Elsevier

Advances in Mathematics

Volume 391, 19 November 2021, 107951
Advances in Mathematics

Beurling-Fourier algebras on Lie groups and their spectra

https://doi.org/10.1016/j.aim.2021.107951Get rights and content

Abstract

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely SU(n), the Heisenberg group H, the reduced Heisenberg group Hr, the Euclidean motion group E(2) and its simply connected cover E˜(2). We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate that “polynomially growing” weights do not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras.

Introduction

Let G be a locally compact group and A(G) its Fourier algebra as defined in [9]. We recall that A(G) is the predual of the von Neumann algebra VN(G) generated by the left regular representation λ:GB(L2(G)), and, moreover, is a dense subalgebra of C0(G). In a sense which is specified in the theory of locally compact quantum groups, the pair (A(G),VN(G)) is the Pontryagin dual object to (L1(G),L(G)), where we purposely suppress mention of the Haar weights. In fact, if G is abelian, then (A(G),VN(G))(L1(Gˆ),L(Gˆ)) where Gˆ is the dual group. Hence we expect aspects of the theory of the convolution algebra L1(G), and related convolution algebras, to be reflected in the theory of A(G).

Beurling algebras, L1(G,ω) for submultiplicative weights, ω:G(0,) satisfying ω(xy)ω(x)ω(y) for x,y in G, are important variants of L1(G), in particular when G is abelian or is compactly generated. Recently, weighted versions of the Fourier algebra have been introduced and investigated by various subsets of the present authors [16], [29], [31] under the name of Beurling-Fourier algebras; see also [38]. These have proved to admit a rich theory. In [29] properties such as Arens regularity are studied. This is followed up by [16] where it is shown that Beurling-Fourier algebras can often be isomorphic to algebras of operators on Hilbert spaces, a property which stands in contrast to A(G) ([30, Prop. 3.1]). In [31], spectral theory and associated properties such as regularity are studied. The present article is really a continuation on the theme of the latter.

The first goal of the present note is to give a unified treatment to all known examples. In broad terms, we proceed as follows.

  • (1)

    We formulate a general definition of a weight W on the dual of G, which allows the definition of the Beurling-Fourier algebra A(G,W) which is commutative. See Section 3.2.

The definition is in terms of unbounded positive operators. In order to show that the definition is meaningful, we must proceed to
  • (2)

    construct examples of weights W on the dual of G.

To do this, we have three fundamental strategies: we construct central weights, typically on connected compact groups (Section 3.3.2); we extend certain weights from subgroups (Section 3.3.3); and we use the Laplacian on certain connected Lie groups, which, in particular, gives us the polynomial weights (Section 3.3.4). The first step in the analysis of a commutative Banach algebra is to understand its spectral theory. Hence for every example that we devise, we
  • (3)

    compute the spectrum of A(G,W).

This turns out to be the most difficult aspect of our theory, and our approach is outlined below.

Beginning with the influential work of Wiener [47], spectral theory has proved to be an essential part of understanding a commutative Banach algebra. Of course, our modern understanding of spectral theory arises from the revolutionary work of Gelfand [15]. Eymard [9], Saitô [41] and Herz [21] have all given different proofs that the spectrum of A(G) is identifiable with G. In both [9], [21], spectral synthesis at points plays a key role in the determination of the spectrum.

Let us consider the case of G abelian. We letGˆC={χ:GC×|χ is continuous and multiplicative}, where C× is the multiplicative group of non-zero complex numbers. Then it is straightforward to see that the spectrum of L1(G,ω) is the set of ω-bounded characters, those χ in GˆC such that |χ(x)|ω(x) for all x in G (see [24, Section 2.8]). Notice for χ in GˆC, the range of any compact subgroup is in T. Hence we expect interesting theory of the spectrum only for G with no compact subgroups, i.e. those groups for which Gˆ is connected. Notice, in the case of connected Lie dual group, i.e. G=Rn×Zm so GˆRn×Tm, and we have GˆCCn×(C×)m is the complexification of Gˆ. We note that especially in the case that G=Rn or Zn, and ω bounded away from 0, the set of Fourier transforms A(G,ω)={fˆ:fL1(G,ω)} have been important in the study of spectral properties of commuting operators on Banach spaces and (systems of) operator equations through A(G,ω)-functional calculus. Here the analytic structure and spectral synthesis in A(G,ω) have been significant issues. See, for example, [33], [44].

When we consider the Beurling-Fourier algebras A(G,W), we simultaneously lose any (a priori) notion of spectral synthesis of points and a straightforward notion of “generalized W-bounded character”. Hence this task of understanding the structure of the spectrum requires a novel approach. We require a means of allowing complexifications of (in the present paper) a connected Lie group G to act as operators on L2(G), affiliated with VN(G), in a manner that interacts nicely with W. Regrettably, there are no general means of doing so in the literature, so we are forced to devise one ourselves. Our approach entails a large amount of hard analysis, specific to examples. Our mix of unbounded operator theory, harmonic analysis, and Lie theory appears to be novel and hence an interesting contribution itself.

