Star product realizations of kappa-Minkowski space
aa r X i v : . [ m a t h - ph ] A p r Star product realizations of κ -Minkowski space B.Durhuus ∗ Department of Mathematical Sciences,University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen, DenmarkA.Sitarz † Institute of Physics,Jagiellonian University,Reymonta 4, 30-059 Krak ´ow, Poland
Abstract
We define a family of star products and involutions associated with κ -Minkowski space.Applying corresponding quantization maps we show that these star products restricted to acertain space of Schwartz functions have isomorphic Banach algebra completions. For twoparticular star products it is demonstrated that they can be extended to a class of polyno-mially bounded smooth functions allowing a realization of the full Hopf algebra structureon κ -Minkowski space. Furthermore, we give an explicit realization of the action of the κ -Poincar´e algebra as an involutive Hopf algebra on this representation of κ -Minkowskispace and initiate a study of its properties. MSC–2000 : 46L65, 53D55, 16T05
The κ -deformation of Minkowski space was originally proposed in [10] as a Hopf algebrawhose underlying algebra is the enveloping algebra of the Lie algebra with generators x , . . . , x d − fulfilling [ x , x i ] = iκ x i , [ x i , x j ] = 0 , i, j = 1 , . . . , d − , (1)where κ = 0 can be viewed as a deformation parameter, since formally, in the limit κ → ∞ one obtains the commutative coordinate algebra of Minkowski space. Of course, to single outthis limit as the Minkowski space requires some additional structure involving the action of the ∗ [email protected] † [email protected] κ [9]. This was, originally, howthe algebra was conceived and we shall return to this issue in Section 4. For the moment weconcentrate on (1).The first object of this paper is to discuss a class of star-products on R d and associated quantiza-tion maps based on the harmonic analysis on the Lie group associated with (1). The motivationoriginates from a similar approach to the standard Weyl quantization map based on its rela-tion to the Heisenberg algebra of quantum mechanics. For the purpose of later reference let usbriefly recall the main steps in this construction. The Heisenberg algebra associated to a particlemoving on the real line is the three-dimensional Lie algebra defined by the relation [ P, Q ] = iC , where C is a central element. The real form of this algebra with basis iP, iQ, iC has a faithfulrepresentation σ in terms of strictly upper triangular matrices: σ ( i ( aP + bQ + cC )) = a c b . (2)The connected and simply connected Lie group of the algebra is by definition the Heisenberggroup , which we denote by H eis . It is the group of upper triangular matrices of the form T ( a, b, c ) = a c + ab b , (3)which is obtained by exponentiation of (2). The group operations, expressed in this parametriza-tion, are seen to be T ( a, b, c ) T ( a ′ , b ′ , c ′ ) = T ( a + a ′ , b + b ′ , c + c ′ + 12 ( ab ′ − a ′ b )) , (4) T ( a, b, c ) − = T ( − a, − b, − c ) . (5)It follows that H eis is a unimodular group with Haar measure equal to dadbdc . Thus the groupalgebra of H eis can be identified with L ( R ) via the parametrization (3). Let ◦ denote theconvolution product on the group algebra.According to the Stone-von Neumann theorem [11] the non-trivial irreducible unitary repre-sentations of H eis are labelled by the value ~ = 0 of the central element C . Fixing ~ , therepresentation π can be expressed in the form π ( T ( a, b, c )) = e i ~ c U ( a, b ) , (6) ( U ( a, b ) ψ )( x ) = e i ~ ab e ibx ψ ( x − ~ a ) , (7)for ψ ∈ L ( R ) . The corresponding representation of the group algebra, also denoted by π , isthen given by π ( F ) = Z R dadbdcF ( a, b, c ) π ( T ( a, b, c )) = Z R F ♯ ( a, b ) U ( a, b ) , (8)2here F ♯ ( a, b ) = Z dcF ( a, b, c ) e − i ~ c . Clearly, F → F ♯ maps L ( R ) onto L ( R ) and a simple calculation yields ( F ◦ G ) ♯ ( a, b ) = Z R da ′ db ′ F ♯ ( a, b ) G ♯ ( a − a ′ , b − b ′ ) e i ( ab ′ − a ′ b ) , (9)where the last expression is a “twisted” convolution product on R that we shall denote by F ♯ ˆ ◦ G ♯ , and where we have set ~ = 1 for the sake of simplicity.According to (8) we may write π ( f ) instead of π ( F ) when f = F ♯ . With this notation the Weylquantization map W is defined as W ( f ) = π ( F f ) , (10)for f ∈ L ( R ) ∩ F − (L ( R )) , where F denotes the Fourier transform on R , ( F f )( a, b ) = 12 π Z dαdβ f ( α, β ) e − i ( aα + bβ ) . (11)Using π ( F ◦ G ) = π ( F ) π ( G ) for F, G ∈ L ( R ) , we obtain from (8) and (9) that W ( f ∗ g ) = W ( f ) W ( g ) , where f and g are functions on R and their Weyl-product is given by f ∗ g ( α, β ) = F − ( F f )ˆ ◦ ( F g )) , (12)which clearly is well defined when f and g are Schwartz functions.From this definition the familiar expressions (see e.g. [7]) for the Weyl product can easily bederived. Likewise, the Weyl operators W ( f ) can be seen to be integral operators for appropriatefunctions f . In particular, it can be shown that W ( f ) is of Hilbert-Schmidt type if and only if f is square integrable, and in this case k W ( f ) k = 2 π Z dαdβ | f ( α, β ) | , (13)where k · k denotes the Hilbert-Schmidt norm. It follows that the Weyl product can be extendedto square integrable functions and W can be extended to an isomorphism between the resultingalgebra and the Hilbert-Schmidt operators on L ( R ) .It is worth emphasizing that the construction outlined here depends on the chosen parametrization(3). An alternative parametrization preserving the invariant measure is, e.g., ( a, b, c ) → T ( a, b, c + ξab ) , where ξ is a real constant. In this case, one obtains a quantization map W ξ and a star product ∗ ξ that are related to ∗ by W ξ ( f ) = W (Ψ ξ f ) , Ψ ξ ( f ∗ ξ g )) = (Ψ ξ f ) ∗ (Ψ ξ g ) , Ψ ξ is defined by (Ψ ξ f )( α, β ) = e iξαβ f ( α, β ) . It follows that Ψ ξ is an isomorphism of star-algebras of Schwartz functions and, moreover, sinceboth F and multiplication by a phase factor preserve the norm in L ( R ) we have that (13) isalso fulfilled with W replaced by W ξ . In particular, one can verify that W − is the so-calledKohn-Nirenberg quantization map, in which case W − ( f ) is the pseudo-differential operatorwith symbol f . The Weyl map is singled our among the maps W ξ by the property W ( f ) ∗ = W ( ¯ f ) , where ¯ f is the complex conjugate of f .The purpose of this paper to is to develop an approach similar to the preceding to quantizationmaps associated with κ -Minkowski space for d = 2 , which we denote by M κ . In Section 2we introduce the κ -Minkowski group G , analogous to H eis , and via harmonic analysis on G we define a family of products, called star products , and involutions for a class B of Schwartzfunctions on R . Explicit expressions for the star products and operator kernels are obtainedwhich are used to show that those involutive algebras have natural isomorphic Banach algebracompletions. In Section 3 two particular star products associated to the left and right invariantHaar measures on G are discussed. It is shown that they have natural extensions to a certain sub-algebra C of the multiplier algebra of B consisting of smooth functions of polynomial growth.Moreover, it is shown that the resulting algebra has a Hopf star algebra structure furnishing astar product representation of κ -Minkowski space. In Section 4 we show how to represent theaction of the κ -Poincar´e algebra P κ on κ -Minkowski space in this particular realization as wellas on the subalgebra B . On the latter we show that the Lebesgue integral is a twisted trace,invariant under the action of P κ . Finally, Section 5 contains some concluding remarks and afew technical details are collected in an appendix. In the following we restrict attention to d = 2 in which case the Lie algebra defined by (1) is theunique noncommutative Lie algebra of dimension and κ -Minkowski space M κ is its universalenveloping algebra. We set x = x and t = κx and consider the real form of the Lie algebrawith generators it, ix fulfilling [ t, x ] = ix. (14)It has a faithful -dimensional representation ρ given by ρ ( it ) = (cid:18) − (cid:19) , ρ ( ix ) = (cid:18) (cid:19) , (15)and the corresponding connected and simply connected Lie group is the group G of × -matrices of the form 4 ( a, b ) = (cid:18) e − a b (cid:19) , a, b ∈ R , (16)obtained by exponentiating ρ : e iρ ( at + b ′ x ) = (cid:18) e − a − e − a a b ′ (cid:19) . (17)The group operations written in the ( a, b ) coordinates become S ( a , b ) S ( a , b ) = S ( a + a , b + e − a b ) , S ( a, b ) − = S ( − a, − e a b ) . (18)An immediate consequence is Lemma 2.1.
The Lebesgue measure da db is right invariant whereas the measure e a da db isleft-invariant on G . In particular, G is not unimodular. Let A denote the convolution algebra of G with respect to the right invariant measure. Identi-fying functions on G with functions on R by the parametrization (16) then A is the involutiveBanach algebra consisting of integrable functions on R with product ˆ ∗ and involution † givenby ( f ˆ ∗ g )( a, b ) = Z da ′ db ′ f ( a − a ′ , b − e a ′ − a b ′ ) g ( a ′ , b ′ ) , (19) f † ( a, b ) = e a ¯ f ( − a, − e a b ) , (20)where f, g ∈ A and ¯ f is the complex conjugate of f . If π is a unitary representation of G (always assumed to be strongly continuous in the following) it is well known (see e.g. [12]) that π gives rise to a representation, also denoted by π , of A by setting π ( f ) = Z dadb f ( a, b ) π ( S ( a, b )) . (21)Thus, we have π ( f ˆ ∗ g ) = π ( f ) π ( g ) and π ( f † ) = π ( f ) ∗ . (22)Following the same procedure as described for the Weyl quantization above we define the Weylmap W π associated with the representation π by W π ( f ) = π ( F f ) for f ∈ L ( R ) ∩ F − (L ( R )) , where F denotes the Fourier transform (11) on R . It then follows from (22) that W π ( f ∗ g ) = W π ( f ) W π ( g ) and W π ( f ∗ ) = W π ( f ) ∗ where the ∗ -product and the ∗ -involution are defined by f ∗ g = F − (( F f )ˆ ∗ ( F g )) . (23)and 5 ∗ = F − ( F ( f ) † ) , (24)respectively. As in the case of the standard Moyal product, one needs to exercise care aboutthe domain of definition for the right-hand sides of (23) and (24). I this section we restrict ourattention to the subset B of Schwartz functions introduced in the following definition, while anextension to a class of polynomially bounded functions will be discussed in subsequent sections. Definition 2.2.
