Dual Convolution for the Affine Group of the Real Line

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on L2(R×,dt/|t|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({{\mathbb {R}}}^\times , dt/ |t|)$$\end{document}. In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on Lp(R×,dt/|t|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p({{\mathbb {R}}}^\times , dt/ |t|)$$\end{document} for p∈(1,2)∪(2,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,2)\cup (2,\infty )$$\end{document}.


Background and Motivation
Given a bounded and SOT-continuous representation π of a topological group G on a Banach space E, one may associate to each ξ ∈ E and φ ∈ E * the coefficient function φ(π( )ξ ) ∈ C b (G). The vector space generated by all coefficient functions of π admits a natural norm, stronger than the uniform norm of C b (G), and its completion in this norm is called the coefficient space of π .
If G is locally compact, we denote by A(G) the coefficient space of the left regular representation λ : G → U(L 2 (G)). Eymard [7] showed that A(G) is actually a Banach algebra with respect to pointwise product, now called the Fourier algebra of G. When G is abelian, the Fourier transform gives an isometric isomorphism between A(G) and the convolution algebra L 1 ( G). Even when G is non-abelian, a well-established theme in abstract harmonic analysis has been to view A(G) as some kind of convolution algebra on a "quantum group" that is dual to G. However, in most cases this "dual convolution" is only defined in a formal or abstract sense.
This article studies a particular case where this notion of dual convolution can be made precise and described explicitly. Consider the group of affine transformations of R, given the natural topology, which we denote by R R × . This group has an unusual property that never occurs for non-trivial compact or abelian groups: writing H = L 2 (R × , dt/|t|), there is an irreducible unitary representation π : R R × → U(H) such that A(R R × ) coincides with the coefficient space of π , which we denote by A π (R R × ). Associated to π is a surjective norm-decreasing map : H ⊗ H → A π (R R × ), which is isometric since π is irreducible.
Since A π (R R × ) = A(R R × ), and since A(R R × ) is a Banach algebra with respect to pointwise product, we can use the surjective isometry : H ⊗ H → A π (R R × ) to equip S 1 (H) = H ⊗H with a commutative Banach algebra structure. In [6,Problème 2.7], after making this observation, Eymard and Terp pose the following challenge:

Interpréter cette multiplication en terme des opérateurs!
The present paper answers their challenge by providing an explicit formula for the new multiplication on S 1 (H) -this is what we refer to as "dual convolution" for R R × . To our knowledge, such a formula has not been recorded before in the literature.
Having established this explicit formula, the rest of our article investigates some applications and variations, described in more detail in Sect. 1.2. These applications and variations are intended to demonstrate that the resulting Banach algebra A can be studied directly, without any prior knowledge of the isomorphism : A → A(R R × ), and to argue that A is an object of intrinsic interest. A loose but instructive parallel is with certain naturally occuring Banach function algebras, such as AC([0, 1]), that can be modelled as L 1 -convolution algebras of certain semigroups.
Informally: by introducing dual convolution on S 1 (H), we are swapping an object where the algebra structure is easy to describe but the norm is complicated, for one where norm estimates are straightforward but the algebra structure is more complicated. This offers an alternative point of view on A(R R × ), which could shed new light on its known properties as a Banach algebra. Moreover, analogous constructions for higher-dimensional semidirect product groups may yield new results for their Fourier algebras.

Outline of Our Paper
In Sect. 2 we set up basic notation and definitions that will be used throughout the paper. We give an explicit definition/description of the group R R × and the key representation π : R R × → U(H), and collect some known facts from the literature for ease of reference.
In Sect. 3 we give an explicit formula for dual convolution as a bilinear map : S 1 (H)×S 1 (H) → S 1 (H). The formula is motivated by showing how one expresses the product of two coefficient functions of π as a continuous average of other coefficient functions (a so-called "fusion formula"). We show by explicit calculations, without invoking the representation π , that is commutative and associative. We also show that if trace-class operators on H are given as integral kernel functions, then can be described on that level also.
Writing A for the Banach algebra (S 1 (H), ): in Sect. 4 we construct a derivation D : A → A * which has interesting operator-theoretic properties as a linear map between Banach spaces (it is cyclic, weakly compact, and "co-completely bounded" in the terminology of [3]). Usually, in constructing derivations on function algebras, it is easy to see that the derivation identity holds on a dense subalgebra, but hard to show that one has a well-defined and bounded map on the whole algebra. By working with dual convolution on A, the situation is reversed: it is easy to check that D is a bounded linear map with the extra properties mentioned above, and the hard part is to verify the derivation identity.
The group R R × is not connected, but has an index 2 subgroup isomorphic to the semidirect product R R × 1 , which is a fundamental example of a non-unimodular connected Lie group. (The notation will be explained in Sect. 2.) Since A(R R × 1 ) cannot be identified with the coefficient space of a single irreducible representation, a direct description of dual convolution for R R × 1 is less straightforward. In Sect. 5 we identify an explicit subalgebra of A that corresponds to A(R R × 1 ), and hence obtain an analogue of dual convolution for R R × . We then show how the construction in Sect. 4 yields a derivation on the Fourier algebra of A(R R × 1 ), which offers a new perspective on some resuts in [1].
In Sect. 6 we consider A p π , the coefficient space of the L p -analogue of π , from the viewpoint of dual convolution. We sketch how our explicit formula for may be extended from into a commutative Banach algebra A p . We then show that A p π is a Banach algebra in its natural norm and is isomorphic to A p (Theorem 6.2). Perhaps surprisingly, for p = 2 there is a crucial difference from the p = 2 case: A p π is not the same as the L p -version of the Fourier algebra (Theorem 6.6), and it appears to be a new Banach function algebra about which we know little at this stage.
Finally, in Sect. 7, we make some remarks about possible directions for future work, and pose some explicit questions about the algebra A p π . In the appendix we show how the tensor product of two induced representations may be expressed as a direct integral of a family of induced representations, and use it to give an alternative proof of the fusion formula for coefficient functions of π .

