ(P0) projects onto the set of Fock vectors in that representation.

The invariant subspacegenerated from a Fock vector is a copy of Fock space, on which the projection P(ϕ), as inProposition 9, vanishes. On the other hand, P(ϕ) ̸= 0, so the Fock sector cannot be thewhole space, and must be zero by irreducibility.

Hence πϕ(P0) = 0, and πϕ(ρ) > 0. Thebound (2) then follows from equation (17).

Moreover, under the mapEH(0)π0−→π0(EH(0))η0−→π0(EH(q))πϕ−→πϕ(EH(q)) ∼= OH(q)1I−Pi v†(ei)v(ei) becomes πϕ(P0) = 0. Hence it lifts to the quotient as bη : OH(0) →EH(0).The properties (1), (4) of this map are readily verified from those proven for the Fockrepresentation.13

To see (3), recall from Lemma 3 that the eigenvalue equation πϕa(ϕ)−1ωϕ = 0 can onlybe satisfied when we also have πϕa†(ϕ)−1ωϕ = 0. Hence with a basis e1 = ϕ, e2, .

. ., ed ∈H we getπϕ(ρ2)Ωϕ =Xiπϕa†(ei)a(ei)Ωϕ = πϕa†(e1)a(e1)Ωϕ = Ωϕ.Since πϕ(ρ) > 0, this entailsπϕbηv(f)Ωϕ = πϕ(ρ)−1 πϕ(a(f))Ωϕ = ⟨f, ϕ⟩πϕ(ρ)−1Ωϕ = ⟨f, ϕ⟩Ωϕ.Therefore, for X ∈EH(0), ωϕbη(Xv(f))= ⟨f, ϕ⟩ωϕbη(X), which proves (3).5.

The boundary points q = ±1Apart from Conjecture C and its special cases, an interesting problem concerning the q-relations (1) is to show that they define a continuous field of C*-algebras EH(q) in theparameter q in the sense of Dixmier [10].If Conjecture C holds, i.e. an isomorphismηq : EH(q) →EH(0) exists, this problem amounts, for q ̸= ±1, to the question whether theelement η−1q (ρ) ∈EH(0) depends continuously on q.

(For |q| <√2 −1, this continuity iseasily verified from the argument in [18]). The interesting questions arise at the boundariesq = ±1.The role of the coherent states is that of a continuous field of states in the followingsense: for any polynomial X in the variables a†(f), a(g), and q (with f, g ∈H), and forevery fixed ϕ ∈H, the coherent expectation ωϕ(X) is a continuous function of q. Thecontinuity of the field q 7→EH(q) is related to the existence of sufficiently many suchcontinuous fields of states.As a first step towards understanding the continuity at q = ±1, we compute thealgebras EH(±1), and their coherent states.

Recall that for q = 1 we have imposed, inProposition 1, the bound Pi a†(ei)a(ei) ≤1I for any family of orthogonal vectors.13 Proposition. Let H be a Hilbert space.

Then EH(+1) is isomorphic to the algebra ofweakly continuous functions on the unit ball of H. A state on this algebra is coherent ifand only if it is pure.Proof : We have to show first that EH(+1) is abelian. Clearly, [a†(f), a(g)] = 0 for all f, g.In particular, each a†(f) is a bounded normal operator.

By Fuglede’s Theorem [14,30],a†(f) and a†(g) also commute, and EH(1) is abelian. A pure state ω must be multiplicative,and is hence determined by its value ω(a†(f)) on the generators.

Since a† is linear, andsince a†(f)a(f) ≤∥f∥2 1I, this expression must be a bounded linear functional, and henceof the form ω(a†(f)) = ⟨ϕ, f⟩with ϕ ∈H, ∥ϕ∥≤1. Hence ω = ωϕ is coherent, and anycoherent state is obtained in this way.

Note that, for all polynomials X in the generators14

a†(f), a(f), the function ϕ 7→ωϕ(X) is weakly continuous on the unit ball. On the otherhand, the algebra of such polynomials is dense in the algebra of all weakly continuousfunctions by the Stone-Weierstraß Theorem.Note that by this Proposition the set of coherent states is faithful at q = +1.

Thissuggests that they may be useful for proving the continuity at q = 1, provided one canshow that collection of coherent representations is also faithful for q < 1. In the followingProposition we see that faithfulness does not hold at the other limit point q = −1, wherethe relations becomea(f)a†(g) + a†(g)a(f) = 2 ⟨f, g⟩1I.

(21)The Proposition is based on well-known results in the theory of Clifford algebras [5,32,25],which arise from these relations either by taking f, g to be in a real Hilbert space, andsetting a†(f) = a(f). In even dimensions this is equivalent to taking (21), and adding therelation that the a†(f) anti-commute with each other.