In [31], three of the present authors considered central weights on compact groups. Here the dual object Gˆ is the set of equivalence classes of irreducible representations. They considered functions w:Gˆ(δ,), δ>0, which satisfy the submultiplicativity conditionw(ρ)w(π)w(π) for any π,π,ρGˆ such that ρππ, i.e. ρ is a subrepresentation of ππ. They, then, letA(G,w)={fC(G):πGˆw(π)dπfˆG(π)1<} where the norm is given by a weighted sum of the trace norms of the matricial Fourier coefficientsfˆG(π)=Gf(g)π(g)dg. In the case that w is the constant function 1, this gives A(G). In this case the subalgebra Trig(G) of finite linear combinations of matrix coefficients of elements of Gˆ forms a dense subspace. McKennon [34] defined the abstract complexification GC of G as the set of non-zero multiplicative functionals SpecTrig(G), and a full analysis was done by Cartwright and McMullen [2]. For a connected compact Lie group, this is exactly the universal complexification due to Chevalley ([1, III.8]), and generally is a pro-Lie group which is an extension of the complexification of the connected component of the identity G0, by the totally disconnected quotient G/G0. Hence computing SpecA(G,w) reduces to the task of determining which elements of GC lie naturally in the dual of A(G,w). This task is interesting only on connected groups and can be easily reduced to connected Lie groups. See Section 5.1.1 for some sample computations in this context.

For the reason just mentioned and other considerations below, we restrict our analysis to certain connected Lie groups. The main theme of this paper is to extend the above scheme to the case of general connected possibly non-compact Lie groups. However, we immediately face an obstacle, namely the absence of the abstract Lie theory applicable for non-compact locally compact groups. There is one model of abstract Lie theory for locally compact groups suggested by McKennon ([35]). However, the authors were not able to find any direct connection between our Beurling-Fourier algebras A(G,W) and McKennon's model (see (2) of Remark 3.16 for more details) at the time of this writing. For this reason, we are forced to establish an “abstract Lie model” suitable for Beurling-Fourier algebras from scratch.

As a motivation for this “abstract Lie model” we consider the case of G=R. Using Pontryagin duality we begin with a weight function (i.e. sub-multiplicative and Borel measurable) w:Rˆ(0,). We assume that w is bounded below (i.e. infxRw(x)>0) to ensure that L1(Rˆ,w)L1(Rˆ). Here, Rˆ is the dual group of R. Then the Beurling-Fourier algebra A(R,w) can be identified with the Beurling algebra L1(Rˆ,w) via the Fourier transform, FRˆ, on the dual group Rˆ. Even though we know RˆR we will keep the notation Rˆ to emphasize the distinction of the two groups. As mentioned above, the spectrum SpecL1(Rˆ,w) is well-understood via the concept of w-bounded characters. However, we require a more subtle route starting with a dense subalgebra A=FRˆ(Cc(Rˆ)) of A(R,w)L1(Rˆ,w), which plays an important role replacing Trig(G) for a compact group G. An element φSpecA(R,w) is determined by its restriction φ|A by the density of AA(R,w), and its transferred version ψ:=φ|AFRˆ:Cc(Rˆ)C is now a continuous multiplicative linear functional with respect to convolution product on Rˆ. This is illustrated in the diagram below. Since ψ arises from a locally integrable function in L(Rˆ,w1), it is continuous on Cc(Rˆ). Hence the Cauchy functional equation for distributions, which will be discussed in Section 6.1.2, shows that (up to normalization of the Lebesgue measure)ψ(f)=Rˆf(x)eicxdx for some c in C. Notice that the Paley-Wiener Theorem tells us that A is an algebra of analytic functions, and hence solving the Cauchy functional equation amounts to saying that such point evaluations comprise SpecA. Returning to SpecA(R,w), we simply need to determine for which c in C does supxRˆ|eicx|w(x)<. Notice that C is the universal complexification RC.

In this paper, we aim to determine SpecA(G,W) for a connected Lie group G by extending the above approach as follows.

  • (Step 1)

    Any dense subalgebra A of A(G,W) gives an injective embeddingSpecA(G,W)SpecA. We require A to satisfy that SpecAGC through an appropriate “abstract Lie” theory. Hence, we require that any element in A extends analytically to GC, so that we can identify a point xGC and the point evaluation functional φxSpecA at x.

  • (Step 2)

    We check which points in GC give rise to a linear functional bounded in the A(G,W)-norm.