Notation and Some General Background
If H 1 and H 2 are Hilbert spaces then H 1 ⊗ 2 H 2 denotes their Hilbert-space tensor product.
Given a complex vector space V , the conjugate vector space V is defined to have the same underlying additive group as V , equipped with the new C-action c ξ = cξ . Note that if H is a Hilbert space then the function H × H → C defined by (ξ, η) → ξ, η is bilinear rather than sesquilinear.
The symbol ⊗ denotes the projective tensor product of Banach spaces. If H is a Hilbert space then there is a standard identification of H ⊗ H with the space S 1 (H) of trace-class operators on H, defined by viewing the elementary tensor ξ ⊗η as the rankone operator α → α, η ξ ; this correspondence is an isometric, C-linear isomorphism of Banach spaces.
Coefficient functions associated to continuous bounded group representations were already defined in the introduction, but we did not give a precise definition of the corresponding coefficient spaces. Most of this article concerns unitary representations on Hilbert spaces, so we review some standard material here in order to fix our notation.
If σ : G → U(H) is a continuous unitary representation and ξ, η ∈ H, we denote the associated coefficient function x → σ (x)ξ, η by ξ * σ η ∈ C b (G). There is a contractive, linear map σ : H ⊗ H → C b (G) defined by σ (ξ ⊗ η) = ξ * σ η. We denote the range of σ by A σ (G), or simply A σ if the group G is clear from context; this is the coefficient space of σ , and we equip it with the quotient norm pushed forward from H ⊗ H/ ker( σ ).
Two special cases should be singled out: 1. If λ denotes the left regular representation G → U(L 2 (G)), then A λ (G) coincides with the Fourier algebra of G, and is usually denoted by A(G

The Affine Group of R
R × denotes the multiplicative group of R, equipped with the subspace topology; it has a Haar measure dt/|t| where dt denotes usual Lebesgue measure on R. We write R × 1 for the subgroup of R × consisting of strictly positive real numbers; the notation is consistent with using G e to denote the connected component of a locally compact group G.
When dealing with L p -spaces on R × , we will usually omit mention of the Haar measure and merely write L p (R × ); this should not be confused with L p (R) which always means the L p -space for the Lebesgue measure on R.
We define R R × to be the set {(b, a) : b ∈ R, a ∈ R × } equipped with the product topology of R × R × and the following multiplication: Inversion in R R × is given by Note that R embeds as a normal closed subgroup of R R × via b → (b, 1), while R × embeds as a closed subgroup via a → (0, a).
In harmonic analysis it is more common to work with the subgroup {(b, a) : b ∈ R, a ∈ R × 1 }. This is a connected Lie group, often referred to in the literature as "the real ax + b group"; we shall return to it in Sect. 5.

The Key Representation and its Coefficient Space
As in the introduction, we let H denote L 2 (R × ). There is a continuous unitary repre- This is a special case of a more general construction: if we consider the character χ 1 on R given by χ 1 (t) = exp(2πit), the previous formula may be written as where we use the explicit realization of an induced representation for a semidirect product group, as described in "Realization III" of [10, Sect. 2.4] (see Appendix A for details). Mackey theory tells us that is irreducible, and is the only infinite dimensional irreducible representation of R R × .
In this article we work not with but with a unitarily equivalent form (which matches the representation defined in [6, Eq. (1.3)]). For a C-valued function on a group G, definef : G → C byf (x) = f (x −1 ). Since Haar measure on R × is invariant under the change of variables t ↔ t −1 , the map ξ →ξ defines an isometric involution J : H → H. We now define π = J (·)J : R R × → U(H). Explicitly, given ξ ∈ H and b ∈ R, a ∈ R × , we have We claimed in the introduction that A π (R R × ) = A(R R × ). This can be seen as follows. The left regular representation λ of R R × can be obtained by inducing the left regular representation of R, which we denote by λ R . Note that λ R is unitarily equivalent to a direct integral (over R × ) of all nontrivial characters of R. Moreover, each such character is induced to a representation of R R × equivalent to π . Since induction and direct integration commute, it follows that λ is equivalent to π ⊗ I H for some separable Hilbert space H. Hence π is weakly equivalent with λ, and A π (R R × ) = A λ (R R × ) = A(R R × ) by the results mentioned in Sect. 2.1.

