Beurling-Fourier Algebras and Complexification

In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska (Adv. Math. 2021) about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to $G$.


INTRODUCTION
Let G be a locally compact group.The Fourier algebra A(G), introduced by Eymard in [5] , is a subalgebra of C 0 (G) consisting of the coefficients of the left regular representation λ of G, i.e.
The natural norm on A(G) that makes it a Banach algebra is given by where infimum is taken over all possible representations u(s) = λ (s)ξ , η .Moreover, A(G) is the unique predual of the group von Neumann algebra V N (G).
The Gelfand spectrum of the algebra is known to be topologically isomorphic to G, giving a non-trivial link between topological groups and Banach algebras.We note that the spectral theory has been an important tool in understanding commutative Banach algebras.
Several authors, including the second author, have been investigating in [19,21,24,13,20,12] a weighted version of the Fourier algebra, by imposing a weight that changes the norm structure.Weighted Fourier algebras for compact quantum groups were studied in [9].Recall that if G is abelian with the dual group Ĝ, the Fourier algebra A(G) is isometrically isomorphic via the Fourier transform to L 1 ( Ĝ).If w : Ĝ → [1, +∞) is a Borel measurable and sub-multiplicative function, i.e. w(st) ≤ w(s)w(t), s,t ∈ Ĝ, (such w is called a weight function) then L 1 ( Ĝ, w) and is a Banach algebra with respect to the norm f w = f w 1 , f ∈ L 1 ( Ĝ, w).Its image under inverse Fourier transform gives a weighted version A(G, w) of A(G).We note that for a weight function w on Ĝ the (unbounded) operator w = ⊕ Ĝ w(s)ds defines a closed positive operator affiliated with L ∞ ( Ĝ) ≃ V N(G) which satisfies Γ( w) ≤ w ⊗ w, where Γ is the comultiplication on V N(G).This model has been taken in [19] and [12] to generalize the notion of weight to general locally compact groups.Accordingly, a weight, called a weight on the dual of G, is a certain unbounded positive operator w affiliated with V N(G), which, if in addition w is bounded below, i.e. ω := w−1 ∈ V N(G), satisfies ω ⊗ ω = Γ(ω)Ω for a contractive 2-cocycle Ω ∈ V N(G × G) (see [12]).One can find numerous examples of non-trivial weights for general compact groups in [21] and certain connected Lie groups in [12].
In this paper, we will work with such weight inverse ω omitting the condition of its positivity.To each ω we will associate a subspace A(G, ω) of A(G) which becomes a commutative Banach algebra with respect to a new weighted norm and the pointwise multiplication and so it is natural to study its Gelfand spectrum, spec A(G, ω).When G is compact and ω is a positive central weight, this question was studied in [21]; specific connected Lie groups, namely SU (N), the Heisenberg group, the reduced Heisenberg group, the Euclidean motion group E (2) and its simply connected cover, were treated in the long paper [12].It has been proved that spec A(G, ω) is closely related to an (abstract) complexification of G. To establish this fact the strategy in [12] was to find a simpler dense subalgebra A so that one could easily identify its spectrum, spec A, and get spec A(G, ω) ⊂ spec A. If G is compact a natural choice is A = Trig G, the algebra of matrix coefficients of finite-dimensional representations of G; spec A is then an abstract complexification of G, introduced by McKennon in [22], which coincides in the case of compact connected Lie groups with the universal complexification of G.For non-compact groups, it seems there is no such natural choice of the subalgebra.In [12] the construction of A is rather technical and each G treated in the paper required an individual approach, which heavily involved in particular the theory of group representations and technique of analytic extensions; the technicalities were an obstacle to develop a general theory applicable to any connected Lie group.
In this paper, we propose a different approach to the problem of identifying the spectrum of A(G, ω) that allows us to realise spec A(G, ω) as a subset of an abstract complexification of G for a wide class of groups and weights.
The key idea is the observation that, identifying the dual of A(G, ω) with V N(G), any multiplicative linear functional corresponds to σ ∈ V N(G) satisfying the same equation as the weight inverse ω, i.e. σ ⊗ σ = Γ(σ )Ω for the contractive 2-cocycle Ω ∈ V N(G × G) associated with ω.A simple formal calculation, which we could make to be rigorous under certain conditions, gives the equality S(σ )σ = S(ω)ω, where S is the antipode on V N(G).That allows us to define a closed operator T σ , affiliated with V N(G) and satisfying Γ(T σ ) = T σ ⊗ T σ (THEOREM 4.5).It is known that the set of all non-zero T ∈ V N(G) with Γ(T ) = T ⊗ T coincides with λ (G) = {λ (s) | s ∈ G}, providing the embedding of G into the spectrum of A(G, ω) through the evaluation u → u(s) = (λ (s), u), s ∈ G; the set G + C,λ of all positive solutions T ∈ V N(G) of Γ(T ) = T ⊗ T is the image of the Lie algebra Λ of derivations C,λ is then the space of all solutions to Γ(T ) = T ⊗ T in V N(G) (PROPOSITION 3.3).Here V N(G) is the set of unbounded operators affiliated with V N(G).In many cases including connected compact and some nilpotent Lie groups G, G C,λ = λ C (G u C ), where G u C is the universal complexification of G and λ C is the extension of the left regular representation to G u C .
The paper is organised as follows.In Section 2, we introduce the notion of a weight inverse ω on the dual of G and use this to define the Beurling-Fourier algebra A(G, ω) as a subalgebra of A(G) with a modified norm and identify its dual with V N(G).As V N(G) has the unique predual, we show that A(G, ω) is isometrically isomorphic to A(G) with a modified product • Ω , depending on the 2-cocycle Ω associated with ω and not the particular weight ω.In PROPOSITION 2.6 we give a necessary and sufficient condition for the inclusion A(G, ω 1 ) ⊂ A(G, ω 2 ).
In Section 3, we review some basic concepts on unbounded operators and operators affiliated with a von Neumann algebra, and define the λ -complexification G C,λ of G as the set of non-zero (unbounded) closed operators T which are affiliated with V N(G) and satisfy the equation Γ(T ) = T ⊗ T.
In Section 4 we investigate the relation that the λ -complexification has to the Gelfand spectrum of A(G, ω).We prove the embedding of spec A(G, ω) into the complexification G C,λ for a wide class of groups and weights; this comes down to verifying that S(σ )σ = S(ω)ω holds for the points σ in spec A(G, ω), considered as a subset of V N(G).We also give a heuristic reason why we conjecture that this holds in general.These arguments give immediately the equality for any virtually abelian group and any weight considered on it.The other cases of G and ω for which the embedding of spec A(G, ω) into G C,λ holds include, for example, compact, discrete and more general [SIN]-groups with arbitrary weights and general locally compact groups with weights extended from weights on the dual of abelian or compact subgroups.Even though we could not establish the inclusion result in full generality our approach allows us to generalise most of the previous results and avoid the main technicalities in [12] to find a dense subalgebra which plays the role of Trig G for the compact case.Moreover, as the main available source of weights on the dual of non-commutative groups are the weights induced from abelian or compact subgroups, THEOREM 4.20 and THEOREM 4.22 cover most of the known Beurling-Fourier algebras.For discrete group G we show that the spectrum of the corresponding Beurling-Fourier algebra is homeomorphic to G.
Finally, in Section 5, we discuss some of the questions that arose during our investigation, as well as some examples that show the necessity of certain conditions.