Let S c denote the space of Schwartz functions on R with compact supportin the first variable, i.e., supp ( f ) ⊆ K × R , where K ⊆ R is compact. Then we define B = F ( S c ) = F − ( S c ) . Proposition 2.3. If f, g ∈ B then f ∗ and f ∗ g also belong to B and are given by f ∗ g ( α, β ) = 12 π Z dv Z dα ′ f ( α + α ′ , β ) g ( α, e − v β ) e − iα ′ v , (25) and f ∗ ( α, β ) = 12 π Z dv Z dα ′ ¯ f ( α + α ′ , e − v β ) e − iα ′ v . (26) respectively.Proof. Invariance of F − ( S c ) under the ∗ -product and the ∗ -involution follows from the factthat S c is an involutive subalgebra of the convolution algebra A as is easily seen from (19) and(20).In order to establish (25), note that its right-hand side equals √ π Z dv ˜ f ( v, β ) g ( α, e − v β ) e iαv , (27)where ˜ f denotes the Fourier transform of f w.r.t. the first variable ˜ f ( a, β ) = 1 √ π Z dα f ( α, β ) e − iaα . Note that the integrand in (27) is a Schwartz function of v, α, β with compact support in v . Thusit suffices to show that the Fourier transform of (27) w. r. t. α, β equals F f ˆ ∗F g . This followsfrom a straightforward calculation using the Plancherel theorem on the β -integral.Concerning (26) we note similarly that the right-hand side equals √ π Z dv ¯˜ f ( − v, e − v β ) e iαv . (28)Here, we note that the integrand is a Schwartz function of v, β, such that Fourier transforming(28) w .r .t . β gives √ π Z dv F f ( − v, − e v b ) e v e iαv = 1 √ π Z dv ( F f ) † ( v, b ) e iαv . (29)Hence, by Fourier inversion and (24) we conclude that the Fourier transform of the right-handside of (26) equals F ( f ∗ ) . This proves (26). 6ote that associativity of the above defined star product on B is an immediate consequence ofassociativity of the convolution product on A . Likewise, f → f ∗ is an involution on B , since f → f † is an involution on A . Thus we have Corollary 2.4. B equipped with the ∗ -product and ∗ -involution defined by (25) and (26) is aninvolutive algebra. It should be noted that the star product and the involution as defined by (23) and (24) areindependent of the choice of representation π of G , while the quantization map W π , that weproceed to discuss next, is indeed representation dependent. G being isomorphic to the identitycomponent of the group of affine transformations on R , its representation theory is well known[8]. In particular, there is a close relationship to the representation theory of the Heisenberggroup [1]. The basic result we shall use is the following, the proof of which is included for thesake of completeness (see also [2]). Proposition 2.5. G has exactly two non-trivial unitary representations π ± . Their action on thegenerators t, x is given by π + ( t ) = − i dds , π + ( x ) = e − s , (30) π − ( t ) = − i dds , π − ( x ) = − e − s , (31) as self-adjoint operators on L ( R ) .All other irreducible unitary representations are one-dimensional of the form π c ( x ) = 0 and π c ( t ) = c for some c ∈ R .Proof. Let π be a unitary representation of G on a Hilbert space H and let v ∈ Dom ( π ( x )) .Differentiating the relation e iαπ ( t ) e ibπ ( x ) e − iaπ ( t ) v = e ibe − a π ( x ) v , (32)which follows from (18), w.r.t. b we get that e − iaπ ( t ) v ∈ Dom ( π ( x )) and e iaπ ( t ) π ( x ) e − iaπ ( t ) v = e − a π ( x ) v , so the two self-adjoint operators e iaπ ( t ) π ( x ) e − iaπ ( t ) and e − a π ( x ) coincide. But since e − a > the spectral subspaces H + , H − and H corresponding to the positive and negative real line and { } , respectively, are identical for π ( x ) and e iaπ ( t ) π ( x ) e − iaπ ( t ) . It follows that those spaces areinvariant under e iaπ ( t ) and e ibπ ( x ) . By irreducibility one of them equals H and the other twovanish.Assume H = H + and define the self-adjoint operator Q by Q = − ln( π ( x )) . Then x = e − Q and by (32) we have exp (cid:0) ibe iaπ ( t ) e − Q e − iaπ ( t ) (cid:1) = exp (cid:0) ibe − a e − Q (cid:1) , e iaπ ( t ) e − Q e − iaπ ( t ) = exp (cid:0) e iaπ ( t ) Qe − iaπ ( t ) (cid:1) = e − Q − a . Taking logarithms gives e iaπ ( t ) Qe − iaπ ( t ) = Q + a and consequently e iaπ ( t ) e ibQ e − iaπ ( t ) = e iab e ibQ . which is recognized as the Weyl form of the canonical commutation relations. Applying theStone-von Neumann theorem [11] we conclude that π = π + . Similarly one shows that π = π − if H = H − , and the case H = H yields the one-dimensional representations as asserted.We will use the notation W ± for W π ± . From the explicit form (6) of the action of the Heisenberggroup in an irreducible representation one obtains the action of G in the representations π ± .Using S ( a, b ) = S (0 , b ) S ( a, the result is π ± ( S ( a, b )) ψ ( s ) = e ± ibe − s ψ ( s + a ) , ψ ∈ L ( R ) . It is now straightforward to determine the action of W ± ( f ) for arbitrary f ∈ L ( R ) ∩F − (L ( R )) .If h ϕ, ψ i denotes the inner product of ϕ, ψ ∈ L ( R ) we get h ϕ, W ± ( f ) ψ i = Z dadbds F f ( a, b ) ϕ ( s ) e ± ibe − s ψ ( s + a )= Z dsdudb ϕ ( s ) F f ( u − s, b ) e ± ibe − s ψ ( u )= √ π Z dsdu ϕ ( s ) ˜ f ( u − s, ± e − s ) ψ ( u ) . Hence we have shown
Proposition 2.6.
For f ∈ L ( R ) ∩ F − (L ( R )) the operators W ± ( f ) are integral operatorson L ( R ) with kernels given by K ± f ( s, u ) = √ π ˜ f ( u − s, ± e − s ) = Z dvf ( v, ± e − s ) e − iv ( u − s ) . As a consequence we can establish the following basic identities.
Proposition 2.7. a) W ± ( f ) is of Hilbert-Schmidt type if and only if the restriction of f to R × R ± is square integrable w.r.t. the measure dµ = | β | − dαdβ , and we have k W ± ( f ) k = 2 π Z R da Z R ± db | f ( α, β ) | dαdβ | β | = 2 π Z R dsdv | f ( v, ± e − s ) | , (33) where k · k denotes the Hilbert-Schmidt norm. ) If W ± ( f ) is trace class then trW ± ( f ) = Z R dsdvf ( v, ± e − s ) . (34) Proof. a) The operator W ± ( f ) is Hilbert-Schmidt if and only if its kernel is square integrable.From Proposition 2.6 we get Z dsdu | K ± f ( s, u ) | = 2 π Z dsdu | ˜ f ( u − s, ± e − s ) | = 2 π Z dsdu | ˜ f ( u, ± e − s ) | . Applying the Plancherel theorem on the u -integral then proves the first assertion as well as (33).b) If W ± ( f ) is trace class, then trW ± ( f ) = Z R dsK ± f ( s, s ) , and (34) follows from Proposition 2.6.We note that although B is not contained in L ( R , dµ ) we have that B ∩ L ( R , dµ ) is densein L ( R , dµ ) . Indeed, let B ′ denote the subspace of B consisting of Fourier transforms ofderivatives w. r. t. the second variable of functions in S c . A function f ( α, β ) in B ′ is then ofthe form βg ( α, β ) where g is a Schwartz function, hence f ∈ L ( R , dµ ) . Moreover, if f isorthogonal to B ′ in L ( R , dµ ) then its Fourier transform, considered as a tempered distribution,vanishes as a distribution, hence also as a tempered distribution. Thus f = 0 and we concludethat B ′ is dense in L ( R , dµ ) .It follows from this remark and (33) that the mappings W ± have unique extensions from B ′ to L ( R , dµ ) such that (33) still holds. In particular, the map W : f → W + ( f ) ⊕ W − ( f ) is injective from L ( R , dµ ) into H ⊕ H , where H denotes the space of Hilbert-Schmidt opera-tors on L ( R ) .On the other hand, it is clear from the proof of Proposition 2.7 that any pair of kernels K ± in L ( R ) originate from an f ∈ L ( R , dµ ) , i.e. W is unitary up to a factor √ π . This proves thefollowing extension result. Theorem 2.8.
Let B ′ and W be as defined above and set ¯ B = L ( R , dµ ) . Then the ∗ -product (25) and involution (26) have unique extensions from B ′ to ¯ B , such that ¯ B becomes a Banachalgebra and W an isomorphism, W ( f ∗ g ) = W ( f ) W ( g ) W ( f ∗ ) = W ( f ) ∗ . If we complete the algebra ¯ B in the operator norm, the resulting C ∗ algebra will be that ofcompact operators. Corollary 2.9.