Remark 2.2 The equality A
is closed under pointwise product. In Sect. 3 we will give an alternative proof of this fact, using dual convolution on H ⊗ H. In Sect. 6 we will see that this alternative proof carries over to the L p -analogue of A π (R R × ), but that this space is not equal to the L p -analogue of A(R R × ).
We shall write rather than π for the canonical quotient map H ⊗ H → A π (R R × ), ξ ⊗ η → ξ * π η. Since π is irreducible, is injective by the remarks in Sect. 2.1, although we shall not use this fact when defining dual convolution in Sect. 3. [6], the map is denoted by F and called "la co-transformation de Fourier" for the group R R × . Note that because R R × is non-unimodular, composing with the operator-valued Fourier transform F :

Bochner Integrals and Related Measure Theory
Our explicit formula for dual convolution is expressed as a Bochner integral, which requires attention to questions of strong measurability (also referred to in the literature It is usually impractical to verify directly that a given Banach-space valued function is strongly measurable. For functions with values in an L p -space an alternative approach is provided by the following result: given two sigma-finite measure spaces ( 1 , μ 1 ) and ( 2 , μ 2 ), and 1 ≤ p < ∞, there is a natural embedding where f ⊗ g is sent to the function ω 1 → f (ω 1 )g. This embedding extends to an isometric isomorphism of Banach spaces L p ( 1 × 2 , μ 1 × μ 2 ) ∼ = L p ( 1 , μ 1 ; L p ( 2 , μ 2 )) (see e.g. [8,Prop. 1.2.24] for the proof of a more general statement). In particular, elements of L p ( 1 × 2 , μ 1 × μ 2 ) define strongly μ 1 -measurable functions 1 → L p ( 2 , μ 2 ).
We use both λ and ρ, even though R × is abelian, because we have in mind possible extensions of the following calculations to semidirect products of the form R n D where D ⊂ GL n (R) need not be abelian.

An Explicit Formula for Fusion of Coefficients
To avoid any doubt we shall pay close attention to issues of convergence and integrability. Let ξ 1 , η 1 , ξ 2 , η 2 ∈ H. For each (b, a) ∈ R R × , (where as usual we treat a measurable function defined on R × as a measurable function defined on R, by prescribing some arbitrary value at 0). Let d(t, s) denote the Haar measure on R 2 . Observe that the function is integrable on R 2 , since by Tonelli's theorem for R 2 followed by Cauchy-Schwarz for H, Therefore, the following changes of variable and order of integration are valid: One can now show that for fixed u ∈ R \ {0, 1}, the inner integral in the last line of Eq. (3.1) can be written as π(b, a)α u , β u for suitable α u , β u ∈ H, and that is a weighted average of explicit coefficient functions α u * π β u as u varies; this is what we mean by a "fusion formula" for coefficient functions.
|h||t| . [Tonelli] If f and g are measurable functions R × → C, let f · g denote their pointwise product (with the usual identifications of functions that agree a.e.).
Then F is equal a.e. to a strongly measurable, (Bochner-)square integrable function R × → H, and Proof We apply Lemma 3.1 with X = ξ 1 ⊗ ξ 2 . As remarked in Sect. 2.4, we may iden- Note that a priori, one only expects the pointwise product of two functions in H to lie in L 1 (R × ). The corollary shows that in fact, and so taking e.g. ξ 1 (t) = ξ 2 (t) = 1 t>1 (t − 1) −1/3 one sees that the RHS can be infinite.

Proposition 3.3 (Explicit fusion for coefficient functions of
By Corollary 3.2, F and G are (after modification on a null subset of R × ) strongly measurable as functions R × → H, and square integrable (with respect to Haar measure on R × ). Therefore, the function h → F(h) * π G(h) is strongly measurable and a.e.
where the final equality follows by using Corollary 3.2 again. Unpacking the definitions of F and G, and comparing them with (3.2), we see that

Remark 3.4
Our direct route to the key formula (3.2) relied on ad hoc manipulations of integrals. There is a more conceptual approach, based on constructing an explicit intertwining map between π ⊗ π and I H ⊗ π . This intertwining map emerges naturally from considering the representation defined in (2.3) and its description as an induced representation; details are given in Appendix A. In fact, this approach was originally how we came up with the formula (3.2), and it motivates the technique used in Lemma 3.1.