BEURLING-FOURIER ALGEBRAS
For a locally compact group G, we let λ : Recall that Γ is the unique normal * -homomorphism satisfying Γ(λ ) is given by the action The coproduct Γ is co-commutative and satisfies the co-associative law: where ι is the identity map.

DEFINITION (WEIGHT INVERSE).
A ω ∈ V N(G) will be called a weight inverse on the dual of G (we usually abbreviate this to a weight inverse) if and ker ω = ker ω * = {0}.
(ii) It follows from (3) that so that ||ω|| 2 ≤ 1 and hence a weight inverse is always a contraction.
(iii) In [12] a (bounded below) weight on the dual of G was defined as an (unbounded) positive operator w which is affiliated with V N(G) and admits an inverse w ) is defined and contractive on a dense subspace, i.e. w −1 is a positive weight inverse, in our terminology.
(iv) A weight inverse was considered in [24] as an element in the multiplier algebra and our definition coincides with the one in [24].(v) The notion of unitary dual 2-cocycle on a compact group was introduced by Landstad [18] and Wassermann [29] in the study of ergodic actions.In the context of quantum groups it was defined by Drinfeld [3].Their 2-cocycle condition is similar and defined as follows: The cocycle of the form (u ⊗ u)Γ(u) −1 is called a coboundary.The inverse of our 2-cocycle satisfies (7).
EXAMPLE 2.2.Let w be a bounded below weight function on G = R or Z given by w and moreover that w The above weight inverses can be extended to , here λ G and λ H are the left regular representations of G and H respectively; the existence of ι H is due to Herz restriction theorem, see for example [15].
For other examples of weights and weight inverses, we refer the reader to [12].
Let A(G) be the unique pre-dual of V N(G).Recall that it can be identified with the space of functions on G: where ȟ(s) = h(s −1 ), s ∈ G, and A(G) becomes a commutative Banach algebra, usually called the Fourier algebra of G, with respect to the pointwise multiplication and the norm given by where the infimum is taken over all possible decomposition f = g * ȟ, see for example [5,17].The duality between V N(G) and A(G) is given by ; here and through the rest of the paper we use •, • to denote the inner product on a Hilbert space and we keep notation (•, •) for duality pairing between M and M * when M is a von Neumann algebra.
For T ∈ V N(G) and f ∈ A(G), we let T f ∈ A(G) be given by If ω is a weight inverse, we define and call it the Beurling-Fourier algebra of G associated to ω.
PROPOSITION 2.3.A(G, ω) is a Banach algebra with respect to the pointwise multiplication and the norm Moreover, A(G, ω) is a predual of V N(G) with the pairing given by Proof.To see that || • || ω is a norm, we should only see that it is well defined.In fact, if f = η * ξ then the closed linear span of λ (s)η , s ∈ G, and let P be the projection onto U.As U is invariant with respect to V N(G), we have Pω = ωP.
From (5) it follows that A(G, ω) is a commutative Banach algebra; in fact, we have and where is the predual of the co-multiplication Γ defined on the operator space projective product of A(G) ⊗A(G) (see [4]).The associativity of the product is clear and the completeness follows from the boundedness of ω: We note that the previous proposition was proved in [12] for positive weight inverses.Similar arguments can be applied to prove the general case.For the reader's convenience, we have chosen to give its full proof.
The next statement shows that we can restrict ourselves to positive weight inverses.Proof.Note that U ∈ V N(G) is unitary by (4).From (5) it is immediate that |ω * | is again a weight inverse.Clearly A(G, ω) = A(G, |ω * |) as subsets of A(G), and the identity is an algebra homomorphism.Moreover We will use the following lemma: then there is c ∈ M such that a 1 = a 2 c.Moreover, we can assume that ker c = ker a 1 , and ker a 2 ⊆ ker c * , and under these assumptions, c is uniquely determined.
Proof.Let a i = S i |a i |, for i = 1, 2, be the polar decompositions, and let P i = S * i S i , so that a i P i = a i .For i = 1, 2, the maps P i M * → a i M * , defined as f → a i f , are bijective linear maps.As a 1 M * ⊆ a 2 M * , there is for every Clearly R is a linear injective map and moreover for b ∈ M, we have R( f b) = R( f )b.Note that P i M * , i = 1, 2, are closed subspaces of M * , thus Banach spaces.We claim that R is closed: let f n be a sequence such that f , and thus a 1 = a 2 c.Clearly, ker c = ker a 1 , ker c * ⊇ ker I −P 2 = (ker a 2 ) ⊥ and that c is the unique element such that a 1 = a 2 c with these properties.PROPOSITION 2.6.Let ω 1 , ω 2 be two weight inverses on the dual of G.The inclusion A(G, ω 1 ) ⊆ A(G, ω 2 ) implies that there is a ∈ V N(G) such that ω 1 = ω 2 a.Furthermore, we have A(G, ω 1 ) = A(G, ω 2 ) if and only if ω 1 = ω 2 a for an invertible element a ∈ V N(G).
Another equivalent model of the Beurling-Fourier algebra, which was given in [12] for positive weights, is defined as follows.For a weight inverse ω and the corresponding 2-cocycle Ω define a new multiplication on A(G) by It follows from ( 9) that (A(G), • Ω ) is a commutative contractive Banach algebra which is isomorphic to A(G, ω), showing that A(G, ω) can be determined by the 2-cocycle Ω rather than the weight inverse ω.Assume A(G, ω 1 ) = A(G, ω 2 ) and let a ∈ V N(G) be the invertible operator such that ω 1 = ω 2 a which exists due to Proposition 2.6.If Ω 1 and Ω 2 are the corresponding 2-cocycles, then and (11) and correspond to weight inverses ω 1 and ω 2 respectively, then u → au, u ∈ A(G), gives the isometric isomorphism (A(G), If a is not assumed to be invertible, the map u → au gives a homomorphism from We note that any 2-cocycle associated with a weight inverse is symmetric, that is invariant under the 'flip' automorphism REMARK 2.7.Let Z 2 sym ( Ĝ) be a category whose objects are injective symmetric 2cocycles and elements a ∈ Hom(Ω 1 , Ω 2 ) are the non-zero operators in V N(G) satisfying (11).With this notation, each weight inverse belongs to the set Hom(Ω, I), where the identity operator I is considered as the identity 2-cocycle: in this case (11) becomes In particular, we have Hom(I, I) = λ (G) ≃ G. Defining the equivalence relation on Z 2 sym ( Ĝ) by Ω 1 ∼ Ω 2 if there is an invertible element a ∈ Hom(Ω 1 , Ω 2 ), we have the embedding of the set of Beurling-Fourier algebras into Z 2 sym ( Ĝ)/ ∼.
We finish this section by defining a representation of (A(G), and define X 13 similarly.Then W satisfies the pentagonal relation For λ Ω ( f where the first equality in the last line can be seen on elementary tensors and using then linearity and density arguments.In fact, if The formula (13) follows from the following calculations 3 COMPLEXIFICATION OF G