Both of these steps are much more involved than in the abelian case illustrated above. In particular, choosing the subalgebra A is a highly non-trivial task since we need to ensure its density in A(G,W) for any general weight W, possibly of “exponential growth”. Thus, for example, an immediate candidate Cc(G), the space of test functions, is not enough for that purpose, as indicated in Remark 6.5 below. To find A in (Step 1) we borrow the “background” Euclidean structure of the given Lie group. This trick drives us to a suitable modification of the Cauchy functional equation, leading us to the points on GC. Thus, the procedure just described could be understood as the “abstract Lie” theory we needed. Moreover, we use the concept of entire vectors for unitary representations to guarantee their analytic extendability.

The technicality of choosing a dense subalgebra A and the associated Cauchy type functional equation forces us not to attempt to establish a general theory applicable for any connected Lie group in this paper. Instead, we will focus on representative examples, namely SU(n) among compact connected Lie groups, the Heisenberg group H among simply connected nilpotent Lie groups and its reduced version Hr, the Euclidean motion group E(2) acting on R2 among solvable non-nilpotent Lie groups and its simply connected cover E˜(2). These are groups which are known to have sufficiently many entire vectors.

In Section 2 we summarize some basic materials we need in this paper. In Section 2.1 we cover basics on unbounded operators, including Borel functional calculus for strongly commuting self-adjoint operators and a general treatment on an extension of ⁎-homomorphisms to certain unbounded operators. In Section 2.2 we provide materials about Lie groups, Lie algebras, and related operators, including complexification models of Lie groups and entire vectors of unitary representations. We include a short Section 2.3.2 on the choice of Fourier transforms since we use various versions of them.

In Section 3 we will provide a general definition of Beurling-Fourier algebras on locally compact groups, and the associated weights on their dual, which replaces the definition in [29]. We begin with motivation from the case of abelian groups in Section 3.1. In Section 3.2 we give a rigorous definition of weights on the dual of locally compact groups and define associated Beurling-Fourier algebras based on it. A more concrete interpretation of Beurling-Fourier algebras is given in the following two subsections 3.2.1 and 3.2.2. In Section 3.3 we introduce three fundamental ways of constructing weights, namely the central weights, the weights extended from subgroups, and the weights obtained from Laplacian on the group.

Starting from Section 4 we examine concrete examples, with the first being compact connected Lie groups, in particular SU(n), the n×n special unitary group. We first provide details of weights on the dual of compact connected Lie groups in Section 5. We then determine the spectrum of Beurling-Fourier algebras in Section 5.1. The cases of central weights and the weights extended from subgroups are fundamentally different, so the corresponding approaches also differ.

In Section 6 we analyze the case of the Heisenberg group H. The technical key observation here is that we borrow the “background” Euclidean structure of H, namely R3 for the choice of dense subalgebra A, playing the role which Trig(G) does in compact theory. Then, we continue to the Cauchy functional equation for distributions on R3 to provide a substitute for the Chevalley style of complexification model. In Section 7 we continue the case of the reduced Heisenberg group Hr, which shares most of the technical details of the Heisenberg group case.

In Section 8 we focus on the case of the Euclidean motion group E(2). The choice of the dense subalgebra becomes more involved, reflecting the structure of the representation theory using the polar form on R2. Moreover, the corresponding Cauchy functional equation is also more involved. In Section 9 we continue the case of the simply connected cover E˜(2) of E(2), which shares most of the technicalities.

Up to this point, we mainly focused on the case of “exponentially growing” weights providing the spectrum of the corresponding Beurling-Fourier algebras strictly larger than the original group. In Section 10, however, we consider the case of “polynomially growing weights”. The main result is that polynomially growing weights do not change the spectrum of the Beurling-Fourier algebras. The proof also provides the regularity of the corresponding Beurling-Fourier algebras. We also provide some non-regular Beurling-Fourier algebras at the end of this section.

In the final section, we collect some questions remaining from our analysis.

Section snippets

Unbounded operators

We collect some of the basic materials on unbounded operators. Our main reference on this matter is the text of Schmüdgen, [42], in particular Chapters 4 and 5.

A linear map T defined on a subspace domT, which we call the domain of T, of a Hilbert space H into another Hilbert space K is called closed if the graph of T, {(h,Th):hdomT}, is closed in HK. We say that another linear map S:domSHK is an extension of T if domTdomS and S|domT=T. In this case we write TS. We say that T is closable

A refined definition for Beurling-Fourier algebras

In [29] the authors suggested a model for a weight W on the dual of a locally compact group G. Regrettably, the suggested set of axioms in [29] was slightly misleading and had limitations in covering variety of examples beyond the ones already covered there, namely the case of “central weights” on compact groups (extending the case of [31]) and the Heisenberg group. Here, we introduce a refined definition of weights and the associated Beurling-Fourier algebras extending the previous definitions