Defining Dual Convolution
The formula in Proposition 3.3 immediately suggests how to define the dual convolution of two rank-one tensors in S 1 (H) = H ⊗ H: given ξ, ξ ∈ H and η, η ∈ H, where the right-hand side is defined as a Bochner integral of a function R × → H ⊗ H. The proof that this function is Bochner integrable is essentially the same as the argument used in proving Proposition 3.3, so we shall not repeat it here; we record for reference that the same calculation yields the upper bound For, under this assumption, λ(a)ξ · ξ vanishes identically whenever |a| is sufficiently small or sufficiently large. It follows (using continuity of translation in H and in C 0 (R × )) that the integrand in (3.4) is a continuous, compactly supported function R × \{−1} → H ⊗H, with no need to worry about various formulas holding only a.e.
We can now extend the operation by linearity and continuity to a contractive bilinear map S 1 (H)×S 1 (H) → S 1 (H), by representing elements of S 1 (H) as absolutely convergent sums of rank-one tensors. To see that this extension is well-defined and independent of how we represent elements of S 1 (H), note that (ξ, η, ξ , η ) → (ξ ⊗ η) (ξ ⊗ η ) defines a contractive multilinear map from H × H × H × H to H ⊗ H, and so by the universal property of ⊗, it extends uniquely to a contractive linear map An alternative integral formula One can rewrite the defining formula (3.4) as after a change of variables 1   In fact, many of the preceding results could have been formulated without the change of variables in (3.2). Both formulations of dual convolution seem to be natural and useful: the formula (3.4) is more closely related to the underlying general principles concerning tensor products of induced representations; but (3.2) is more enlightening for certain calculations, such as (3.8) below.

An abstract definition of
An alternative way to think of our construction of , viewed as a bounded linear map from S 1 (H) ⊗ S 1 (H) to S 1 (H), is by constructing it as the composition of the maps shown in Fig. 1.
We now explain briefly what each of these maps is.
• The "shuffle" map interchanges the second and third factors in the tensor product, i.e. it sends ξ ⊗ η ⊗ ξ ⊗ η to ξ ⊗ ξ ⊗ η ⊗ η . • The "embed" map is self-explanatory, and V is from Lemma 3.1. The map "identify" is the same identification described in Sect. 2.4 and used in Corollary 3.2. • The "diagonal" map is given as follows: for Banach spaces E 1 and E 2 there is a canonical contraction then the trace map is the same as slicing in the first variable against the constant function 1 ∈ L ∞ (R × ).) The advantage of this approach is that all issues concerning strong measurability, or showing that various maps are well-defined and do not depend on how an element of H ⊗ H is represented as an infinite sum of tensors, are automatically taken care of by the formal identifications between various Banach spaces. Moreover, this approach also generalizes easily to the L p -setting, or to settings with additional operator space structure. The disadvantage is that this definition of is rather abstract, and is less suited to concrete calculations.

Basic Properties of Dual Convolution
Clearly is commutative: this follows directly from a change of variable h → h −1 in (3.4). Proving that is associative requires more work. (Recall that even when considering the usual convolution of two L 1 -functions on a locally compact group G, checking associativity directly by attempting to interchange integrals requires careful use of Fubini's theorem to justify treating identities that only hold a.e. as if they hold everywhere.) To show that is associative, it suffices by linearity and continuity to show that In the following calculations, we shall adopt the following notational convention to make our formulas more manageable. Given a function in C c (R × ) which is obtained from ξ 1 , ξ 2 and ξ 3 by some explicit formula The expression in the inner integral is measurable and Bochner integrable as a function One can use similar arguments to expand (ξ 1 ⊗ η 1 ) (ξ 2 ⊗ η 2 ) (ξ 3 ⊗ η 3 ) as a double integral with values in H ⊗ H, and show by appropriate changes of variable that this is equal to the right-hand side of (3.8). Alternatively, observe that since is commutative, then observe that the value of the last integral in Eq. (3.8) is unchanged if one swaps ξ 1 ⊗ η 1 with ξ 3 ⊗ η 3 (since this corresponds to interchanging the variables u and v in the integral). Note that Proposition 3.3 can be rephrased as and so by linearity and continuity, it follows that (T T ) = (T ) (T ) for all T , T ∈ S 1 (H). This gives an independent proof that A π (R R × ) is closed under pointwise product. It could also have been used to prove associativity of , by transferring it from associativity of pointwise product in A π (R R × ). We believe that the direct proof given above has independent interest, especially in light of the symmetry displayed by the formula in Eq. (3.8).
We sum up the results of this section in the following theorem.

Dual Convolution at the Level of Functions
Trace-class operators on H = L 2 (R × ) are often given not as explicit sums of rank-one tensors, but as integral operators defined by certain kernel functions R × × R × → C.
In this section we provide a description of dual convolution that may be easier to apply in such cases. We may view elements of S 1 (H) as measurable functions on R × × R × , as follows. First note that complex conjugation of functions defines a C-linear isometric isomorphism of vector spaces from L 2 (R × ) onto L 2 (R × ), which extends to an isometric isomorphism (3.10) Furthermore, the natural map is linear and norm-decreasing, and it is injective since Hilbert spaces have the approximation property. Finally, note that we may identify Thus, up to a.e. equivalence 2 we can view any T ∈ S 1 (H) as a measurable function on R × ×R × which is square-integrable (with respect to the measure |s| −1 |t| −1 d(s, t)). For ease of notation, we shall denote this function also by T , suppressing mention of the embeddingι. With this convention, which is the usual form in which an integral operator is given.