PRELIMINARIES ON UNBOUNDED OPERATORS
We start with some basic material on unbounded operators which will be used in the paper.Our main reference is [25].
Let H be a Hilbert space with the inner product •, • .Recall that a linear operator T defined on a subspace D(T ) ⊂ H, called a domain of T , is said to be closed if the graph of T , {(ξ , T ξ ) | ξ ∈ H}, is closed in H⊕ H. Given linear operators T and S, we write T ⊂ S if D(T ) ⊂ D(S) and S| D(T ) = T ; we say that S is an extension of T .We have The minimal closed extension of a closable T exists and will be denoted by T .We say that a subspace If T is an operator with a dense domain it has a well-defined adjoint operator Any selfadjoint T has a spectral measure E T on the σ -algebra B(R) of Borel subsets of R, and if f is a Borel measurable function, we write f (T ) for the operator If T is a closed operator with dense domain, then T * T is positive and T has the polar decomposition T = U |T |, where |T | = (T * T ) 1/2 and U is a partial isometry; |T |, T and U have the identical initial projections.
We say that a closed operator T defined on a dense domain D(T ) ⊆ H is affiliated with a von Neumann algebra M of B(H) if U T ⊂ TU for any unitary operator U ∈ M ′ , where M ′ as usually stands for the commutant of M.
Let A be a linear operator on H.A vector ϕ in H is called analytic for A if ϕ ∈ D(A n ) for all n ∈ N and if there exists a constant M (depending on ϕ) such that We write D ω (A) for the set of all analytic vectors of A. If A is selfadjoint with E A (•) being the spectral measure of A, then E A (∆)ϕ is analytic for A for any ϕ ∈ H and any bounded ∆ ∈ B(R), as It is known (see e.g.[25,PROPOSITION 10.3.4]) that if T is a symmetric operator, i.e.T ⊂ T * , with a dense set of analytic vectors, then T is essentially selfadjoint.
If U and V are subspace of H we write U⊙ V for the algebraic tensor product of the subspaces; H 1 ⊗ H 2 is the usual Hillbertian tensor product of two Hilbert spaces H 1 and H 2 .
If T 1 , T 2 are closed densely defined operator with the domains D(T 1 ) and D(T 2 ) ⊂ H respectively, then the operator is dense in H⊗ H, showing that η = 0 and that T 1 ⊗ T 2 is closable.Unless otherwise stated we will write T 1 ⊗ T 2 for the corresponding closure.
We say that two selfadjoint operators T 1 , T 2 strongly commute, if Let T i be selfadjoint operators, i = 1, 2. Then T 1 ⊗ 1 and 1 ⊗ T 2 are selfadjoint operators that commute strongly.Then T 1 ⊗ T 2 is selfadjoint and For closed densely defined operators S 1 , S 2 with polar decomposition