The integral w. r. t. dµ over R × R ± is a positive trace on ¯ B in the followingsense: for any f, g ∈ ¯ B , Z duds ( f ∗ f ∗ )( u, ± e − s ) ≥ and Z duds ( f ∗ g )( u, ± e − s ) = Z duds ( g ∗ f )( u, ± e − s ) . roof. If f ∈ ¯ B then W ( f ) is Hilbert-Schmidt and the first inequality follows from (33). If f, g ∈ ¯ B then W ( f ) W ( g ) is trace class and the second identity follows from Theorem 2.8 and(34).For later use we note the following identities. Proposition 2.10. a) If f, g ∈ ¯ B then Z dαdβ | β | − ( f ∗ g ∗ )( α, β ) = Z dαdβ | β | − f ( α, β ) ¯ g ( α, β ) . b) If f, g ∈ B then Z dαdβ ( f ∗ g ∗ )( α, β ) = Z dαdβ f ( α, β ) ¯ g ( α, β ) , (35) Z dαdβf ∗ ( α, β ) = Z dαdβ ¯ f ( α, β ) . (36) Proof. a) Follows immediately from Proposition 2.7 and Theorem 2.8.b) Using (23) and (24) as well as (19) and (20) we have Z dαdβ ( f ∗ g ∗ )( α, β ) = F ( f ∗ g ∗ )(0) = ( F ( f )ˆ ∗F ( g ∗ ))(0)= ( F ( f )ˆ ∗F ( g ) † )(0) = Z dadb F f ( − a, − e a b ) F g ( − a, − e a b ) e a = Z dadb F f ( a, b ) F g ( a, b ) = Z dαdβ f ( α, β ) ¯ g ( α, β ) . Similarly, we have Z dαdβ f ∗ ( α, β ) = F ( f ∗ )(0) = ( F ( f ) † )(0) = F ( f )(0) = Z dαdβ ¯ f ( α, β ) . In particular, it follows that Z dαdβ ( f ∗ f ∗ )( α, β ) ≥ , f ∈ B , but in general R dαdβ f ∗ g ( α, β ) = R dαdβ g ∗ f ( α, β ) , i.e. R dαdβ is not a trace on B .However, we shall see in Proposition 4.7 that R dαdβ satisfies a twisted trace property. The above procedure can be also applied to the convolution algebra of the left invariant measureon G instead of the right invariant one. It is then convenient to use the parametrization R ( a, c ) = S ( a, e − a c ) , a, c ∈ R , (37)10n which the left invariant measure is dadc by Lemma 2.1. Given a unitary representation π of G , the corresponding quantization map ˜ W π is defined by ˜ W π ( f ) = Z dadc F f ( a, c ) π ( R ( a, c )) = Z dadc F f ( a, c ) π ( S ( a, e − a c )) , for f ∈ L ( R ) ∩ F − (L ( R )) . More generally, let us consider the map W ϕπ given by W ϕπ ( f ) = Z dadc F f ( a, c ) π ( S ( a, ϕ ( a ) c )) , (38)where ϕ is a smooth, positive function on R . Defining ( U f )( a, b ) = f ( a, η ( a ) b ) η ( a ) , for any function f of two variables, where η ( a ) = ϕ ( a ) − , a change of variables in (38) gives W ϕπ ( f ) = π ( U F f ) . (39)The corresponding star-product ∗ φ and involution ∗ ϕ are given by f ∗ ϕ g ( α, β ) = 12 π F − U − (( U F f ) ˆ ∗ ( U F g )) (40)and f ∗ ϕ ( α, β ) = F − U − (( U F f ) † ) , (41)which are easily seen to be well defined for f, g ∈ B . More explicitly, the following resultholds. Proposition 2.11. If f, g ∈ B and ϕ is positive and smooth then f ∗ ϕ g ( α, β ) = 12 π Z dadb ˜ f ( b, ω ( a, b ) e a − b β )˜ g ( a − b, ω ( a, a − b ) β ) e iαa , (42) and f ∗ ϕ ( α, β ) = 12 π Z dv Z dα ′ ¯ f ( α + α ′ , ω ( a, − a ) e a β ) e − iα ′ v , (43) where ω ( a, b ) = η ( a ) ϕ ( b ) e b − a . In particular, the star product ⋆ for the left-invariant measure, obtained for ϕ ( a ) = e − a , be-comes f ⋆ g ( α, β ) = 12 π Z dv Z dα ′ f ( α, e v β ) g ( α + α ′ , β ) e − iα ′ v , (44) and the involution ⋆ for the left-invariant product is f ⋆ ( α, β ) = 12 π Z dv Z dα ′ ¯ f ( α + α ′ , e v β ) e − iα ′ v . (45)11 roof. The first two identities follow by straightforward computation using (40) and (41) andFourier inversion. The last two identities follow from the first two after a change of variablescombined with Fourier inversion. Details are left to the reader.
Definition 2.12. By B ϕ we shall denote the involutive algebra obtained by equipping B with theproduct ∗ ϕ and involution ∗ ϕ . Remark 2.13.
In [6] a star product is obtained by a somewhat different approach involving areducible representation of G acting on functions of two variables. Although the explicit formof that star product is not given in [6], it can be verified that it indeed coincides with (25) .The star product considered in [2] (and in [3–5]) corresponds to the case ϕ ( a ) = − e − a a aboveand has the property that the involution equals complex conjugation. However, this propertydoes not determine the star product uniquely among the products ∗ ϕ , as it holds more generallyif ϕ satisfies the relation ϕ ( − a ) = e a ϕ ( a ) , a ∈ R . The form of the Weyl operators W ϕ ± ( f ) for π = π ± is obtained from (39) and Proposition 2.6by an easy computation that we omit. The result is the following. Proposition 2.14.
Assume ϕ is positive and smooth. For f ∈ L ( R ) ∩ F − (L ( R )) theoperators W ϕ ± ( f ) are integral operators on L ( R ) with kernels given by K ± f ( s, u ) = √ π ˜ f ( u − s, ± ϕ ( u − s ) e − s ) = Z dvf ( v, ± ϕ ( u − s ) e − s ) e − iv ( u − s ) . It can now be seen that the norm and trace formulas (33) and (34) hold independently of thechoice of ϕ . Proposition 2.15. a) W ϕ ± ( f ) is Hilbert-Schmidt if and only if the restriction of f to R × R ± is square integrable w.r.t. the measure dµ and we have k W ϕ ± ( f ) k = 2 π Z R dsdv | f ( v, ± e − s ) | . (46) b) If W ϕ ± ( f ) is trace class then trW ϕ ± ( f ) = Z R dsdvf ( v, ± e − s ) . (47) Proof. a) Using Proposition 2.14 we get k W ϕ ± ( f ) k = 2 π Z duds | ˜ f ( u − s, ± ϕ ( u − s ) e − s ) | = 2 π Z dvdsf ( v, ± ϕ ( v ) e − s ) | = 2 π Z dvdr | f ( v, ± e − r ) | which coincides with (33).b) Similarly, we have trW ϕ ± ( f ) = √ π Z ds ˜ f (0 , ± ϕ (0) e − s ) = Z dvdsf ( v, ± e − s ) as claimed. 12 efinition 2.16. By ¯ B ϕ we denote the Banach algebra obtained by equipping L ( R , dµ ) withthe product and involution defined by (42) and (43) and extended from B ′ using (46) in the samemanner as for the case ϕ = 1 treated previously. Theorem 2.17.
The involutive algebras B ϕ , resp. ¯ B ϕ , where ϕ is positive and smooth, areisomorphic.Proof. For the case of B ϕ we note that F − U F maps B ϕ onto B = B ϕ =1 and is by constructiona homomorphism by (23),(24),(40) and (41). The inverse map is obtained by replacing ϕ by η .The same argument applies to ¯ B ϕ since Theorem 2.15 shows that F − U F is an isometry on B ′ and therefore its extension is an isometry from B ϕ onto B . Remark 2.18.
The quantization maps W ± were also considered in [1] and relations (33) and (34) were likewise derived.For the particular case ϕ ( a ) = e a − a , relations (46) and (47) also appear in [2]. The star algebras B ϕ or ¯ B ϕ defined in the previous section obviously do not contain the coordi-nate functions α and β . Hence, to obtain a representation of M κ with α and β representing thegenerators t and x in (14) we need an extension of the domain of definition for the star productand involution. It is the purpose of this section to exhibit such an extension.As originally mentioned in [9] and developed in [10], M κ has a natural Hopf algebra structure,which arises by dualization of the momentum subalgebra of the κ -Poincar´e Hopf algebra. Thecoalgebra structure ( △ , ε ) and antipode S are defined by △ t = t ⊗ ⊗ t , △ x = x ⊗ ⊗ x ,ε ( t ) = ε ( x ) = 0 ,S ( t ) = − t , S ( x ) = − x . As will be seen in Theorem 3.6 below the extension we present allows a realization of the fullHopf algebra structure of κ -Minkowski space. Unless stated explicitly otherwise we work withthe star product associated with the right invariant measure because of its simple form (25).Analogous results for the ⋆ -product (44) are obtained similarly. C Using standard notation ∂ nα = Q ki =1 ∂∂α i for α = ( α , . . . , α k ) ∈ R k , n = ( n , . . . , n k ) ∈ N k , N = { , , , , . . . } , and with | · | denoting the Euclidean norm on R k we introduce thefollowing function spaces. Definition 3.1.
Let C k be the space of smooth functions f ( α, β ) on R k satisfying polynomialbounds of the form | ∂ nα ∂ mβ f ( α, β ) | ≤ c n,m (1 + | α | ) N n (1 + | β | ) M n,m , (48)13 or all α, β ∈ R k and such that the Fourier transform ˜ f of f (as a tempered distribution) w.r.t. α has compact support in α . Here, n, m ∈ N k are arbitrary and N n , M n,m are constants ofwhich the former is independent of m , and c n,m is a positive constant.Given f ∈ C k we denote by K f the smallest compact subset of R k such that supp ( f ) ⊆ K f × R k and we call K f the α -support of f .For k = 1 we set C = C and we have the canonical inclusion C ⊗ C ֒ → C given by: ( f ⊗ g )( α , α , β , β ) = f ( α , β ) g ( α , β ) , for f, g ∈ C . Remark 3.2.