The algebra arising in this way isthe Fock representation of (21), and we will refer to it as the CAR-algebra [8]. The pointof the following Proposition is that no anti-commutation relation is added, but that sucha relation automatically holds in every irreducible representation.14 Proposition.

Let H be a Hilbert space, and consider the C*-algebra EH(−1) as definedin Proposition 1. Then(1) The elementsa†(f)a†(g) + a†(g)a†(f) = 2bθ(f, g),for f, g ∈H generate the center of EH(q).

(2) The center of EH(−1) is isomorphic to C(S), where S is the set of all symmetric bilinearforms θ : H × H →C such that|θ(f, g)| ≤∥f∥∥g∥for all f, g ∈H, with the coarsest topology making the functions θ 7→θ(f, g) continu-ous. (3) Let θ be a symmetric bilinear form satisfying the above bound, and let N (θ) denote thereal subspace of vectors f ∈H such that θ(f, f) = ∥f∥2.

Let r(θ) denote the dimensionof the complement of N (θ) in H, taken as a real Hilbert space. Let EH(−1, θ) denotethe quotient of EH(−1) by the relations bθ(f, g) = θ(f, g)1I.

Then(a) If r(θ) is finite and even, EH(−1, θ) is isomorphic to the algebra of 2r(θ)/2-dimensional matrices. (b) If r(θ) is finite and odd, EH(−1, θ) is isomorphic to the direct sum of two copiesof the algebra of 2(r(θ)−1)/2-dimensional matrices.

(b) If r(θ) is infinite, EH(−1, θ) is isomorphic to the CAR-algebra on an infinitedimensional Hilbert space.15

Proof :(1) By an elementary computation one verifies that bθ(f, g) commutes with alla(h). Hence bθ(f, g) is normal, and by Fuglede’s Theorem [30] it must also commute witha†(h).

Hence bθ(f, g) is in the center for all f, g ∈H. Let C(S) ⊂EH(−1) denote theC*-algebra generated by the bθ(f, g).

Its spectrum space S is the set of those symmetricbilinear forms θ, which may arise in an irreducible representation of EH(−1), i.e. those θfor which the relationsa(f)a†(g) + a†(g)a(f) = 2 ⟨f, g⟩1Ia†(f)a†(g) + a†(g)a†(f) = 2θ(f, g)1I(∗)have a solution by a bounded linear operator a† : H →B(R) for some Hilbert space R.The rest of the proof depends on the analysis of this set of relations.The unique feature of the relations (1) in at q = −1 is that the symmetry with respect toexchange of a and a† (up to questions of linearity/antilinearity).

Therefore we will considerH now as a real Hilbert space of dimension dimIR(H) = 2 dimC(H), and introduce thehermitian generatorss(f) := 12a†(f) + a(f),which are real linear in f ∈H. From s(f) and the complex structure on H we can recoverthe original generators by the formula a†(f) = s(f)−i s(if).

In terms of the new generatorswe get the relationss(f)s(g) + s(g)s(f) = ℜen⟨f, g⟩+ θ(f, g)o=: 2Θ(f, g)1I. (∗∗)Clearly, Θ is a symmetric, real valued form on the real Hilbert space H. Since Θ(f, f) =s(f)2 ≥0, the positivity of Θ is necessary for the existence of a representation, and hencefor θ ∈S.We claim that the positivity of Θ is equivalent to the inequality in item (2) of theProposition.

Clearly, |θ(f, f)| ≤∥f∥2 is sufficient for Θ(f, f) ≥0. Conversely, assume2Θ(f, f) = ∥f∥2 + ℜeθ(f, f) ≥0.

Substituting f 7→if in this inequality we get that|ℜeθ(f, f)| ≤∥f∥2. Hence by the Schwarz inequality in the real Hilbert space H we have|ℜeθ(f, g)| ≤∥f∥∥g∥, and the result follows by replacing f in this inequality by a complexmultiple of f. It is also easy to see that the rank of Θ is equal to r(θ), as defined in item(3).In order to prove the characterization of S, the joint spectrum of the central elementsbθ(f, g), it remains to be shown that for every Θ ≥0 there is some representation of(∗).

We will simultaneously prove (3) by constructing all such representations (assumingΘ ≥0), and showing that they have the form given in (3) with r(θ) = rank Θ.We can find an orthonormal basis {ei} ⊂H such that Θ(ei, ej) = Θiδij. The generatorss(ei) with Θi = 0 have to be zero, and the remaining ones can be multiplied by Θ−1/2i, sothat (∗∗) becomes equivalent to the relationssisj + sjsi = 2 δij 1I,(∗∗∗)where the si = s∗i , and i = 1, .