λ -COMPLEXIFICATION OF A LOCALLY COMPACT GROUP
Let G be a locally compact group and let W be the fundamental multiplicative unitary on Clearly, the unbounded operator Γ(T ) is closed.If T * = T and E T (•) is the spectral measure of T , then both operators 1 ⊗ T and Γ(T ) are selfadjoint with 1 ⊗ E T (•) and Γ • E T (•) being the corresponding spectral measures.In particular, DEFINITION.By the λ -complexification G C,λ of G we shall mean the set of all nonzero (unbounded) operators T ∈ V N(G) such that We note that , see e.g.[27,Chapter 11,THEOREM 16] As for α ∈ Λ, the operators iα ⊗ 1 and 1 ⊗ iα are selfadjoint and strongly commute, the sum iα ⊗ 1 + 1 ⊗ iα, defined via the functional calculus, gives a selfadjoint operator; in (15) we require it to be equal to the selfadjoint operator Γ(iα).
Proof.It follows from the functional calculus and definition of Γ that Proof.That exp iα is positive and affiliated with V N(G) follows from the functional calculus.
Let T ∈ G C,λ ∩V N(G) + .Using arguments similar to those in the proof of the previous proposition we obtain By Stone's theorem about infinitisimal generator of a strongly continuous unitary group, we obtain Proof.As it was noticed before, if are the polar decompositions of Γ(T ) and T ⊗ T respectively.Hence, by uniqueness of the polar decomposition, the equality Let G be a connected Lie group and g its Lie algebra with the exponential map exp It is known that idπ(X ) is essentially self-adjoint.We denote its self-adjoint closure by i∂ π(X ) which is the infinitesimal generator of the strongly continuous one-parameter unitary group t → π(exp G (tX )), i.e.
PROPOSITION 3.4.Let G be a connected Lie group with Lie algebra g.
and {exp (tα)| t ∈ R} is continuous in the weak * topology on V N(G) with the weak *limit w * − lim t→0 exp (tα To see the reverse inclusion, we note that ∂ λ Since lim t→0 t −1 [exp (tV )ϕ − ϕ] = V ϕ for any closed skew adjoint operator V and ϕ ∈ D(V ), we can easily obtain that ∂ λ (X ) ∈ Λ. [22] and Cartwright and McMullen [1], where they developed an abstract Lie theory for general, not necessarily Lie, compact groups.If we choose representatives π j : G → B(H j ) of the isomorphism classes of irreducible (finite-dimensional) unitary representations of G and identify † , where Trig(G) † is the linear dual of the span of coefficients of irreducible unitary representations of G.In this case we have that G C,λ coincides with the complexification G C from [1,22] .We have that G C is a group and as

REMARK. Our definition is motivated by the work of McKennon
The concept of complexification was later generalised from compact to general locally compact groups in [23] by McKennon, where the group W * -algebra W * (G) was used instead of V N(G).Our construction is an adaptation of McKennon's idea to the group von Neumann algebra setting.We have chosen this approach as it fits better our purpose to describe the spectrum of Beurling-Fourier algebras.As for the compact group case McKennon's complexification G C admits a factorisation , where G γ is the image of G under the canonical monomorphism γ from G to the group of unitary elements of W * (G) (compare this to the factorisation in PROPOSITION 3.4).However unlike the compact case, the unboundedness of elements in G + C and also + causes a problem in considering G C and G C,λ as groups, see [23, section 4].A relation to the universal complexification of G, when G is a Lie group, is also unclear in general.However, in many interesting examples considered in [12] we have where G u C is the universal complexification of G and λ C is the exten- sion of the left regular representation to G u C ; the equality means that for any ϕ ∈ G C,λ there exists g ∈ G u C such that such that ϕ = λ C (g), see the discussion in [12, section 2.3]; in those cases one also has the Cartan decomposition , where exp C is the extension of the exponential map to the complexification g C of the Lie algebra g of G.It seems an interesting question to investigate the group structure of G C,λ but it diverges from the main purpose of this paper.

THE SPECTRUM OF BEURLING-FOURIER ALGEBRA AND COMPLEXIFICATION
In this section we establish sufficient conditions in terms of groups and weight inverses for the inclusion of the Gelfand spectrum of A(G, ω) into the λ -complexification G C,λ of G, generalising some earlier results from [21] and [12].

Point-spectrum Correspondence
Let φ : A(G, ω) → C be a character of A(G, ω).By the duality (8), there is a unique and moreover, every σ ∈ V N(G) satisfying ( 16) gives rise to a unique point in the spectrum spec A(G, ω).In fact, for u, v ∈ A(G), on the other hand giving (16).
We can thus identify spec A(G, ω) as the set of all non-zero elements σ ∈ V N(G) satisfying ( 16), i.e.
Note that spec A(G, ω) depend on the 2-cocycle Ω rather than the weight inverse ω.Moreover, by (16), for any σ ∈ A(G, ω) thus satisfying condition (3) in the definition of a weight inverse.It is a question whether σ also satisfies (4).We will see in this section that in many cases (though we conjecture all) the elements in spec A(G, ω) are again weight inverses.
We let S be the antipode of V N(G); this is an anti-isomorphism of V N(G) given by We refer to [6] for background on the theory of Hopf-von-Neumann algebras but warn that our notations may differ from those in [6].
Throughout the rest of this section, we use H for L 2 (G) and write ψ ξ ,η to denote the normal functional on B(H) given by ψ ξ ,η (x) = xξ , η , x ∈ B(H).
REMARK.It has been known for compact groups ( [21]) and some Lie groups with certain weights ( [12]) that the operators σ ω −1 , σ ∈ spec A(G, ω), are "points" of the complexification G C .From this, the claim of the proposition becomes intuitively quite clear.Formally, if there is an element T ∈ G C such that σ = T ω then, as S(T ) = T −1 (the antipode "inverts" the elements of G and G C ), we would have S(σ Proof.Let η and ζ in H be such that σ * ζ = ω * η = 0.By LEMMA 4.1, we have Multiplying both hand sides of the equality from the left by Ω * W * and using the equality Ω * Γ(σ ) * = σ * ⊗ σ * which holds for all σ ∈ spec A(G, ω) and in particular for ω, we conclude that and hence ω * S(ω) * = σ * S(σ ) * .