Note that
B ⊆ C and, additionally, if f ∈ C and p is a polynomial in α, β , then p and pf are in C . In order to extend the ∗ -product to C let f, g ∈ C and define for fixed α, β ∈ R , g α,β ( v ) = g ( α, e − v β ) e iαv , v ∈ R . (49)Motivated by (27) we then set ( f ∗ g )( α, β ) = 1 √ π Z dv ˜ f ( v, β ) g α,β ( v ) (50)which is well-defined since ˜ f ( v, β ) has compact support in v and g α,β is a smooth function. Weshow below that f ∗ g ∈ C and that K f ∗ g ⊆ K f + K g . (51)In fact, viewing the product (27) as a linear map on C ⊗C , it extends to a linear map m ∗ : C → C by setting ( m ∗ F )( α, β ) = 12 π Z dα ′ Z dv χ F ( v ) F ( α ′ , α, β, e − v β ) e i ( α − α ′ ) v = 12 π Z dα ′ Z dv χ F ( v ) F ( α + α ′ , α, β, e − v β ) e − iα ′ v , F ∈ C , (52)where χ F denotes a smooth function on R of compact support such that χ F ( v ) ˜ F ( v , v , β , β ) = ˜ F ( v , v , β , β ) as distributions, that is χ F equals on a neighborhood of the projection of K F on the first axis.Note that (52) coincides with (50) if F = f ⊗ g, f, g ∈ C . Convergence of the double integralin (52) is a consequence of the polynomial bounds (48) for F , as can be seen as follows. Let ζ be a smooth function of compact support on R that equals on a neighborhood of and writethe integral in (52) as a sum of two terms F ( α, β ) and F ( α, β ) where F ( α, β ) = 12 π Z dα ′ Z dv ζ ( α − α ′ ) χ F ( v ) F ( α ′ , α, β, e − v β ) e i ( α − α ′ ) v . (53)14bviously, this latter integral is absolutely convergent and by repeated differentiation w. r. t. α, β it is seen that F is smooth and satisfies polynomial bounds of the form (48). For F , givenby formula (53) with ζ replaced by − ζ , one obtains after N partial integrations w. r. t. v theexpression F ( α, β ) == i N π Z dα ′ Z dv ( α − α ′ ) − N (1 − ζ ( α − α ′ )) ∂ N ∂v N (cid:0) χ F ( v ) F ( α ′ , α, β, e − v β ) (cid:1) e i ( α − α ′ ) v . (54)Choosing N large enough one obtains an absolutely convergent integral as a consequence of(48), using that N n is independent of m . Applying the same argument to derivatives of theintegrand w. r. t. α, β it follows easily that F is smooth and satisfies the bounds (48). In theAppendix it is proven that m ∗ F is independent of the choice of χ F with the mentioned propertyand that supp ( m ∗ F ) ⊆ { v + v | ( v , v ) ∈ K F } × R , (55)of which (51) is a special case. In particular, m ∗ ( F ) has compact α -support and hence we mayconclude that m ∗ ( F ) ∈ C for all F ∈ C .More generally, we can define maps C k +1 → C k by letting m ∗ act on any two pairs of variables ( α i , β i ) , ( α j , β j ) while keeping the other variables fixed. We shall use the notation m ∗ ⊗ and ⊗ m ∗ for the maps C → C where m ∗ acts on ( α , β ) , ( α , β ) and ( α , β ) , ( α , β ) ,respectively. Using arguments similar to those above one proves associativity of m ∗ , that is m ∗ ( m ∗ ⊗
1) = m ∗ (1 ⊗ m ∗ ) . (56)Details are given in the appendix.Next, we proceed to define the involution on C . A convenient form is obtained from (28) which,after a simple change of variables, yields Z dαdβ f ∗ ( α, β ) ˜ φ ( α, β ) = Z dvdβ ¯˜ f ( v, β ) χ f ( v ) φ ( − v, e − v β ) e − v , (57)for φ ∈ S ( R ) , where χ f is an arbitrary smooth function of compact support that equals on aneighborhood of K f . Defining ( R f φ )( v, β ) = χ f ( v ) φ ( − v, e − v β ) e − v , (58)for φ ∈ S ( R ) , it is clear that R f is a continuous mapping from S ( R ) into itself. Hence, itfollows that (57) defines f ∗ as a tempered distribution for any f ∈ C . We refer to the appendixfor a proof that f ∗ is independent of the choice of function χ f with the asserted property. TheFourier transform of f ∗ w. r. t. α is given by e f ∗ ( φ ) = ¯˜ f ( R f φ ) , (59)from which it is clear that the α -support of f ∗ fulfils K f ∗ ⊆ − K f . (60)15ence we can choose χ f ∗ ( v ) = χ f ( − v ) . It is then easily verified that R f R f ∗ φ = χ f φ which by use of (59) gives f f ∗∗ ( φ ) = ¯ e f ∗ ( R f ∗ φ ) = ˜ f ( R f R f ∗ φ ) = ˜ f ( φ ) , since χ f equals on a neighbourhood of K f . This shows that f ∗∗ = f , f ∈ C . In order to show that f ∗ ∈ C for any f ∈ C we first note that f ∗ is, in fact, a function given bythe following generalization of (26), f ∗ ( α, β ) = 12 π Z dα ′ Z dv χ f ( − v ) ¯ f ( α + α ′ , e − v β ) e − iα ′ v . (61)Indeed, convergence of this double integral is a consequence of (48), which can be seen byarguments similar to those for m ∗ as follows. Let again ζ be a smooth function on R of compactsupport that equals on a neighborhood of and write the integral in (61) as a sum of two terms f ∗ ( α, β ) and f ∗ ( α, β ) , where f ∗ ( α, β ) = 12 π Z dα ′ Z dv ζ ( α ′ ) χ f ( − v ) ¯ f ( α + α ′ , e − v β ) e − iα ′ v . (62)Clearly, this latter integral is absolutely convergent and by repeated differentiation w. r. t. α, β itfollows easily that f ∗ is smooth and satisfies (48). For f ∗ one obtains after N partial integrationsw. r. t. v the expression f ∗ ( α, β ) = ( − i ) N π Z dα ′ Z dv (1 − ζ ( α ′ )) ∂ N ∂v N (cid:0) χ f ( − v ) ¯ f ( α + α ′ , e − v β ) (cid:1) e − iα ′ v . (63)By choosing N large enough this integral is absolutely convergent by (48). Applying the sameargument to arbitrary derivatives of the integrand w. r. t. α, β it follows easily that f ∗ is smoothand satisfies (48). Having proven convergence of the integral (61), its coincidence with f ∗ follows easily. Hence, we have shown that f ∗ belongs to C and is given by (61).Evidently, the preceding arguments can be extended to show that, more generally, an involution ∗ on C k is obtained by setting F ∗ ( α, β ) = = (2 π ) − k Z d k α ′ Z d k vχ F ( − v ) ¯ F ( α + α ′ , e − v β , . . . , e − v k β k ) e − i ( α ′ v + ··· + α ′ k v k ) , (64)for F ∈ C k , where χ F is a smooth function that equals on a neighbourhood of K F . We thenhave the following relation, whose proof is given in the appendix, ( m ∗ F ) ∗ = m ∗ ( F ∗ ) ∧ , F ∈ C , (65)16here ∧ denotes the flip operation on C , F ∧ ( α , α , β , β ) = F (( α , α , β , β ) . As a special case we get ( f ∗ g ) ∗ = g ∗ ∗ f ∗ (66)for all f, g ∈ C .To summarize, we have established the following result. Proposition 3.3. C equipped with the ∗ -product (50) and ∗ -involution (61) is an involutivealgebra. This algebra can be viewed as an involutive subalgebra of the multiplier algebra of B : Corollary 3.4. If f ∈ C and g ∈ B then both f ∗ g and g ∗ f are in B .Proof. It suffices to show the result only for f ∗ g , since both B and C are involutive algebras.We know that f ∗ g is in C so it suffices to show that it is a Schwartz function whenever g is.Using (50) we have | f ∗ g ( α, β ) | = 1 √ π (cid:12)(cid:12)(cid:12)(cid:12)Z dα ′ f ( α ′ , β ) F ( χ f g α,β )( α ′ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + | β | ) M kF ( χ f g α,β ) k≤ C ′ (1 + | β | ) M k χ f g α,β k ′ , where k · k , k · k ′ are appropriate Schwartz norms, C, C ′ are constants and we have used (48). If g is a Schwartz function we clearly have k χ f g α,β k ′ ≤ C N ′ ,M ′ (1 + | α | ) − N ′ (1 + | β | ) − M ′ for arbitrary N ′ , M ′ ≥ and suitable constants C N ′ ,M ′ . Hence f ∗ g is of rapid decrease. Similararguments apply to derivatives of (50) w. r. t. α, β , thus proving that f ∗ g is a Schwartz functionif g is. Example 3.5.