. ., rank Θ.Hence the isomorphism type of EH(−1, θ)depends only on rank Θ.

One readily verifies that rank Θ = r(θ), as defined in (3).For finite r(θ), (3) follows from the standard results of the representation theory of Clifford16

algebras (see e.g. Theorems 2 and 3 in §9,No.4 of [5]).

The CAR-algebras are a specialcase of these arguments with θ = 0. For infinitely many generators si we therefore get aninductive limit which we can take along the simple algebras of even numbers of generators[32], and which is identical with the inductive limit defining the CAR-algebra over aninfinite dimensional Hilbert space.

We note that in the case (3b) the center is generatedby the odd elementbs = s1 · · · sr(θ),which is unitary and satisfies bs2 = ±1I, depending on r(θ) modulo 4 [25]. In any case, bshas two eigenvalues ±1 or ±i, which label the two irreducible representations with givenθ, and are exchanged by the parity automorphism defined by a†(f) 7→−a†(f).We have shown that the algebra generated by the elements bθ is isomorphic to C(S) with Sas described in item (2).

It remains to be shown that this algebra coincides with the centerof EH(−1). Let the center of EH(−1) be C(eS) for some compact space eS.

Since C(S) is asubalgebra, we have a canonical continuous surjection p : eS →S. Whenever r(θ) is eventhe relations (∗) have only one irreducible representation, which implies that p−1({θ}) is asingle point.

Otherwise, p−1({θ}) may consist of at most two points, corresponding to thetwo irreducible representations of (∗). The parity automorphism induces a homeomorphismF : eS →eS which leaves all points with even or infinite r(θ) fixed.

Whenever r(θ) is odd,and p−1({θ}) consists of two points, these two points are exchanged by F.Since H has even or infinite real dimension, r(θ) is odd or infinite for a dense subset ofθ ∈S. Now consider some θ with odd r(θ), and let θα ∈S be a net with θα →θ, andr(θα) even for all α.

Let eθα be the net in eS uniquely defined by peθα= θα. Since F iscontinuous, and F eθα = eθα any cluster point eθ of this net must also be fixed under F, andsince p is continuous, we must have p(eθ) = θ.

But the only way eθ ∈p−1({θ}) can be fixedby F is that p−1({θ}) is a single point. It follows that p : eS →S is a bijection, and thecenter of EH(−1) coincides with the algebra C(S) generated by the bθ(f, g).It is clear that the coherent representations of EH(−1) are precisely those for whichθ(f, g) = ⟨ϕ, f⟩⟨ϕ, g⟩(22)is a rank one operator.

The set N (θ) of vectors f with ∥f∥2 = θ(f, f) is either null, when∥ϕ∥< 1, or is the the one-dimensional real subspace spanned by ϕ when ∥ϕ∥= 1. Hencefor the peripheral coherent states on EH(−1) with dim H < ∞, r(θ) is odd.When dim H = 1, all symmetric bilinear forms on H are of the form (21).

Hence in thiscase the set of coherent states provides an everywhere faithful family of continuous fieldsof states. Accordingly, q 7→EH(q) is a continuous field of C*-algebras [28].

For dim H > 1an interesting problem arises here: since the rank one bilinear forms are a low dimensionalsubset of S it is clear that many irreducible representations of EH(−1) are not coherentrepresentations. Is it possible to embed states on such non-coherent representations ofEH(−1) into a continuous field of states for the field q 7→EH(q)?17

AcknowledgementsThis paper grew out of a collaboration [18] with Lothar Schmitt (Osnabr¨uck). It hasbenefited from conversations with R. Speicher, B. K¨ummerer, A. Nica, and K. Dykema,whom we also thank for making their work available to us.

R.F.W. acknowledges financialsupport from the DFG (Bonn) through a scholarship and a travel grant.References[1] R. Askey:“Continuous q-Hermite polynomials when q > 1”, pp.

151-158in: D. Stanton(ed.

):“q-series and partitions”, Springer-Verlag, New York 1989[2] J.C. Baez:“R-commutative geometry and quantization of Poisson algebras”,Adv.Math. 95(1992) 61-91[3] G. Bergmann:“The diamond lemma for ring theory”, Adv.Math.

29(1978) 178-218[4] L.C. Biedenharn:“The quantum group SUq(2) and a q-analogue of the boson opera-tors”, J.Phys.A.

22(1989) L873-L878[5] N. Bourbaki: ´El´ements de math´ematique, Alg`ebre, Chapitre 9, Hermann, Paris 1959[6] M. Bo˙zejko and R. Speicher:“An example of a generalized Brownian motion”,Commun.Math.Phys. 137(1991) 519-531 ;part II in: L.

Accardi(ed. ):“Quantum Probability & related topics VII”, World Sci-entific, Singapore 1992[7] O. Bratteli, D.E.