REMARK.
The following formal calculations support the idea that the above proposition might be true for any σ ∈ spec A(G, ω).
Here comes the main result of this section that establishes a connection between (a part of) the spectrum spec A(G, ω) and G C,λ .
Write T σ also for the closure.Then T σ is affiliated with V N(G), and even more, it is affiliated with the von Neumann algebra N(ω, σ ) generated by ω and σ .In fact, let V ∈ N(ω, σ ) ′ be a unitary.Then for any ξ ∈ H of the form ξ = ωη, we have The only claim left to prove is that Γ(T σ ) = T σ ⊗ T σ .Observe first that and We have By convention, T σ ⊗T σ is the closure of the operator T σ ⊙T σ defined on D(T σ )⊙ D(T σ ) or, equivalently, on ω(H) ⊙ ω(H), as ω(H) is a core of T σ .Hence To see the equality, we must prove that Γ(T σ )| ω(H)⊙ω(H) = Γ(T σ ).To do this we note first that Γ(ω)(H⊗ H) is a core for Γ(T σ ) and hence the linear subspace is dense in the graph of Γ(T σ ).Therefore, it is enough to see that the closure of As Ω(H⊙ H) is dense in H⊗ H, we have that for any Γ(ω)ξ , ξ ∈ H⊗ H, there exists (ξ n ) n ⊂ H⊙ H such that Ωξ n → ξ and hence Γ(ω)Ωξ n → Γ(ω)ξ .Moreover, for σ ∈ spec A(G, ω) means that the domain D(T * σ ) of the operator T * σ = (σ ω −1 ) * = (ω * ) −1 σ * is not zero.The theorem says that in this case D(T * σ ) is large enough to be dense in H, as the latter is equivalent to the closability of T σ .
In what follows we shall use the notation T σ for the closed operator Tσ when there is no risk of confusion.
We derive now a number of consequences from the previous theorem.We assume that ker Ω * = {0}.COROLLARY 4.6.For σ ∈ spec A(G, ω) as in THEOREM 4.5, there is a natural isometric isomorphism A(G, σ ) ∼ = A(G, ω), Proof.This is immediate from the definitions of the norm and product on the corresponding spaces: We remark that the above corollary is also clear from the discussion after the proof of PROPOSITION 2.6.
Proof.The "only if" part follows from THEOREM 4.5.If σ = T ω for T ∈ G C,λ then σ * ⊃ ω * T * giving the "if" part.REMARK 4.8.In [12] the dual A(G, ω) * is identified with the weighted space V N(G, ω) given by V N(G, ω) Then the spectrum of A(G, ω) is considered as a subset of V N(G, ω) instead of V N(G).
Clearly we have the isometry Next, we prove a 'partial converse' of THEOREM 4.5, which shows that every element in G C,λ can be seen as coming from a weight inverse.
The image of C 0 (R) is clearly non-degenerate, and hence we can extend ϕ in a unique way to a homomorphism ϕ : , then it is easy to see from the uniqueness of the extensions that the diagram is commutative; here we write Γ R for the restriction of the coproduct to C b (R).
Next we derive some further properties of spec A(G, ω) ∩ G C,λ ω.
Proof.By the functional calculus, we have ω(H It follows that T t β ω is bounded for every t ∈ [0, 1], and hence T t β ω and λ (s)T t β ω belong to V N(G), and the function t → T t β ω is strongly continuous; to see the latter we observe that if P n = E([0, n]), where E(•) is the spectral measure of T β , then t → T t β P n ωξ is continuous for every ξ ∈ H.Moreover, as P n → I strongly.Basic approximation arguments give now that T t β ωξ must depend continuously on t ∈ [0, 1] for each ξ ∈ H. From this we conclude that t → ψ(t) is continuous as the map from [0, 1] to V N(G) ≃ A(G, ω) * with the weak * topology.
It follows from the functional calculus that Γ(T t β ) = T t β ⊗ T t β for all t ∈ [0, 1], and thus so that T t β ω and hence λ (s)T t β ω are in spec A(G, ω).As the kernel of T β is trivial, there is n ∈ N such that the orthogonal projection The restriction of T t β to the invariant subspace P H is then invertible for every t ∈ [0, 1] and as Pλ (s) * ψ(t) = PT t β Pω, t ∈ [0, 1], we have The last results concern a deformation retraction of weight inverses.LEMMA 4.12.Assume that a weight inverse ω is positive.For every s ∈ [0, 1], the operator ω s is again a weight inverse.
(iii) If σ ∈ spec A(G, ω s ), then by ( 24) as the kernel of ω t−s is trivial.The injectivity of σ → σ ω t−s follows from the fact that the range of ω t−s is dense in H.