It is useful to note the following instances of the ∗ -product.a) If f, g ∈ C where g ( α, β ) = g ( α ) depends only on α then f ∗ g ( α, β ) = f ( α, β ) g ( α ) .b) If f, g ∈ C and f ( α, β ) = f ( β ) depends only on β then f ∗ g ( α, β ) = f ( β ) g ( α, β ) .c) If f ( α, β ) = α and g ∈ C depends only on β then ( f ∗ g )( α, β ) = αg ( β ) + iβg ′ ( β ) , ( g ∗ f )( α, β ) = g ( β ) α. In particular, the constant function is a unit of C and α ∗ g ( β ) − g ( β ) ∗ α = iβg ′ ( β ) . For g ( β ) = β this relation yields a representation of the defining relation (14) in terms ofa ∗ -commutator with t, x corresponding to α, β . Note also that, α ∗ = α and β ∗ = β by (61) .As a result we see that the star algebra C furnishes a representation of κ -Minkowskispace, to be further developed in Theorem 3.6 below. .2 The Hopf algebra C We now proceed to discuss the coalgebra structure on C using the same notation for the coprod-uct, counit and antipode as for M κ . The coproduct is of the standard cocommutative form ( △ f )( α , α , β , β ) = f ( α + α , β + β ) . (67)We show below that this defines a map △ : C → C . The maps △ ⊗ and ⊗ △ have naturalextensions to maps from C to C , for which we shall use the same notation: ( △ ⊗ f ( α , α , α , β , β , β ) = f ( α + α , α , β + β , β ) , (1 ⊗ △ ) f ( α , α , α , β , β , β ) = f ( α , α + α , β , β + β ) . It is evident that △ is coassociative, that is ( △ ⊗ △ = (1 ⊗ △ ) △ , as maps from C to C .The counit ε : C → C and antipode S : C → C are defined by ε ( f ) = f (0 , , (68)and ( Sf )( α, β ) = f ∗ ( − α, − β ) , (69)respectively, for f ∈ C . Obviously, ε and S are linear maps and it easily verified that S = Id C . We now state the main result of this section.
Theorem 3.6. C equipped with the ∗ -product, the ∗ -involution, the coproduct △ , the counit ε and the antipode S defined above is a Hopf star algebra. Moreover, the algebra homomorphism ι from M κ to C defined by ι ( t ) = α , ι ( x ) = β is compatible with the Hopf algebra structure: △ ι = ( ι ⊗ ι ) △ , ε ι = ε , S ι = ι S . (70) Proof.
To show that C is a Hopf algebra we need to check that the coproduct and counit are welldefined, that they are algebra homomorphisms and that S satisfies the conditions of an antipode,where the ∗ -product on C is defined by a straightforward generalisation of (50) to functions of variables, that is ( F ∗ G )( α , α , β , β ) = 1(2 π ) Z dv Z dv Z dα ′ Z dα ′ F ( α + α ′ , α + α ′ , β , β ) G ( α , α , e − v β , e − v β ) e − i ( α ′ v + α ′ v ) , (71)18or F, G ∈ C .That ε is a well-defined homomorphism follows by inserting α = β = 0 in (50), which gives ε ( f ∗ g ) = 1 √ π Z dv ˜ f ( v, g (0 ,
0) = f (0 , g (0 ,
0) = ε ( f ) ε ( g ) . Concerning the coproduct it is evident that △ f is a smooth function satisfying the polynomialbounds (48). Noting that g △ f ( v , v , β , β ) = √ πδ ( v − v ) ˜ f ( v , β + β ) , (72)where δ is the Dirac delta function, it follows that g △ f has compact α -support with K △ f = { ( v, v ) | v ∈ K f } . Hence △ f ∈ C .To show that △ is a homomorphism we write (71) as ( △ f ) ∗ ( △ g )( α , α , β , β ) == 12 π Z dv dv g △ f ( v , v , β , β ) △ g ( α , α , e v β , e v β )) e i ( α v + α v ) . Using (72) this yields ( △ f ) ∗ ( △ g )( α , α , β , β ) = 1 √ π Z dv ˜ f ( v, β + β ) g ( α + α , e − v ( β + β )) e i ( α + α ) v , which is seen to be identical to △ ( f ∗ g )( α , α , β , β ) by (50) and (67), as desired.For the antipode we need to demonstrate the relation m ∗ ( S ⊗ △ = ε Id C = m ∗ (1 ⊗ S ) △ , (73)where S ⊗ and ⊗ S denote the natural extensions to C of the corresponding operators on C ⊗ C . First, we use (61) and (67) to write ( S ⊗ △ f ( α , α , β , β ) = 12 π Z dα ′ Z dv ′ χ f ( v ′ ) f ( α ′ , − e v ′ β + β ) e i ( α − α − α ′ ) v ′ . Since this expression depends on ( α , α ) only through α − α it follows immediately from(52) that m ∗ ( S ⊗ △ f is independent of α . Hence we may set α = 0 and obtain m ∗ ( S ⊗ △ f ( α, β ) =1(2 π ) Z dα ′′ Z dv ′′ Z dα ′ Z dv ′ χ f ( − v ′′ ) χ f ( v ′ ) f ( α ′ , − e v ′ β + e − v ′′ β ) e − iα ′′ ( v ′ + v ′′ ) e − iα ′ v ′ . (74)Let now ζ be a smooth function on R of compact support which equals on a neighborhoodof , and define the functions ζ R , R > , by ζ R ( v ) = ζ ( vR ) , v ∈ R . Then insert ζ R ( α ′′ ) + (1 − ζ R ( α ′′ ))( ζ R ( α ′ + α ′′ ) + (1 − ζ R ( α ′ + α ′′ )) v ′ , v ′′ in the three terms containing at least one factor (1 − ζ R ( α ′′ )) or (1 − ζ R ( α ′ + α ′′ )) , weobtain absolutely convergent integrals that vanish in the limit R , R → ∞ . In other words, theintegral (74) can be obtained as the limit for R , R → ∞ of the absolutely convergent integralsdefined by inserting an extra convergence factor ζ R ( α ′′ ) ζ R ( α ′ + α ′′ ) . For the regularizedintegrals we then obtain, after integrating over α ′′ , the expression π ) / Z dv ′ dv ′′ dα ′ χ f ( − v ′′ ) χ f ( v ′ ) R F ( ζ )( R ( v ′ + v ′′ )) ζ ( α ′ R ) f ( α ′ , − e v ′ β + e − v ′′ β ) e − iα ′ v ′ . It is now easy to verify that the limit for R → ∞ equals π Z dv ′ dα ′ χ f ( v ′ ) ζ ( α ′ R ) f ( α ′ , e − iα ′ v ′ . Finally, letting R → ∞ gives the result √ π Z dα ′ F (( χ f ) )( α ′ ) f ( α ′ ,
0) = f (0 , , where we have used that ( χ f ) equals on a neighborhood of K f . This proves the first equalityin (73). The second one follows similarly.So far, we have demonstrated that C is a Hopf algebra. To show that it is an involutive Hopfalgebra it remains to verify that △ and ε fulfil ( △ f ) ∗ = △ ( f ∗ ) and ε ( f ∗ ) = ε ( f ) , (75)where the involution on C is given by (64). The last relation in (75) is obvious. To establish theformer we note that for f ∈ C and τ ∈ S ( R ) we get from (67) by a simple change of variablesthat the action of g △ f as a distribution on τ is given by g △ f ( τ ) = Z dαdα ′ dβdβ ′ f ( α, β )˜ τ ( α − α ′ , α ′ , β − β ′ , β ′ ) . Clearly, R dα ′ dβ ′ ˜ τ ( α − α ′ , α ′ , β − β ′ , β ′ ) is a Schwartz function of ( α, β ) whose Fourier trans-form w. r. t. α at ( v, β ) equals R dβ ′ τ ( − v, − v, β − β ′ , β ′ ) . Thus, setting ( T τ )( v, β ) = Z dβ ′ τ ( v, v, β − β ′ , β ′ ) , we have g △ f ( τ ) = ˜ f ( T τ ) , and hence by (59) ^ △ ( f ∗ )( τ ) = e f ∗ ( T τ ) = ¯˜ f ( R f T τ ) . Using (58) we have 20 f ( v )( R f T τ )( v, β ) = χ f ( v ) Z dβ ′ τ ( − v, − v, e − v β − β ′ , β ′ ) e − v = χ f ( v ) Z dβ ′ τ ( − v, − v, e − v ( β − β ′ ) , e − v β ′ ) e − v = T ( R f ⊗ R f ) τ ( v, β ) . Inserting this into the previous equation yields ^ △ ( f ∗ )( τ ) = ¯˜ f ( T ( R f ⊗ R f ) τ ) = g △ f (( R f ⊗ R f ) τ ) = ^ ( △ f ) ∗ ( τ ) , which proves the claim.Finally, knowing that △ , ε are homomorphisms and S an antihomomorphism, the compatibilityrelations (70) follow by verifying their validity for the generators t, x . For △ and ε this is trivialto verify. As noted in Example 3.5 we have α ∗ = α and β ∗ = β such that S ( α ) = − α and S ( β ) = − β . On the other hand, the antipode on M κ fulfils St = − t , Sx = − x , which shows that the relationfor S in (70) also holds for the generators. Remark 3.7.
The homomorphism property of △ proven above holds more generally in the form M ∗ ( △ ⊗ △ ) F = △ ( m ∗ F ) , F ∈ C , (76) where M ∗ : C → C denotes the canonical extension of (71) given by M ∗ H ( α , α , β , β ) = 1(2 π ) Z dv Z dv Z dα ′ Z dα ′ χ H ( v ) χ G ( v ) H ( α + α ′ , α + α ′ , α , α , β , β , e − v β , e − v β ) e − i ( α ′ v + α ′ v ) , (77) for H ∈ C , where χ H , χ H denote smooth functions of compact support that equal on a neigh-borhood of the projection of K H onto the first and second axis, respectively. The verification of (77) is left to the reader. Remark 3.8.