Proposition 3.7 (Pointwise formulas for dual convolution
where both of the integrals above are absolutely convergent for a.e. (s, t) ∈ R × × R × .
Proof When T 1 and T 2 are rank-one tensors, this follows from the definition of .
Hence it is true when T 1 and T 2 are finite rank operators. Every trace class operator is the limit in trace-norm of finite rank operators, and by going down to a subsequence we can assume that the convergence holds pointwise a.e. Now observe that if T = ∞ n=1 f n ⊗ g n where ∞ n=1 f n H g n H < ∞, the traceclass operator R = ∞ n=1 | f n | ⊗ |g n | satisfies R(s, t) ≥ |T (s, t)| a.e. The result now follows using the Lebesgue dominated convergence theorem, replacing T 1 and T 2 in (3.11a) or (3.11b) with "dominating operators" R 1 and R 2 .

An Explicit Derivation from A to its Dual
In this section we construct an explicit derivation D : A → A * and study some of its operator-theoretic properties. We will relate D to constructions in [1] in the next section.
We briefly review some general definitions. For a Banach algebra A, each ∈ (A ⊗ A) * corresponds to a bounded linear map A → A * defined by a → (a ⊗ ). This map A → A * is a derivation if the following identity holds: a 0 a 1 ) + (a 1 ⊗ a 2 a 0 ) for all a 0 , a 1 , a 2 ∈ A. (4.1) The derivation is said to be cyclic if (a ⊗ b) = − (b ⊗ a) for all a, b ∈ A.
Although we do not consider coefficient functions in this section, note that for every ξ, η ∈ H we have ξ * π η = Rξ * π Rη.
Constructing our derivation Define a multilinear map : We view as a bilinear form on A, and define D to be the corresponding operator A → A * . For i = 0, 1, if we write T i = ξ i ⊗ η i and use the convention T i (s, t) = ξ i (s)η i (t) as in Sect. 3.4, then we can rewrite (4.2) as:

Proposition 4.2 D : A → A * is cyclic and weakly compact.
Proof The identity (4.3) shows that (T 1 ⊗ T 0 ) = − (T 0 ⊗ T 1 ) when T 0 and T 1 are rank-one tensors; by linearity and continuity it holds for all T 0 , T 1 ∈ A. It also follows from (4.3), using the Cauchy-Schwarz inequality, that extends to a bounded bilinear form on L 2 (R × × R × ). Hence the operator D : A → A * factors through the embedding of A into L 2 (R × × R × ); in particular D is weakly compact.
Next, we show that D is a derivation by showing that satisfies the identity (4.1).

Theorem 4.3 (Derivation identity) For every T
(

4.4)
Proof By linearity and continuity, it suffices 3 to verify (4.4) in the special case where T 0 , T 1 and T 2 are rank-one tensors in C c (R × ) ⊗ C c (R × ). We now consider the three terms in (4.4), using (4.3) and Fubini's theorem. In each case the integral is taken over (R × ) 3 : Also, where the last equality used the change of variables s → (1 + h)s, t → (1 + h)t. A similar calculation yields For every s, t ∈ R × , define To prove that (4.4) holds, it suffices to show that I(s, t) = II(s, t) + III(s, t) for (almost) every s, t ∈ R × . For II(s, t): the change of variables h → − 1 1+h sends dh |h| to dh |1+h| and sends 1 + h to 1 − 1 1+h = h 1+h , so that For III(s, t): the change of variables h → −(1 + 1 h ) sends dh |h| to dh |1+h||h| and sends 1 Adding ( * * ) and ( * * * ) and recalling that s |s| = sign(s), we obtain I(s, t) as required.
Finally, we show that D : A → A * is completely bounded after composing with the transpose map on A * = B(H).
To verify complete boundedness of D : A → A * we use the following characterization. Recall that we have natural injective maps where "shuffle" swaps the second and third factors in the tensor product. Now define (4.5) We have since S ⊗ R and R ⊗ id are unitary operators on H ⊗ 2 H, it follows that • s −1 extends continuously to an element of (S 1 (H ⊗ 2 H)) * .

Remark 4.5
In the language of [3], D : A → A * is co-completely bounded, since "reverses the operator space structure" on B(H).

1
The semidirect product R R × 1 may be viewed as an open subgroup of R R × , by identifying it with {(b, a) : b ∈ R, a ∈ R × 1 }. General results on Fourier algebras of open subgroups then allow us to identify A(R R × 1 ) with a closed subalgebra of A(R R × ). More precisely, let us introduce the non-standard notation: )(b, a) = f (b, a) for a > 0 and P e ( f )(b, a) = 0 for a < 0. Then P e is a (completely) contractive projection from A(R R × ) onto A e (R R × ), and the composition of the maps is a completely isometric bijection. We now proceed to identify the subalgebra of A that corresponds to A e (R R × ) and hence models A(R R × 1 ). Recall the operator S : H → H given by (Sξ)(t) = sign(t)ξ(t). S is an isometric involution, so it has eigenvalues ±1, and H decomposes as an orthogonal direct sum of the corresponding eigenspaces, H = H + ⊕ 2 H − . The spaces H ± have the following explicit description: Let P ± be the orthogonal projection of H onto H ± , and define P diag : H ⊗H → H ⊗H by

Lemma 5.1 P diag is a norm-one projection.
Proof Since P + and P − are complementary projections, a direct calculation yields (P diag ) 2 = P diag . Therefore, it suffices to show that P diag is contractive:

[Pythagoras in H]
By the definition of the projective tensor norm, it follows that P diag (T ) ≤ T for all T ∈ H ⊗ H.