CONDITIONS GUARANTEEING COMPLEXIFICATION
In this section we will investigate conditions on the group G and the weight inverse ω for which the inclusion ( 21) of the spectrum of A(G, ω) into the complexification G C,λ holds true.First, we present some sufficient conditions for ker Ω * = {0}.
Recall that if H is a closed subgroup of G and λ H and λ G are the left regular repre- sentations of H and G respectively, then there is a canonical injective * -homomorphism We say that a weight inverse ω on the dual of G is central if ω is in the center of V N(G).PROPOSITION 4.14.Let ω be a weight inverse on the dual of G. Then ker Ω * = {0} holds provided that any of the following is satisfied: where ω H is a central weight inverse on the dual of a closed sub- group H of G.
Proof.As K is invariant and ω| K is invertible, ω * (K) = K.We have where the latter is non-zero by LEMMA 4.4.
Using LEMMA 4.15 we can now list groups and weights for which the spectrum of the associated Beurling-Fourier algebra is in the complexification G C,λ , meaning that we identify A(G, ω) * with V N(G, ω) as in REMARK 4.8; with a slight abuse of notation we write spec A(G, ω) ⊂ G C,λ .
(1) G is compact and ω is arbitrary.If G is compact then it is known that the left regular representation λ on G is a direct sum of irreducible (finite-dimensional) representations and hence there exists a finite-dimensional invariant subspace K ⊆ H.As ker ω = {0}, ω is invertible on K.By PROPOSITION 4.14, ker Ω * = {0}.By COROLLARY 4.7, spec A(G, ω) ⊂ G C,λ .In [21] and [12] the result was derived from the "abstract Lie" theory developed in [1,22] showing that the multiplicative linear functionals on Trig(G), the algebra of coefficient functions with respect to irreducible representations, can be identified with the complexification G C,λ .As Trig(G) ⊂ A(G, ω), the statement is clear.
(2) G is an extension of a compact group by abelian group and ω is a weight inverse such that ker Ω * = {0}.If K is a non-trivial compact normal subgroup, let P K ∈ B(L 2 (G)) be the projection onto the (non-trivial) subspace of functions which are constant on the cosets xK, x ∈ G.As P K commutes with λ G (g), g ∈ G, the sub- space P K L 2 (G) is invariant with respect to λ G , and as G/K is abelian and P K f are constant on the cosets, λ G (g 1 g 2 )P K f = λ G (g 2 g 1 )P K f , i.e. the von Neumann algebra generated by λ G (g)P K , g ∈ G, is commutative.As ω K := ω| P K L 2 (G) belongs to the von Neumann algebra, there exists a subspace K (e.g. (3) G is a separable Moore group and ω is arbitrary.If G is a Moore group, i.e. any irreducible representation of G is finite dimensional, then G is a type I group with the unitary dual Ĝ being a standard Borel space.Moreover, there is a standard Borel measure µ and a µ-measurable cross section ξ → π ξ from Ĝ to concrete irreducible unitary representation acting on H ξ such that λ is quasi-equivalent to where (x n ) n is a sequence such that (x n (ξ )) n is total in H ξ for any ξ .As ker ω = ker ω * = {0}, there exists a null set M ⊂ Ĝ such that ker ω ξ = ker ω * ξ = {0} for any ξ ∈ Ĝ \ M.Then, as H ξ is finite-dimensional, for each ξ ∈ Ĝ \ M, we have |ω ξ | ≥ c ξ I ξ for some c ξ > 0. Hence µ(∆ ε ) > 0 for some ε > 0 and and G × G is Moore, we can argue as above to conclude that ker Ω * = {0}.Therefore, by COROLLARY 4.7, we have the inclusion of the spectrum of A(G, ω) into G C,λ as in the previous paragraph.
(4) G is a separable type I unimodular group and ω = ⊕ Ĝ ω ξ dµ(ξ ) with ω ξ invertible on a set N ⊂ Ĝ of positive measure.We define an invariant subspace K such that ω| K is invertible as above and get the statement of COROLLARY 4.7 in this case as well if ker Ω * = {0}.Central weights fall in this class.Any weight on G such that the set N = {ξ ∈ Ĝ | dim H ξ < ∞} has positive µ-measure also satisfies that condition.
Recall that a locally compact group G is called an [IN]-group if it has a compact conjugation-invariant neighbourhood of the identity.It is called a [SIN]-group if it has a base of conjugate-invariant neighbourhoods of e.We note that any [SIN]-group is [IN].Typical [SIN]-groups are discrete, compact and abelian groups.
The following result is likely known, but we could not find a reference.