We have above used the ∗ -product and ∗ -involution associated with the rightinvariant Haar measure on G to equip C with a Hopf algebra structure compatible with thatof κ -Minkowski space. The reader may easily check that by similar arguments one obtains analternative realization of M κ on the basis of the ⋆ -product and ⋆ -involution associated with theleft invariant Haar measure. A salient feature of κ -Minkowski space is the existence of an action on it of a deformation ofthe Poincar´e Lie algebra [9, 10], the so-called κ -Poincar´e algebra. In two dimensions, the latter21s usually presented as the Hopf algebra with generators E, P and N , the energy, momentumand Lorentz boost, respectively, fulfilling the relations [ P, E ] = 0 , [ N, E ] = P, △ E = E ⊗ ⊗ E , △ P = P ⊗ e − E ⊗ P , [ N, P ] = 12 (1 − e − E ) − P , △ N = N ⊗ e − E ⊗ N , (78)and with counit annihilating the generators whereas the antipode acts according to S ( E ) = − E, S ( P ) = − e E P, S ( N ) = − e E N. As mentioned previously, the algebraic κ -Minkowski space M κ can be defined as the dual ofthe Hopf subalgebra generated by E and P [10]. It is the purpose of this section to exhibitexplicitly the action of the κ -Poincar´e algebra in terms of linear operators on the realization C of M κ developed in the previous section. Moreover, we shall find that by restriction we obtainan action of the κ -Poincar´e algebra on the smaller algebra B .To avoid the appearance of the exponential of E in (78) we prefer to introduce it as an invertiblegenerator E and define the κ -Poincar´e algebra P κ accordingly as the Hopf algebra generated by E, P, E , N fulfilling [ P, E ] = [ P, E ] = [ E, E ] = 0 , [ N, E ] =
P , [ N, E ] = −E P , [ N, P ] = 12 (1 − E ) − P , △ E = E ⊗ ⊗ E , △ P = P ⊗ E ⊗
P , △E = E ⊗ E , △ N = N ⊗ E ⊗
N , (79)and with counit and antipode given by ε ( E ) = ε ( P ) = ε ( N ) = 0 , ε ( E ) = 1 , (80) S ( E ) = − E , S ( E ) = E − , S ( P ) = −E − P, S ( N ) = −E − N . (81)We also observe that, although the κ -Poincar´e algebra was originally introduced without invo-lution, it is easy to verify that E ∗ = E, P ∗ = P, N ∗ = − N, E ∗ = E , (82)defines an involution on P κ making it a Hopf star algebra. Note, however, that the involutiondoes not commute with S . C In order to define the action of
P, E, E on C we first make a slight digression on imaginarytranslations of elements in C . 22et f ∈ C . Since ˜ f has compact α -support it follows (see e.g. [13]) that f can be analyticallycontinued to an entire function of α . The analytic continuation will likewise be denoted by f and is given by f ( α + iγ, β ) = 1 √ π ˜ f ( χ f ( v ) e i ( α + iγ ) v , β ) = 1 √ π Z dα ′ f ( α + α ′ , β ) F ( e − γv χ f ( v ))( α ′ ) . (83)For fixed γ ∈ R we claim that the function T γ f defined by ( T γ f )( α, β ) = f ( α + iγ, β ) (84)belongs to C . Indeed, since F ( e − γv χ f ( v ))( α ′ ) is a Schwartz function of α ′ we get immediatelyfrom (83) that the derivatives of T γ f are obtained by differentiating the integrand and, combin-ing this with (48), it follows easily that T γ f fulfils polynomial bounds of the form (48). That g T γ f has compact α -support follows from g T γ f ( v, β ) = e − γv ˜ f ( v, β ) . (85)From this relation or, alternatively, from the uniqueness of the analytic continuation of f in α we conclude that the imaginary translation operators T γ : C → C form a one-parameter group, T γ + η = T γ T η , γ, η ∈ R , T = Id C . Similarly, n -parameter groups of imaginary translation operators T γ are defined on C n for any n ∈ N and γ ∈ R n . We shall write T γ for T ( γ,...,γ ) , independently of n , for γ ∈ R .We next note the following two properties of these maps. Proposition 4.1.
For fixed γ ∈ R the map T γ : C → C is an algebra automorphism, that is T γ ( m ∗ F ) = m ∗ ( T γ F ) , F ∈ C . (86) Moreover, T γ ( f ∗ ) = ( T − γ f ) ∗ , f ∈ C . (87) Proof.
Since T − γ = T − γ , it is sufficient to verify (86) and (87). By (52) we have m ∗ T γ F ( α, β ) = Z dα ′ Z dvχ F ( v ) F ( α ′ + α + iγ, α + iγ, β, e − v β ) e − iα ′ v . (88)That this is an entire function of z = α + iγ for fixed β is seen as follows. By inserting aconvergence factor ζ R ( α ′ ) into the integrand we have, as seen previously, that the regularizedintegrals converge to the integral (88) as R → ∞ for fixed z . It is easy to see that the conver-gence is uniform in z on compact subsets of C . Since the regularized integrals are obviouslyanalytic in z it follows that the same holds for (88). Hence this is the unique entire functionwhose restriction to R coincides with m ∗ F ( α, β ) for fixed β . But this function is by definitionequal to the lefthand side of (86) for z = α + iγ . This concludes the proof of (86).23oncerning (87) we note that by (61) ( T − γ f ) ∗ ( α, β ) = 12 π Z dα ′ Z dv χ f ( − v ) ¯ f ( α − iγ + α ′ , e − v β ) e − iα ′ v , (89)which by similar arguments as those above is seen to be an entire function of z = α + iγ forfixed β . Since it coincides with f ∗ ( α, β ) for γ = 0 we conclude that it equals the lefthand sideof (87) for all z ∈ C . This proves (87).By the preceding analyticity argument we obtain T γ ( f ∗ )( α, β ) = 12 π Z dα ′ Z dv χ f ( − v ) ¯ f ( α ′ , e − v β ) e − γv e i ( α − α ′ ) v , (90)for f ∈ C , since the right hand side is seen to be an analytic function of z = α + iβ thatcoincides with the right hand side of (89) for γ = 0 .Now we can state the main result of this subsection on the action of the Hopf subalgebra gener-ated by E, P, E , called the extended momentum algebra , on C . Theorem 4.2.
The algebra C is an involutive Hopf module algebra with respect to the followinglinear action of the extended momentum algebra on C : E ⊲ f = − i ∂f∂α , P ⊲ f = − i ∂f∂β , E ⊲ f = T f . (91) Proof.
It is clear that the actions of
E, P, E defined by (91) are linear on C and are mutuallycommuting. Therefore, it only remains to verify the compatibility of the action with the ∗ -product and involution. That E ⊲ ( m ∗ F ) = m ∗ ( E ⊗ ⊲ F + m ∗ (1 ⊗ E ) ⊲ F is obvious from (52) since differentiation w.r.t. α in the integrand is permitted by a standardconvergence argument. For the action of E we have that E ⊲ ( m ∗ F ) = m ∗ ( E ⊗ E ) ⊲ F, which is a special case of (86). Finally, for the action of P we have ( P ⊲ ( m ∗ F )) ( α, β ) = − i (2 π ) Z dα ′ Z dv χ F ( v ) (cid:18) ∂F∂β ( α ′ , α, β, e − v β ) + e − v ∂F∂β ( α ′ , α, β, e − v β ) (cid:19) e i ( α − α ′ ) v , (92)where it is seen that the contribution from the first term in parenthesis evidently equals m ∗ ( P ⊗ F ( α, β ) .On the other hand, from (83) we get (( E ⊗ ⊲ F ) ( α , α , β , β ) = 12 π Z dα ′ Z dv χ F ( v ) F ( α + α ′ , α , β , β ) e − v e − iα ′ v m ∗ (( E ⊗ ⊲ F ) ( α, β ) = 12 π Z dα ′ Z dv Z dα ′ Z dv χ F ( v ) χ F ( v ) F ( α + α ′ + α ′ , α, β, e − v β ) e − v e − iα ′ v + α ′ v . By introducing convergence factors ζ R ( α ′′ ) ζ R ( α ′ + α ′′ ) as in the proof of (73) above we obtainafter integrating over α ′′ and taking the limit R , R → ∞ that m ∗ (( E ⊗ ⊲ F ) ( α, β ) = Z dα ′ Z dv ′ χ F ( v ′ ) F ( α + α ′ , α, β, e − v ′ β ) e − v e − iα ′ v ′ . Using that ( χ F ) equals on a neighborhood of the projection of K F onto the first axis we seethat the second term in parenthesis in (92) yields the contribution m ∗ ( E ⊗ P ) F ( α, β ) . Hence,we have shown that P ⊲ ( m ∗ F ) = m ∗ ( P ⊗ ⊲ F + m ∗ ( E ⊗ P ) ⊲ F , F ∈ C , (93)which concludes the argument that the action of the extended momentum algebra is compatiblewith multiplication on C .Compatibility of the action with the involution is the statement that ( h ⊲ f ) ∗ = ( Sh ) ∗ ⊲ f ∗ , (94)for f ∈ C and h in the extended momentum algebra. It suffices to verify this for the generators E, P, E . For E it follows directly from (87), whereas for E it is a consequence of (61) bydifferentiating both sides with respect to α .Differentiating (61) with respect to β and using (90) we obtain P ⊲ f ∗ = −E ( P ⊲ f ) ∗ , which gives ( P ⊲ f ) ∗ = − ( E − P ) ⊲ f ∗ . Since S ( P ) ∗ = ( −E − P ) ∗ = −E − P it follows that (94) is satisfied for h = P . This completesthe proof of the theorem. P κ on C To represent the boost operator N by a linear action on C we introduce the operators of multi-plication by α and β as ( L α f )( α, β ) = αf ( α, β ) , ( L β f )( α, β ) = βf ( α, β ) , for f ∈ C . 25 emma 4.3. L α and L β are linear operators on C which satisfy the following rules with respectto the product and involution on C : L α ( m ∗ F ) = m ∗ (1 ⊗ L α ) F = m ∗ ( L α ⊗ F + m ∗ (1 ⊗ L β P ) F , (95) L β ( m ∗ F ) = m ∗ ( L β ⊗ F = m ∗ ( E − ⊗ L β ) F , (96) ( L α f ) ∗ = L α f ∗ − L β P f ∗ and ( L β f ) ∗ = E L β f ∗ . (97) Proof.