Remark 5.2
The decomposition H = H + ⊕ 2 H − gives a decomposition of T ∈ S 1 (H) as a 2 × 2 block matrix T = P + T P + P + T P − P − T P + P − T P − If we identify H ⊗ H with S 1 (H), then P diag corresponds to "compression to the diagonal".

Proposition 5.3 P diag is a homomorphism.
Proof It suffices to prove that P diag (T 1 ) To simplify our formulas slightly, we write ξ + 1 = P + ξ 1 , etc., and for functions f , g defined on the set We have To analyze each of these four terms, we consider the effect of λ(1+h) and λ(1+h −1 ) on vectors in H + or H − , as h varies over R × . Note that if α ∈ H ± and β ∈ H ∓ (i.e. α and β have "different parity") then α · β = 0 as elements of H. Also: if a > 0 then λ(a)(H ± ) = H ± ; and if a < 0 then λ(a)(H ± ) = H ∓ . Using these facts, • if h < 0 and h = −1, then precisely one of 1 + h or 1 + h −1 is negative, and so • if h > 0, then 1 + h and 1 + h −1 are both positive, and so Therefore, considering the four terms in (5.4), we obtain where we have used the same notational convention as in Sect. 3.3 to simplify the formulas. Now we consider P diag (T 1 T 2 ); since the Bochner integral commutes with bounded linear maps, this equals Therefore, splitting the integral in (5.6) into three pieces, and recalling that P diag = ± P ± ⊗ P ± , we obtain Comparing this with the combination of (5.4), (5.5a) and (5.5b), we have shown that P diag (T 1 ) P diag (T 2 ) = P diag (T 1 T 2 ) as required.
Definition 5.4 (The diagonal subalgebra) We define A diag := P diag (A). Note that by Proposition 5.3, A diag is a subalgebra.
We now examine the image of A diag under the map : A → A(R R × ).
both have the same "parity") then ξ * π η vanishes outside R R × 1 . (iv) If ξ ∈ H ± and η ∈ H ∓ (i.e. they have different "parity") then ξ * π η vanishes on R R × 1 . The claims in the lemma follow easily from the definitions of π and H ± , so we leave the details to the reader. Proof By linearity and continuity, it suffices to verify this identity on rank-one tensors in H ⊗ H. Let ξ, η ∈ H; then (ξ ⊗ η) = P + ξ ⊗ P + η + P + ξ ⊗ P − η + P − ξ ⊗ P + η + P − ξ ⊗ P − η = (P + ξ) * π (P + η) Hence by Lemma 5.5(iv), for every (b, a) ∈ R R × 1 , Thus, P e (ξ ⊗ η) and P diag (ξ ⊗ η) agree on R R × 1 . By definition, P e (ξ ⊗ η) vanishes outside R R × 1 ; and by Lemma 5.5(iv), so does P diag (ξ ⊗ η). We conclude that P e (ξ ⊗ η) = P diag (ξ ⊗ η) as required. Since is an algebra isomorphism, this provides an alternative proof that P diag is a homomorphism and A diag is a subalgebra of A. Moreover, by the remarks at the start of this section, we may identify A e (R R × ) with A(R R × 1 ). Thus (A diag , ) may be viewed as a realization of dual convolution for R R × 1 . Remark 5.7 Lemma 5.5(ii) shows that the restriction of π to R R × 1 splits as π + ⊕π − where π ± : R R × 1 → U(H ± ). Up to unitary equivalence, π + and π − are the only two infinite-dimensional unitary representations of R R × 1 ; they can also be constructed directly as induced representations. Attempting to construct dual convolution for R R × 1 directly requires consideration of π + ⊗ π + , π + ⊗ π − , π − ⊗ π + and π − ⊗ π − , and the fusion rules for the "mixed parity" cases are not so straightforward. Indeed, π + ⊗ π − is not quasi-equivalent to an irreducible representation of R R × 1 . Finally, we consider derivations on A diag and hence on A(R R × 1 ). Let D : A → A * be the derivation constructed in Sect. 4. Composing with the inclusion ι : A diag → A and the restriction ι * : A * → (A diag ) * , we obtain a derivation D 1 = ι * Dι : A diag → (A diag ) * . Cyclicity and weak compactness of D are inherited by D 1 , just from the definition. Now let be the transpose operator from Definition 4.4. Since ι is a complete isometry and * ι = ι * , complete boundedness of D : It remains to check that D 1 is not identically zero. Recall that by definition where (Sξ)(t) = sign(t)ξ(t) and Rξ(t) = ξ(−t). Fix some non-zero vector α ∈ H + and put β = Rα ∈ H − , so that α ⊗ α and β ⊗ β belong to A diag . Since Sα = α and R 2 = id we have Intertwining D 1 with yields a non-zero derivation D : A e (R R × ) → A e (R R × ) * . This derivation turns out to coincide, up to a scaling factor, with the derivation constructed in [1]; the proof requires the orthogonality relations for π ± or the Plancherel theorem for R R × 1 . Since is a completely isometric algebra isomorphism, D inherits the properties of D 1 . In particular D is weakly compact and "co-completely bounded" (using the terminology of [3]), properties which were less obvious from the original construction in [1].