PROPOSITION 4.16. G is an [IN]-group if and only if V N(G) admits a normal tracial state.
Proof.Assume that tr ∈ V N(G) * is a tracial state.Then the function f (g) = tr(λ (g)) ∈ A(G) ⊆ C 0 (G), and thus we have a conjugate-invariant compact neighbourhood, e.g.
Conversely, assume that K is a compact neighbourhood which is conjugate-invariant, and let ξ K be the Proof.As ker Ω * = {0} if follows from LEMMA 4.4 that for all σ ∈ spec A(G, ω) we have σ (H) is dense in H. Consider the following two inequalities Letting R = (σ * σ +ω * ω) 1 2 , we can deduce from (25) -similar to the proof of LEMMA 2.1that there exist Moreover, we have so that the density of the range of R (implied by the density of the range of ω) gives In particular, we obtain that U * U and V * V commute.Assume towards contradiction that σ * (H) ∩ ω * (H) = {0}.Then by (26) and the injectivity of R we can deduce that also U * (H) ∩V * (H) = {0}.Thus (27) implies that U,V are partial isometries.As ker Ω * = {0}, it follows from LEMMA 4.4 that ker σ * = ker ω * = {0}, and by (26), kerU * = kerV * = {0}.Thus U * and V * are isometries in V N(G) such that (27) holds, i.e. (U,V ) is a representation of the Cuntz algebra O 2 in V N(G).This contradicts the claim that V N(G) admits a tracial state φ : Proof.By [2, 13.10.5],G is a [SIN]-group if and only if V N(G) is finite.Therefore, as ker Ω = {0}, we have ker Ω * = {0}, giving, by LEMMA 4.4, ker σ * = {0} and hence by finiteness of V N(G), ker σ = {0} for any σ ∈ spec A(G, ω).As both (σ * ) −1 and (ω * ) −1 are densely defined and affiliated with V N(G), and the set of affiliated elements is an algebra, we obtain that σ * (L 2 (G)) ∩ ω * (L 2 (G)) = {0} as the domain of We remark that V N(G) is finite for all Moore groups G and hence any such G is Proof.G clearly does not contain any non-trivial image of a homomorphism R → G, and we can deduce that the complexification is trivial, i.e.G C,λ = G.Moreover, as G is a [SIN]-group, by PROPOSITION 4.18 the spectrum of A(G, ω) is the smallest possible, that is G.
An important class of weights that has been studied in the literature are weights extended from closed abelian or compact subgroups, see [12,Proposition 3.25].The next statements show that for all such weights we have the inclusion of the spectrum into the complexification.We first recall the construction of a so called central weight on the dual of a compact group following [12], see also [21].
Recall the conjugate representation π of π ∈ H which is defined as follows: we denote the linear dual space of H π by H π and for A ∈ B(H π ), let A t in B(H π ) be its linear adjoint; for s ∈ H we define π(s) = π(s −1 ) t ; it is a unitary irreducible representation on H π and π = π, as equivalence classes.
If H is abelian, the weight inverse condition for positive weight inverse ω H can be also equivalently written as (28) since in this case ω H (s −1 )ω H (st) ≤ ω H (t) for almost all s,t ∈ H. Proof.Let Ω be the 2-cocycle associated to ω.By PROPOSITION 4.14 ker Ω * = {0} and hence ker σ * = {0} for every σ ∈ spec A(G, ω) by LEMMA 4.4.To show that σ is a weight inverse, it is enough to see the equality (29), which will imply ker σ = {0}.To prove (29), let us without any loss of generality assume that ω H is positive.Then ω H satisfies (28).It is clearly preserved by ι H giving As in the proof of LEMMA 2.1 we can conclude that there is a unique element Φ ∈ V N(G) ⊗V N(G) such that If W is the fundamental multiplicative unitary, the latter equality gives Let ξ , η, ξ , η ∈ H.We retain the notation ψ x,y for the normal functional ψ x,y (T ) = T x, y , T ∈ B(H).By LEMMA 4.1, for any σ ∈ spec A(G, ω).In particular, it holds for ω which combined with (38) gives Fix σ ∈ spec A(G, ω).As the range of ω * is dense in H, there exists {η n } n ⊂ H such that ω * η n → σ * η.From (39) and (40) we obtain Reasoning as in the remark after the proof of PROPOSITION 4.2, we have that the operator M := (ΦW * )W Ω satisfies for some s ∈ G and σ ∈ spec A(H, ω H ).
Proof.The condition spec A(G, ω) ⊂ G C,λ implies that any σ ∈ spec A(G, ω) admits a factorisation σ = T ω for some Moreover, as T * = (σ ω −1 ) * = (ω −1 ) * σ * and ker showing that σ is a weight inverse.It follows that and is independent of particular σ ∈ spec A(G, ω).Applying a slice map ι ⊗ f , f ∈ A(G), to (35) and using the fact that the elements of the form f σ form a dense subspace in A(G) (as the range of σ is dense) we obtain Consider the subspace (the weak * closure).By (36), we have Aσ ⊆ ι H (V N(H)).Let I A ⊆ A(G) be the preannihilator of A, i.e.
We claim that I A is equal to the subspace and indeed, this follows from the action of A(G) on V N(G) being commutative.Moreover, the same argument shows that I A ⊆ A(G) is a non-trivial closed ideal, as σ = 0.By duality, we have As I A = A(G), there is at least one s ∈ G such that λ G (s) * annihilates I A , and hence λ G (s) * ∈ A. It follows that and moreover that the pre-image σ = ι −1 H (λ G (s) * σ ) ∈ spec A(H, ω H ).This gives the statement of the theorem.
Combining methods in the proofs of THEOREM 4.20 and THEOREM 4.21 we obtain a generalisation of THEOREM 4.20 to weights induced from non-central weights of compact subgroups of G. THEOREM 4.22.Let H ⊆ G be a compact subgroup, and ω H ∈ V N(H) be a weight inverse on the dual of H. Then with ω = ι H (ω H ), we have Proof.Let F ⊂ H be finite and set PF to be the central projection in V N(H) given by PF = ⊕ π∈ Ĥ χ F (π)I π , where χ F is the indicator function of F. Set Then using arguments as in [12, 3.3.2],we obtain PF ⊗ PF ≤ Γ( PC F ) and hence ( PF ⊗ PF )Γ( PC F ) = PF ⊗ PF , which gives we can apply S ⊗ι to the last equality to obtain In V N(H × H), we consider the element Let Ω be the 2-cocycle associated with ω.As in the proof of THEOREM 4.20 we have for σ ∈ spec A(G, ω) and where ξ , η, ξ , η ∈ H. Take ξ ∈ S(P F )L 2 (G) and η ∈ P F L 2 (G).Then the right-hand side of (40) becomes 39) and (40) together with Γ(P F )Φ * (S(P F ) ⊗I) ∈ V N(G ×G), we get Let M be in the commutant of ι H (V N(H)) ⊗V N(G).Clearly, M commutes with the left-hand side of (41) and as PF → I weak * , we obtain that it commutes with Γ(σ * )(S(σ ) * ⊗ I).Therefore, If we let f ∈ V N(G) * be arbitrary, then it follows from ( 42) We now proceed in a similar way as in the proof of THEOREM 4.21 and let We can argue as before that the ideal I A := A ⊥ = A(G) and hence there is s ∈ G such that f (s) = 0 for all f ∈ I A .As I ⊥ A = A, λ G (s) ∈ A and therefore S(σ )λ G (s) ∈ ι H (V N(H)) and λ G (s −1 )σ ∈ ι H (V N(H)).It follows that there is an σ ∈ spec A(H, ω H ) such that λ G (s −1 )σ = ι H ( σ ), and hence Let G be a connected simply connected Lie group and g its associated Lie algebra.We also fix the symbol H and h for a connected closed Lie subgroup of G and its Lie algebra respectively.We write λ G and λ H for the left regular representations of G and H respectively.The next statement generalizes [12, THEOREM 5.9, THEO-REM 6.19, THEOREM 7.11, THEOREM 8.20 and THEOREM 9.11], where it was proved for compact connected Lie groups with a weight induced from a closed Lie subgroup, the Heisenberg group, the reduced Heisenberg group, the Euclidean motion group on R 2 , and the simply connected cover of it with weights induced from abelian connected Lie subgroups.We note that the proofs of the latter theorems from [12] required lengthy and specific arguments for each particular group.We also answer [12,Question 11.4] as our technique does not require the existence and density of entire vectors for the left regular representation which was essential to prove the mentioned results in [12].
Proof.By THEOREM 4.20, THEOREM 4.22 and the remark after LEMMA 4.15 we have spec A(G, ω) ⊂ G C,λ , and hence by PROPOSITION 3.4 for any σ ∈ spec A(G, ω) there is a unique s ∈ G and X ∈ g such that Ran(ω) ⊂ D(exp i∂ λ G (X )), exp i∂ λ G (X )ω is bounded and for all u ∈ A(G, ω).By THEOREM 4.21 and THEOREM 4.22, we have that for some σ ∈ spec A(H, ω H ) and t ∈ G.By assumption of the theorem, there exist s ∈ H and X ∈ h such that and σ = λ H ( s) exp i∂ λ H ( X).As ι H (exp i∂ λ H ( X)) = exp i∂ λ G ( X), we obtain by applying [12,PROPOSITION 2.1] that ι H ( σ) = λ G ( s) exp i∂ λ G ( X) from which we get the inclusion "⊂".
Similarly, we can start with a bounded below weight w : A ≃ R → (0, ∞) and consider ω = ι A ( M w −1 ).We have ω(σ ± ) = F −1 w−1 (−x)Fand if w = β |x] , then We note that by [14] the left regular representation does not admit a dense subset of entire vectors, the fact that was an obstacle in [12] for the study of the spectrum of A(G, ω).The density of the set H w (λ ) of entire vectors was also important for the identification of A(G, ω) as a subset of the complexification G C of G: letting λ C (exp X ) = exp ∂ λ (X ), X ∈ g C , one obtains a representation of G C on H w (λ ), see [14,COROLLARY 2.2]; in general and in particular for the "ax + b"-group, it seems there is no natural way for λ to be continued to a global representation of the complexified group and hence to see G C,λ as a group.
We refer the reader to [12] for other specific examples of weights and precise descriptions of the spectrum of the associated Beurling-Fourier algebras (Examples 6.21, 7.13 and 8.22).