More on representations of compact connected Lie groups

In this section we recall some representation theory of compact connected Lie groups starting with the highest weight theory from [46] and [31, Section 5]. We consider the decomposition g=z+g1, where z is the center of g and g1=[g,g]. Let t be a maximal abelian subalgebra of g1 and T=expt. Then there are fundamental weights λ1,,λr,Λ1,,Λlg with r=dimz and l=dimt such that any πGˆ is in one-to-one correspondence with its associated highest weight Λπ=i=1raiλi+j=1lbjΛj with (ai)i=1rZr and (b

Weights on the dual of compact connected Lie group G

In this section, we present three types of weights on the dual of a compact connected Lie group G: central weights, weights extended from subgroups, and weights derived from the Laplacian. Note first that a weight W on the dual of G can be understood as a collection of matrices (W(π))πGˆπGˆMdπ with the dense subspace S=Φ1(Trig(G)) in Section 3.2.2. We begin with central weights.

Example 5.1

For α0 we have dimension weights wα given bywα(π):=(dπ)α,πGˆ. The sub-multiplicativity comes from the fact

The Heisenberg group

The Heisenberg group isH={(y,z,x)=[1xz1y1]:x,y,zR}=(R×R)R with the group law being the matrix multiplication or equivalently(y,z,x)(y,z,x)=(y+y,z+z+xy,x+x). Here, we use the notation (y,z,x) instead of (x,y,z) in order to keep the semi-direct product structure of H. The Haar measure of H is given by d(y,z,x)=dxdydz, where dx, dy, and dz denote the Lebesgue measure on R.

For any aR, we have an irreducible unitary representation given byπa(y,z,x)ξ(t)=eia(tyz)ξ(x+t),ξL2(R). Now the

The reduced Heisenberg group

The reduced Heisenberg group Hr isHr=(R×T)R with the group law(y,z,x)(y,z,x)=(y+y,zzeixy,x+x). The order of variables (y,z,x) is again from the semi-direct product structure of Hr. Using Mackey machinery, we can completely describe the unitary dual of Hr as follows:Hrˆ={χr,s:(r,s)R2}{πn:nZ{0}}, where for every nZ{0}, the irreducible unitary representation πn is defined on the Hilbert space L2(R), whereas χr,s is a one-dimensional representation for every (r,s)R2. The precise

The Euclidean motion group on R2

The Euclidean motion group on R2 isE(2)={(x,y,z)=[zx+iy01]:x,yR,zT}=R2T with the group law by the matrix multiplication or equivalently(x,y,z)(x,y,z)=((x,y)T+ρ(z)(x,y)T,zz) where ρ(z)=[RezImzImzRez] and (x,y)T refers to the transposed column vector. Here, we use the notations (x,y,z)=([xy],z)=((x,y)T,z).

The unitary dual E(2)ˆ of E(2) can be described as follows. For any r>0 we define an irreducible unitary representation πr acting on L2[0,2π] byπr(x,y,z)F(θ)=eir(xcosθ+ysinθ)F(θt)

The simply connected cover E˜(2) of the Euclidean motion group

The simply connected cover E˜(2) of the Euclidean motion group E(2) on R2 isE˜(2)=R2R with the group law(x,y,t)(x,y,t)=((x,y)T+ρ(t)(x,y)T,t+t) where ρ(t)=[costsintsintcost] and we use the notation (x,y,t)=([xy],t)=((x,y)T,t).

The representation theory of E˜(2) can be described as follows. For any r>0 and zT we consider the Hilbert spaceHr,z={FLloc2(R):F(θ+2π)=zF(θ)for almost every θR} with the inner productF,G:=12π02πF(θ)G(θ)dθ where Lloc2(R) refers to the space of locally

The spectrum under polynomial weights and regularity of Beurling-Fourier algebras

In this section we will demonstrate that a “polynomially growing” weight W does not change the spectrum, i.e. SpecA(G,W)G and prove regularity of the associated algebra A(G,W). Recall that a subalgebra A of C0(Σ) for a locally compact Hausdorff space Σ is called regular on Σ if for each proper, closed subset E of Σ and each xΣE there exists fA with f(x)=1 and f0 on E. A commutative Banach algebra A is regular if its algebra of Gelfand transforms is regular on SpecA. When the weight is

Questions

In this final section we collect relevant questions that we were not able to answer at the time of this writing.

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  • 1

    M. Ghandehari was partially supported by University of Delaware Research Foundation, and partially by NSF grant DMS-1902301, while this work was being completed.

    2

    H.H. Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) Grant NRF-2017R1E1A1A03070510 and the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIT) (Grant No. 2017R1A5A1015626).

    3

    N. Spronk was partially supported by NSERC grant 312515-2020.

    4

    L. Turowska was partially supported by “Stiftelsen G S Magnussons Fond” and the Department of Mathematical Sciences, Chalmers University of Technology through a guest research program.

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