A New Banach Algebra Structure on L p (R × ) ⊗ L q (R × )
Throughout this section, we assume 1 < p < ∞ and denote by q the conjugate index to p. We denote the usual pairing between L p (R × ) and L q (R × ) by , p,q ; note that ξ, η 2,2 = ξ, η .
For sake of precision, recall that there is an isometric, C-linear isomorphism of Banach spaces By intertwining withι, we may transfer the Banach algebra structure defined on . Moreover, one can use the natural analogue of the formula (3.4) to equip L p (R × ) ⊗ L q (R × ) with a Banach algebra structure, in a way that extends the p = q = 2 case.
That is: for ξ, ξ ∈ L p (R × ) and η, η ∈ L q (R × ), we claim that is a well-defined Bochner integral taking values in L p (R × ) ⊗ L q (R × ), and that extends to a bounded bilinear map which is commutative and associative. We denote the resulting commutative Banach Most of the steps needed to justify this claim consist of routine modifications of the arguments in Sect. 3.2, so we shall not give full details here. We highlight some of the relevant technical points.
(S1) There is an L p -analogue of Lemma 3.1, with an isometry V p : L p (R × ×R × ) → L p (R × × R × ) defined by the formula The L p -analogue of Fig. 1 As before, this shows that for ξ, ξ ∈ L p (R × ) the function One then performs the same construction with p replaced by q. (S2) However, one has to be careful taking pointwise products of two functions in L p (R × ) or L q (R × ). The expression defining F(h) a priori only takes values in L p/2 (R × ), which for 1 < p < 2 is not a Banach space (it is complete and quasi-normed, but not locally convex). (S3) Once one has shown that is well-defined and contractive as a bilinear map can prove it is commutative and associative by repeating the arguments of Sect. 3.3 almost verbatim; the key point is that C c (R × ) is still norm-dense in both L p (R × ) and L q (R × ). (S4) The abstract description of for H ⊗ H, shown in Fig. 1, has a natural and straightforward generalization to the L p -setting, which is sketched in Fig. 2.
For p = 2,ι : A → A 2 is an isometric isomorphism of Banach algebras. Since : A → C 0 (R R × ) is an injective homomorphism, it follows that A 2 ∼ = A is semisimple, and that we can identify A 2 with a Banach function algebra on R R × . We now show that the same is true for A p .
The formula π(b, a)ξ(t) := e 2πibt ξ(ta) still defines an isometric, SOT-continuous representation of R R × on L p (R × ). Hence, for each ξ ∈ L p (R × ) and η ∈ L q (R × ) there is an associated coefficient function: This formula defines a contractive linear map p : is an algebra homomorphism, by a direct calculation using the L p -analogue of Proposition 3.3. Hence A p π is a Banach function algebra, which in the case p = 2 is just A π (R R × ).
Proof By linearity and continuity it suffices to prove that for each ξ, η ∈ C c (R × ) the coefficient function f := p (ξ ⊗ η) belongs to C 0 (R R × ). This now follows because ξ, η ∈ L 2 (R × ) and A 2

An alternative argument, which does not rely on the equality
is integrable (use the Riemann-Lebesgue lemma for the Fourier transform on R). By a standard compactness argument, whose details we omit, we conclude that f ∈ C 0 (R × K ) ⊂ C 0 (R R × ).
So far everything has been a straightforward translation of what was done for the p = 2 case. In contrast, the next result seems to require extra work.
is an isometric isomorphism of Banach algebras.
For p = 2 this is a special case of general results already mentioned in Sect. 2. For general p, we make use of results from [6] that are particular to π and R R × . Consider the following space: V 0 is a linear subspace of A(R); standard properties of A(R) imply that V 0 is normdense in L p (R × ) for every p ∈ (1, ∞). The following lemma is a special case 4 of a result from [6], restated in a more direct form to avoid possible clashes of notational conventions.