The two left identities in (95) and (96) follow immediately from (52). From the lastexpression in (52) we obtain L α ( m ∗ F )( α, β ) = m ∗ ( L α ⊗ F ( α, β ) − Z dα ′ Z dvχ F ( v ) α ′ F ( α + α ′ , α, β, e − v β ) e − iαv = m ∗ ( L α ⊗ F ( α, β ) + m ∗ (1 ⊗ L β P ) F ( α, β ) , where the last step follows by a partial integration w. r. t. v . This proves the second identity in(95). Similarly, the second identity in (96) is obtained from L β ( m ∗ F )( α, β ) = Z dα ′ Z dvχ F ( v ) e v F ( α + α ′ , α, β, e − v β ) e − v βe − iα ′ v = m ∗ ( E − ⊗ L β ) F ( α, β ) , where the last step follows by the same argument as in the proof of (93) above.The second identity of (97) follows immediately from (61) and (90). For the first one wemultiply both sides of (61) by α and obtain after a partial integration L α f ∗ ( α, β ) = ( L α f ) ∗ ( α, β ) − ( L β P f ) ∗ ( α, β ) . Using the the second identity of (97) and (94) for h = P the first identity of (97) follows.We are now in a position to extend Theorem 4.2 as follows. Theorem 4.4.
Defining the linear action of N on C by N = − iL α P − i − E ) L β + i L β P , (98) and the action of E, P, E as in (91) then C becomes an involutive Hopf module algebra of P κ .Proof. That
N, P, E and E satisfy the commutation relations of (79) is easily seen by inspection.It remains to check that the action of N on C is compatible with the product and involution on C using the coproduct of (79). By (93) and Lemma 4.3 one gets, for F ∈ C , N ⊲ m ∗ F = (cid:18) − iL α P − i − E ) L β + i L β P (cid:19) m ∗ F = − iL α m ∗ ( P ⊗ F − iL α m ∗ ( E ⊗ P ) F − i L β m ∗ F + i E m ∗ ( L β ⊗ F + i L β m ∗ (cid:0) ( P ⊗ F + 2( E P ⊗ P ) F + ( E ⊗ P ) F (cid:1) . − im ∗ ( L α P ⊗ F − im ∗ ( P ⊗ L β P ) F ) − im ∗ ( E ⊗ L α P ) F − i m ∗ ( L β ⊗ F + i m ∗ ( E L β ⊗ E ) F + i m ∗ ( L β P ⊗ F + im ∗ ( P ⊗ L β P ) F + i m ∗ ( E ⊗ L β P ) F .
Here, two terms are seen to cancel, and using the relation m ∗ ( E L β ⊗ F − m ∗ ( E ⊗ L β ) F = 0 , which follows from (96), we can rewrite the last expression in the form m ∗ (cid:18) − i ( L α P ⊗ F − i L β ⊗ F + i E L β ⊗ F + i L β P ⊗ F ) (cid:19) + m ∗ (cid:18) − i ( E ⊗ L α P ) F − i E ⊗ L β ) F + i E ⊗ E L β ) F + i E ⊗ L β P ) F (cid:19) )= m ∗ (( N ⊗ ⊲ F + ( E ⊗ N ) ⊲ F ) . This proves compatibility of the action of N with the product on C ..Using (94) for h = P and h = E and (97) we get ( N ⊲ f ) ∗ = (cid:18)(cid:18) − iL α P − i − E ) L β + i L β P (cid:19) f (cid:19) ∗ = (cid:18) − iL α E − P + iL β P E − P + i − E − ) L β E − i L β E E − P (cid:19) f ∗ = (cid:18) − iL α P − i − E ) L β + i L β P (cid:19) E − f ∗ = N E − ⊲ f ∗ , Noting that ( S ( N )) ∗ = − ( E − N ) ∗ = N E − , this proves compatibility of the action of N withinvolution.In view of the obvious fact that ∂∂α , ∂∂β , L α , L β and T all map B into itself, the following is aconsequence of Theorem 4.4. Corollary 4.5.
The subalgebra B of C is an involutive Hopf module algebra for P κ with actiondefined by (91) and (98) . Remark 4.6.
By inspection of (79) , (80) , (81) it is seen that setting Λ q ( E ) = E , Λ q ( P ) = P , Λ q ( E ) = E , Λ q ( N ) = N + qP , defines a Hopf algebra automorphism Λ q of P κ for each q ∈ C . As a consequence, one obtainsan involution on P κ for any q ∈ R by replacing N ∗ = − N in (82) by N ∗ = − N + qP . (99)27 or this involution Theorem 4.4 is still valid if N as given by (98) is replaced by N ′ = N + q P .
The particular choice q = 1 ensures that the operator N ′ is antisymmetric w. r. t. the L -innerproduct on B , as is easily verified. More generally, it follows that the action of h ∗ on B in thiscase coincides with that of the adjoint of h w. r. t. the L -inner product on B , for any h ∈ P κ , see Proposition 4.7 below. On B the integral w. r. t. dαdβ is well defined as a linear form that we shall denote by R . In thefollowing proposition we collect some basic properties of R in relation to the module algebrastructure on B . Proposition 4.7. a) The integral w. r. t. the uniform measure on R is invariant under the actionof P κ on B defined above in the sense that, for any h ∈ P κ and f ∈ B , Z h ⊲ f = ε ( h ) Z f . (100) b) R is a left and right invariant integral on the Hopf algebra B in the sense that (cid:18)Z ⊗ Id (cid:19) △ f = Z f = (cid:18) Id ⊗ Z (cid:19) △ f , f ∈ B . (101) c) Z Sf = Z f and Z f ∗ ( Sg ) = Z g ∗ ( Sf ) . (102) d) For any f, g ∈ B and h ∈ C we have Z ( h ⊲ f ) ∗ g ∗ = Z f ∗ ( h ∗ ⊲ g ) ∗ , (103) if the involution on P κ is defined by (99) for q = 1 and the action of E, P, E , N on B are givenby (91) and N ⊲ f = ( − iL α P − i − E ) L β + i P L β P ) f , f ∈ B . (104) e) For any f, g ∈ B we have Z f ∗ g = Z ( E ⊲ g ) ∗ f . (105) which means that R is a twisted trace.Proof. a) It suffices to verify (100) for the generators E, P, E and N . First, since both E and P act on f as partial derivatives Z dαdβ ( P ⊲ f )( α, β ) = 0 = Z dαdβ ( E ⊲ f )( α, β ) . (106)28or E we have Z dαdβ ( E ⊲ f )( α, β ) = Z dαdβf ( α + i, β ) = Z dαdβf ( α, β ) as a consequence of Cauchy’s theorem. Finally, for the action of N , one uses the identities L β P = P L β − P, L α P = P L α , to deduce from the preceding results that Z dαdβ ( N ⊲ f )( α, β ) = 0 . This finishes the proof of a).b) Identities (101) follow trivially from the translation invariance of the measure dαdβ .c) The first identity of (102) follows from (36) and (69): Z dαdβ ( Sf )( α, β ) = Z dαdβ ( Sf )( − α, − β ) = Z dαdβ f ∗ ( α, β ) = Z dαdβ f ( α, β ) . The second identity follows from the former by using that S is an antihomomorphism and S = Id on B .d) By (35) we see that (103) is equivalent to the statement that the action of h ∗ ∈ P κ on B asa linear operator on B ⊂ L ( R ) equals the action of the adjoint of h w. r. t. the standard innerproduct on L ( R ) . That this holds for E and P is clear form (91). For E we have Z ( E ⊲ f ) ∗ g ∗ = Z E ⊲ ( f ∗ ( E − ⊲ g ∗ ) = Z f ∗ ( E ⊲ g ) ∗ , by (86), (87) and (100). This proves (103) for h = E . Since L α and L β are symmetric op-erators on B ⊂ L ( R ) one can now check by direct computation that N as given by (98) isantisymmetric.e) Using Cauchy’s theorem and a change of variables we get from (25) that Z ( E ⊲ g ) ∗ f = 12 π Z dαdβ Z dv Z dα ′ g ( α + α ′ + i, β ) f ( α, e − v β ) e − iα ′ v = 12 π Z dαdβ Z dv Z dα ′ g ( α ′ , β ) f ( α, e − v β ) e − v e i ( α − α ′ ) v = Z dβdv ˜ g ( v, β ) ˜ f ( − v, e − v β ) e − v . A change of variables shows that the last expression equals R dβdv ˜ f ( v, β )˜ g ( − v, e − v β ) whichby reversing the steps above yields R f ∗ g . This completes the proof.29 .3 Explicit dependence on the kappa parameter For the sake of completeness we end this section by reintroducing the κ -parameter which weeliminated at the outset by rescaling the t generator of M κ . The correct dependence on κ for both M κ and P κ is obtained by simply rescaling the variables α, β by κ , i. e. set ( α, β ) = ( κ ˆ α, κ ˆ β ) and express the (co)algebra operations in terms of the dimensionful variables ˆ α, ˆ β , and thenrename the latter ( α, β ) . Explicitly, the ∗ -product on B is replaced by f ∗ κ g ( α, β ) = 12 π Z dα ′ dv f ( α + α ′ , β ) g ( α, e − vκ β ) e − iαv , (107)and the involution is changed to f ∗ ( α, β ) = 12 π Z dα ′ dv ¯ f ( α + α ′ , e − vκ β ) e − iα ′ v , (108)whereas the coproduct and counit are unchanged. Furthermore, the action of the operators E, P, E , N on B are redefined as E ⊲ f = − i ∂f∂α , P ⊲ f = − i ∂f∂β , E ⊲ f = T κ f ,N = − iL α P − iκ (cid:0) − E (cid:1) L β + i κ L β P , where L α , L β denote multiplication by α, β , respectively, as before. With these definitions weobtain a function algebra realization B of M κ and a representation of the involutive Hopf algebra P κ on B , as displayed in e. g. [10].Finally, we note the following series representation of the ∗ κ -product for sufficiently regularfunctions. For simplicity we consider a rather restricted class of functions but the proof can beadapted to more general situations. Proposition 4.8. If f, g ∈ B and g ( α, β ) is an entire function of β then ( f ∗ κ g )( α, β ) = ∞ X n =0 i n κ n n ! ∂ nα f ( α, β ) ( β∂ β ) n g ( α, β ) , for all ( α, β ) ∈ R .Proof. First rewrite (107) as ( f ∗ κ g )( α, β ) = 1 √ π Z dv ˜ f ( v, β ) g ( α, e − vκ β ) e iαv . By analyticity of g ( α, e − v β ) in v we have g ( α, e − vκ β ) = ∞ X n =0 ( − n κ n n ! v n ( β∂ β ) n g ( α, β ) . K f we get ( f ∗ κ g )( α, β ) = 1 √ π ∞ X n =0 ( − n κ n n ! Z dv ˜ f ( v, β ) v n ( β∂ β ) n g ( α, β ) e iαv . Now, use √ π Z ˜ f ( v, β ) v n e iαv = ( − i∂ α ) n f ( α, β ) , to conclude the proof. The star product formulation of the κ -Minkowski algebra presented in this paper has potentialadvantages with regard to future developments. It is a basis-independent construction realizedas a function space with a richer structure than the algebraic version, and with a simpler analyticform of the product than in previous approaches.We consider it as first step towards the construction of a geometry on κ -Minkowski space inthe sense of spectral triples. A primary goal will be to study the equivariant representationsof the algebra B and to look for equivariant Dirac operators. The existence of the invarianttwisted trace on B suggests that the geometry of κ -Minkowski space might be closer to thecase of quantum groups ( q -deformations) than originally believed. In particular, the failure ofthe spectral triple construction for the compactified version of κ -Minkowski space is possiblyrelated to this fact, and the remedy might be to look for twisted spectral geometries.Furthermore, there are interesting relations between the star product formulation of κ -Minkowskispace and the deformations of Rieffel [14] determined by actions of R d . We postpone the dis-cussion of these issues, as well as extensions to higher dimensions, to a future publication. The purpose of this appendix is to show that the definitions (52) and (61) of multiplication andinversion on C are independent of the choice of the functions functions χ f and χ F satisfying thestated properties and to prove (55), (56) and (65). The supportof m ∗ F .