SOME REMARKS AND OPEN QUESTIONS
In this section, we list open questions and make some remarks.The most pressing question that we left unanswered is of course whether we can extend the point spectrum correspondence to general locally compact groups.Namely, QUESTION 5.1.Does spec A(G, ω) ⊂ G C,λ hold for any locally compact group G and any weight inverse ω?
As it was noticed in Section 2 the definition of the product in A(G, ω) depends on the 2-cocycle Ω rather than the weight inverse ω, and A(G, ω) ≃ A(G, Ω), where A(G, Ω) is A(G) (as a Banach space) with the modified product We note that the 2-cocycle Ω associated with a weight inverse is always symmetric, i.It is easy to see that ϒ(x)ϒ(y) ≤ ϒ(x + y) for x, y ≥ 0 and hence Ω(x, y) ≤ 1.Thus Ω(x, y) ∈ L ∞ (R) ⊗L ∞ (R).Moreover, it is not hard to see that Ω is a symmetric 2cocycle.Hence we have a well-defined algebra A(R, Ω).Using that A(R) ∼ = L 1 (R) via the Fourier transform, the Ω-modified product between f , g ∈ L 1 (R) is Notice that if x < 0, then Ω(x − y, y) = 0 for all y ∈ R and hence f * Ω g = 0 a.e. on (−∞, 0), in particular, B := L 1 (R + ) is a subalgebra of (L 1 (R), * Ω ).
Next we will see that spec B empty.Let B ′ = L 1 (R + , 1 ϒ ) with the convolution product ( f * g)(x) = ∞ 0 f (x − y)g(y)dy.Then is an isometric isomorphism.Let φ be a linear multiplicative functional on B ′ .Then there is m ∈ L ∞ (R + ) such that m(x)ϒ(x) ∈ L ∞ (R + ) and As φ is multiplicative, m(x) = e ax for some a ∈ C. As lim x→∞ |e ax ϒ(x)| = lim x→∞ |e ax (1+ x) x | → ∞ for any a ∈ C, the spectrum of B ′ and hence of B is empty.To see that this carries over to the actual algebra A(R, Ω), we use that f * Ω g ∈ B, for all f , g ∈ L 1 (R), and hence if we would have a multiplicative linear functional φ such that φ ( f ) = 1 for some f ∈ L 1 (R), then φ ( f * Ω f ) = 1 and hence φ ∈ spec B, a contradiction.We modify the previous example slightly to obtain a continuous 2-cocycle.Consider the function ν(x) = e − 1 x , for x ≥ 0, 0, otherwise.
It is easy to see that ν(x + y) ≥ ν(x)ν(y) for all x, y ∈ R. Now let L(x) = ν(x)ϒ(x), for x ≥ 0, 0, for x < 0. and Θ(x, y) = L(x)L(y) L(x+y) , for x, y ≥ 0, 0, otherwise, then Θ(x, y) ≤ 1 for all x, y ∈ R. Furthermore, we have Θ(x, y) ∈ C b (R 2 ).It is not that hard to see that also A(R, Θ) has empty spectrum (the argument is more or less the same as above).If G C,λ = G then the homomorphism φ : C b (R) → V N(G) from the proof of PROPOSITION 4.9 intertwines the coproducts and the image ( φ ⊗ φ)(Θ) is then also a 2-cocycle.It seems reasonable to expect that the resulting algebra would also have properties similar to the one above (i.e.not very nice spectrum-vice).
QUESTION 5.4.What happens if we remove the condition ker ω = ker ω * = {0} from the definition of weight inverse?
We call such ω a partial weight inverse.A classification of partial weight inverses for discrete G will be given in a separate paper.
λ coincides with the well-known construction of the universal complexification of G due to Chevalley and the Lie algebra of G C is the complexification g C of the Lie algebra g ≃ Λ of G, where the usual Lie bracket

THEOREM 4 . 21 .
Let H ⊆ G be a closed subgroup, ω H ∈ V N(H) be a weight inverse and ω

THEOREM 4 . 23 .
Let G be a connected simply connected Lie group and let H be either abelian or compact connected closed subgroup of G. Suppose ω H is a positive weight inverse on the dual of H and ω = ι H (ω H ) is the extended weight inverse on the dual of G. Then spec spec A(G, ω) ≃ {λ G (g) exp i∂ λ G (sH) | g ∈ G, s ∈ R, |s| ≤ ln β } = {exp (t∂ λ G (E)) exp (s∂ λ G (H)) | t ∈ R, |Im s| ≤ ln β }.