Lemma 6.3 (Eymard-Terp)
Let ξ, η ∈ V 0 . Then there exists f ∈ L 1 (R R × ) such that, for every α, β ∈ C c (R × ), where the integrals over R × are taken with respect to the Haar measure of this group.
Since we only need a subset of Eymard and Terp's result, we include a proof of the lemma for the reader's convenience. Proof (following [6,Prop. 1.13

])
Let ξ, η ∈ V 0 . The right-hand side of Eq. (6.3) is equal, after a change of variables a → ta, to Proof of Theorem 6.2 It suffices to prove that p : is injective; the rest of the theorem follows from earlier observations. Let ξ, η ∈ V 0 and let f ∈ L 1 (G) be as provided by Lemma 6.3. Let j p : L p (R × ) ⊗ L q (R × ) → B(L p (R × )) be the map which sends an elementary tensor α ⊗ β to the rank-one operator γ → γ, β p,q α. Then we may rewrite Eq. 6.3 as: Hence, by linearity and continuity of j p and p , In particular, suppose w ∈ ker( p ). Then j p (w)ξ, η = 0. Since this holds for all ξ, η ∈ V 0 , and since V 0 is norm-dense in L p (R × ) and in L q (R × ), it follows that j p (w) = 0. Since L p (R × ) has the approximation property, j p is injective, and we conclude that w = 0 as required. a)ξ, η p,q dμ(b, a). (6.4) p is a contractive, weak * -weak * continuous algebra homomorphism, and it can be identified with the adjoint of p . Hence injectivity of p is equivalent to weak *density of p (M(R R × )) in B(L p (R × )). In effect, our proof of Theorem 6.2 works by showing that p (L 1 (R R × )) contains a norm-dense subspace of K(L p (R × )) and hence is weak * -dense in B(L p (R × )). While this formulation of the proof is more intuitive, it does not seem to make the argument significantly simpler. Note that for p = 2 the weak * -density result would follow from general facts about unitary representations of locally compact groups, but the proofs of those facts use C * -algebraic tools which are not available for general representations on L p -spaces.
Since both Banach spaces embed continuously in C 0 (R R × ), the closed graph theorem would then imply that the inclusion map A p (R R × ) → A p π is continuous. But this contradicts Proposition 6.5.

Remark 6.7 Since A
p π is the coefficient space of an isometric group representation on an L p -space, it is contained in the multiplier algebra of A p (R R × ). This follows from an L p -version of Fell's absorption principle (valid for any locally compact group), which appears to be folklore and goes back to the 1960s/70s. It would be interesting to study the relationship between A p π and A p (R R × ) in greater detail.

Concluding Remarks
We finish by suggesting some avenues for further exploration.
Affine groups of other local fields Much of [6] works in the general setting of a field K which is locally compact, second-countable and non-discrete, together with the corresponding affine group K K × . All the calculations of Sect. 3 and Sect. 6 should remain true for such a K , provided that one replaces the exponential function in the definition of π with a nontrivial character of (K , +). However, Sects. 4 and 5 use certain special features of R × that are not shared by K × , and we do not expect them to generalize to Q p , for instance.

Constructing explicit derivations on Fourier algebras
The question of which groups G allow non-zero derivations A(G) → A(G) * has been intensively studied in recent years. The calculations in Sect. 4 may give new ideas or techniques for constructing derivations on Fourier algebras of other (Lie) groups.
A concrete model for LCQG questions By enhancing the decomposition in Fig. 1 with operator-space structure, using row and column Hilbert spaces in the appropriate places, one can show that extends to a completely contractive map S 1 (H) ⊗ op S 1 (H) → S 1 (H), where ⊗ op denotes the projective tensor product of operator spaces. The adjoint of this map is a * -homomorphism : B(H) → B(H⊗ 2 H), which is coassociative since is associative. Moreover, the adjoint of : coincides with the canonical * -homomorphism VN(R R × ) → B(H) obtained by  sending λ(b, a) to π(b, a). Because is a homomorphism, * intertwines with the canonical comultiplication on VN(R R × ).
It might be interesting to study various general constructions for Hopf von Neumann algebras, using (B (H), ) as our concrete model of (VN(R R × ), ). In particular, to our knowledge it remains an open question if the operator systems WAP( G) and LUC( G) are subalgebras of VN(G) for non-abelian G; our concrete model may provide a new angle of attack when G = R R × .
One note of warning: the transpose operator : B(H) → B(H) is not intertwined with the canonical involution on VN(R R × ), because (π(b, a)) = π(b, a) * . If we wish to also introduce Kac algebra structure on (B(H), ), the antipode is given not by but by a unitarily similar operator.
Questions regarding A p and A p π . Let p ∈ (1, 2) ∪ (2, ∞).  Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Tensor Products of Induced Representations
Consider a semidirect product G = N H . The left action of H on N is denoted by h · n; if σ ∈ N then the corresponding left action of H on N is defined by h · σ : n → σ (h −1 · n).
Given a (continuous, unitary) representation σ : N → U(H σ ) we define the induced representation Ind G N σ : G → U(L 2 (H , H σ )) by the formula  [10], we get the following theorem: Theorem A.1 Let G = N H and let π 1 and π 2 be representations of N . For i = 1, 2 let i = Ind π i be the induced representation of G on L 2 (H , H π i ). Then where ρ is the right regular representation (ρ(h) f (k) = f (kh)), and φ f ⊗g ∈ L 2 (H , .
Proof A direct calculation shows that W preserves the inner product: Using (A.1), it is easy to verify that ρ(h) ⊗ g)).
As an application, we now derive an alternative proof of Proposition 3.3. We use the same notation as defined in Sect. 2. For r ∈ R, define χ r : R → C by χ r (t) = exp(2πirt), so that r → χ r is a group isomorphism R → R.