Let F ∈ C ⊗ C . First, observe that by the definition (52) of m ∗ and the ensuing convergencearguments we have, for ϕ ∈ S ( R ) , m ∗ F ( ϕ ) = Z dα ′ Z dv ′ Z dβdαχ F ( v ′ ) F ( α + α ′ , α, β, e − v ′ β ) ϕ ( α, β ) e − iα ′ v ′ . (109)For fixed v ′ , β ∈ R and ξ, η ∈ S ( R ) we have Z dα ′ dαF ( α + α ′ , α, β, e − v ′ β ) F ξ ( α ′ ) F η ( α ) = Z dudu ′ ˜ F ( u, u ′ , β, e − v ′ β ) ξ ( u ) η ( u + u ′ ) . η ( u + u ′ ) = 0 for all ( u, u ′ ) ∈ K F . Since this holds for arbitrary ξ ∈ S ( R ) itfollows that Z dαF ( α + α ′ , α, β, e − v ′ β ) ˜ ϕ ( α, β ) = 0 , if ϕ ( u + u ′ , β ) = 0 for all ( u, u ′ ) ∈ K F . Hence we get from (109) that ] m ∗ F ( ϕ ) = m ∗ F ( ˜ ϕ ) = 0 if ϕ ( u + u ′ , β ) vanishes for ( u, u ′ ) ∈ K F for arbitrary β . This proves (55). Independence of the χ -functions. Let f ∈ C and write f ∗ = f ∗ R + f ∗ R , where f ∗ R and f ∗ R are given by (62) and (63), respectively, with ζ ( α ′ ) replaced by ζ R ( α ′ ) = ζ ( α ′ R ) , and where ζ is a smooth function of compact support that equals on a neighborhood of . Choosing N in (63) sufficiently large, it follows from (48) that f ∗ R converges to uniformlyon compact subsets of R as R → ∞ . Hence, f ∗ R converges uniformly to f ∗ on compactsubsets of R . As the reader may easily verify, this also holds if we set ζ = F ( ζ ) , where ζ is asmooth function with support contained in [ − , such that R ∞−∞ ζ ( v ) dv = 1 , since in this case ζ R ( v ) = R F ( ζ ( Rv )) converges uniformly to on compact subsets of R as R → ∞ . With this choice of ζ R we have f ∗ R ( α, β ) = 1 √ π Z dvχ f ( − v ) Z du Rζ ( R ( v − u )) ¯˜ f ( − u, e − v β ) e iαu . Since the support of u → ζ ( Ru ) is contained in [ − R , R ] it follows that the last integral vanishesfor all v outside any given distance δ > from − K f if R > δ . This proves that the integraldefining f ∗ only depends on the values of χ f in any neighborhood of K f as desired.The proof that m ∗ F, F ∈ C , only depends on the values of χ F in any neighborhood of theprojection of K F onto the first axis is essentially identical to the preceding argument and weskip further details. Associativity of the product.
We consider m ∗ given by (52) and want to verify the relation (56). For G ∈ C we have by (52) ( m ∗ ⊗ G ( α , α , β , β ) = 12 π Z dα ′ Z dv χ G ( v ) G ( α + α ′ , α , α , β , e − v β , β ) e − iα ′ v and m ∗ ( m ∗ ⊗ G ( α, β ) = 1(2 π ) Z dα ′ Z dv Z dα ′ Z dv χ G ( v ) χ ++ G ( v ) G ( α + α ′ + α ′ , α + α ′ , α, β, e − v β, e − v β ) e − i ( α ′ v + α ′ v ) , (110)where χ ++ G is a smooth function of compact support that equals on a neighborhood of the set { v + v | ( v , v , v ) ∈ K G for some v ∈ R } . Similarly, we get m ∗ (1 ⊗ m ∗ ) G ( α, β ) = 1(2 π ) Z dα ′ Z dv Z dα ′ Z dv χ G ( v ) χ G ( v ) G ( α + α ′ , α + α ′ , α, β, e − v β, e − ( v + v ) β ) e − i ( α ′ v + α ′ v ) (111)32ow, rewrite (110) as m ∗ ( m ∗ ⊗ G ( α, β ) = 1(2 π ) Z dα ′ Z dv Z dα ′ Z dv χ G ( v ) χ ++ G ( v ) G ( α + α ′ , α + α ′ , α, β, e − v β, e − v β ) e − i ( α ′ v + α ′ ( v − v )) and insert convergence factors ζ R ( α ′ ) ζ R ( α ′ ) to justify interchange of integrations to obtain m ∗ ( m ∗ ⊗ G ( α, β ) = 1(2 π ) Z dα ′ Z dv Z dα ′ Z dv χ G ( v ) χ ++ G ( v + v ) G ( α + α ′ , α + α ′ , α, β, e − v β, e − ( v + v ) β ) e − i ( α ′ v + α ′ v ) . By an argument similar to the one proving independence of f ∗ on the choice of χ f above,we may in this integral replace the function χ G ( v ) χ ++ G ( v + v ) by any smooth functionof compact support that equals on a neighborhood of the set { ( v , v ) | ( v , v , v ) ∈ K G for some v ∈ R } . Since this holds for the function χ G ( v ) χ G ( v ) we conclude that theintegrals (110) and (111) are equal as desired. 33 he ∗ -operation is an antihomomorphisn. Let F ∈ C and let χ + F , χ F , χ F denote smooth functions of compact support that equal onthe α -support of m ∗ F and on the projections of K F onto the first and second coordinate axis,respectively. Using definitions (52) and (61) we then have ( m ∗ F ) ∗ ( α, β ) = 1(2 π ) Z dα ′ Z dv Z dα ′ Z dv χ F ( v ) χ + F ( − v ) ¯ F ( α + α ′ , α + α ′ , e − v β, e − ( v + v ) β ) e iα ′ v − iα ′ ( v + v ) (112)For the right-hand side of (65), on the other hand, we get m ∗ (( F ∗ ) ∧ )( α, β ) =1(2 π ) Z dα ′ Z dv Z dα ′ Z dv Z dα ′ Z dv χ F ( − v ) χ F ( − v ) χ F ( − v )¯ F ( α + α ′ , α ′ , e − ( v + v ) β, e − v β ) e − i ( α ′ v + α ′ v )+ iαv e iα ′ ( v − v ) . (113)Inserting convergence factors ζ R ( α ′ ) ζ R ( α ′ ) ζ R ( α ′ ) into the last integral we recover its valuein the limit R, R , R → ∞ by the same arguments as above. By performing the α ′ -integrationfirst in the regularized integral we obtain π ) / Z dv Z dα ′ Z dv Z dα ′ Z dv χ F ( − v ) χ F ( − v ) χ F ( − v ) ζ R ( α ′ ) ζ R ( α ′ ) ¯ F ( α + α ′ , α ′ , e − ( v + v ) β, e − v β ) e − i ( α ′ v + α ′ v )+ iαv F ( ζ R )( v − v ) , and in the limit R → ∞ this gives π ) Z dα ′ Z dv Z dα ′ Z dv χ F ( − v ) χ F ( − v ) ζ R ( α ′ ) ζ R ( α ′ ) ¯ F ( α + α ′ , α ′ , e − ( v + v ) β, e − v β ) e − i ( α ′ v + α ′ v )+ iαv . A simple change of variables now yields m ∗ (( F ∗ ) ∧ )( α, β ) = lim R ,R →∞ π ) Z dα ′ Z dv Z dα ′ Z dv χ F ( − v − v ) χ F ( v ) ζ R ( α ′ ) ζ R ( α ′ ) ¯ F ( α + α ′ , α + α ′ , e − v β, e − ( v + v ) β ) e iα ′ v − iα ′ ( v + v ) , Repeating previous arguments we see by choosing ζ R such that F ( ζ R ) has support in [ R , R ] thatin the limit above the function χ F ( − v − v ) χ F ( v ) can be replaced by any smooth functionof compact support that equals on a neighborhood of the set { ( v , v ) | ( v , − v − v ) ∈ K F } without changing the value of the limit. Since this holds, in particular, for the function χ F ( v ) χ + F ( − v ) we conclude that the limit equals (112). This proves (65). Acknowledgement
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