Evans, F.M. Goodman, and P.E.T.

Jørgensen:“A dichotomy forderivations on On”, Publ. RIMS, Kyoto Univ.22(1986) 103–117[8] O. Bratteli and D.W. Robinson: Operator algebras and quantum statistical mechanics,volume II, Springer Verlag, Berlin, Heidelberg, New York 1981[9] J. Cuntz:“Simple C*-algebras generated by isometries”,Commun.Math.Phys.

57(1977) 173-185[10] J. Dixmier: C*-algebras, North-Holland, Amsterdam 1982[11] N. Dunford and J.T. Schwarz: Linear Operators, I: General Theory,Interscience,New York 1958[12] K. Dykema and A. Nica:“On the Fock representation of the q-commutation relations”,Preprint Berkeley 1992[13] D.I.

Fivel:“Interpolation between Fermi and Bose statistics using generalized com-mutators”,Phys.Rev.Lett. 65(1990) 3361-3364.

Erratum: Phys.Rev.Lett. 69(1992)2020[14] B. Fuglede:“A commutativity problem for normal operators”,Proc.Nat.Acad.Sci.USA 36(1950) 35-40[15] O.W.

Greenberg:“Particles with small violations of Fermi or Bose statistics”,Phys.Rev.D. 43(1991) 4111-4120[16] O.W.

Greenberg and R.N. Mohapatra:“Difficulties with a local quantum field theoryof possible violation of the Pauli principle”, Phys.Rev.Lett.

62(1989) 712-71418

[17] R. Honegger and A. Rieckers:“The general form of non-Fock coherent Boson states”,Publ.RIMS, Kyoto Univ. 26(1990) 945-961[18] P.E.T.

Jørgensen, L.M. Schmitt, and R.F.

Werner:“q-canonical commutation rela-tions and stability of the Cuntz algebra”,Preprint Osnabr¨uck 1992; to appear inPacific.J.Math. [19] P.E.T.

Jørgensen, L.M. Schmitt, and R.F.

Werner:“q-relations and stability of C*-isomorphism classes”,To appear in the proceedings of the Great Plains operatortheory symposium, Iowa 1992[20] P.E.T. Jørgensen, L.M.

Schmitt, and R.F. Werner:“Positive representations of generalWick ordering commutation relations”, In preparation[21] J. Katriel and A.I.

Solomon:“A q-analogue of the Campbell-Baker-Hausdorffexpan-sion”, J.Phys.A. 24(1991) L1139-L1142[22] J.R. Klauder and E.C.G.

Sudarshan: Fundamentals of Quantum Optics, Benjamin,New York 1968[23] B. K¨ummerer and R. Speicher:“Stochastic integration on the Cuntz algebra O∞”,J.Funct.Anal. 103(1992) 372-408[24] B. K¨ummerer, R. Speicher, and M. Bo˙zejko, unpublished results[25] H.B.

Lawson, Jr.and M.-L. Michelson: Spin geometry,Princeton Univ.Press,Princeton 1989[26] Y-Q. Li and Z-M. Sheng:“A deformation of quantum mechanics”, J.Phys.A 25(1992)6779-6788[27] A.J.

Macfarlane:“On q-analogues of the quantum harmonic oscillator and the quantumgroup SU(2)q”, J.Phys.A. 22(1989) 4581-4588[28] G. Nagy and A. Nica:“On the ‘quantum disk’ and a ‘non-commutative circle’ ”,Preprint, Berkeley 1992.

To appear in the proceedings of the Great Plains opera-tor theory symposium, Iowa 1992[29] B. Sz.-Nagy and C. Foia¸s: Harmonic analysis of operators on Hilbert space, North-Holland Publ., Amsterdam 1970[30] W. Rudin: Functional analysis, McGraw-Hill, 1975[31] S. Sakai: C*-algebras and W*-algebras,Springer-Verlag, Ergebnisse 60, New York1971[32] D. Shale and W. Stinespring:“States of the Clifford algebra”, Ann.Math. 80(1964)365-381[33] R. Speicher:“A new example of ‘Independence’ and ‘white noise’ ”,Probab.Th.Rel.Fields 84(1990)141-159[34] R.F.

Werner:“The Free Quon Gas Suffers Gibbs’ Paradox”, Preprint, Osnabr¨uck 1993[35] S.L. Woronowicz:“Twisted SU(2) group.

An example of non-commutative differentialcalculus”, Publ.RIMS, Kyoto Univ. 23(1987)117-181[36] D. Zagier:“Realizability of a model in infinite statistics”,Commun.Math.Phys.

147(1992) 199-21019

---

출처: arXiv:9303.002 • 원문 보기