Representations of the Orlicz Figa-Talamanca Herz Algebras and Spectral subspaces

Let G be a locally compact group. In this note, we characterise non-degenerate *-representations of A_\Phi(G) and B_\Phi(G). We also study spectral subspaces associated to a non-degenerate Banach space representation of A_\Phi(G).


Introduction
Let G be a locally compact group. It is well known that there is a one to one correspondence between the unitary representations of G and the non-degenerate * -representations of L 1 (G) [5, p. 73]. Similarly, if X is any locally compact Hausdorff space, then there is a one to one correspondence between the cyclic * -representations of C 0 (X) and positive bounded Borel measures on X [8, p. 486]. The corresponding result for the Fourier algebra A(G) of a locally compact group is due to Lau and Losert [10]. For more on the Fourier algebra see [4,9]. Recently, Guex [11] extended the result of Lau and Losert to Figà-Talamanca Herz algebras. We refer the readers to [2] for more on Figà-Talamanca Herz algebras.
It is shown in [14] that A Φ (G) is a regular, tauberian, semisimple commutative Banach algebra with the Gelfand spectrum homeomorphic to G. This paper has the modest aim of characterising the non-degenerate * -representations of A Φ (G) in the spirit of [10]. This characterisation is given in Corollary 3.4. In Section 4, we show that any non-degenerate * -representation of A Φ (G) can be extended uniquely to a non-degenerate * -representation of B Φ (G). In Section 5, we provide an application to ergodic sequences. Godement in his fundamental paper [6] on Wiener Tauberian theorems studied spectral subspaces associated to a certain Banach space representations. This result was extended to the Fourier algebra A(G) by Parthasarathy and Prakash [12]. In Section 6, we also study spectral subspaces of A Φ (G).

Preliminaries
Let Φ : R → [0, ∞] be a convex function. Then Φ is called a Young function if it is symmetric and satisfies Φ(0) = 0 and lim x→∞ Φ(x) = +∞. If Φ is any Young function, then define Ψ as Then Ψ is also a Young function and is termed as the complementary function to Φ. Further, the pair (Φ, Ψ) is called a complementary pair of Young functions.
Let G be a locally compact group with a left Haar measure dx. We say that a Young function Φ satisfies the ∆ 2 -condition, denoted Φ ∈ ∆ 2 , if there exists a constant K > 0 and x 0 > 0 such that Φ(2x) ≤ KΦ(x) whenever x ≥ x 0 if G is compact and the same inequality holds with x 0 = 0 if G is non compact.
The Orlicz space, denoted L Φ (G), is a vector space consisting of measurable functions, defined as The above norm is called as the Luxemburg norm or Gauge norm. If (Φ, Ψ) is a complementary Young pair, then there is a norm on L Φ (G), equivalent to the Luxemberg norm, given by, This norm is called as the Orlicz norm. Let C c (G) denote the space of all continuous functions on G with compact support. If a Young function Φ satisfies the ∆ 2 -condition, then C c (G) is dense in L Φ (G). Further, if the complementary function Ψ is such that Ψ is continuous and Ψ(x) = 0 iff x = 0, then the dual of (L Φ (G), N Φ (·)) is isometrically isomorphic to (L Ψ (G), · Ψ ). In particular, if both Φ and Ψ satisfies the ∆ 2 -condition, then L Φ (G) is reflexive.
For more details on Orlicz spaces, we refer the readers to [13]. Let Φ and Ψ be a pair of complementary Young functions satisfying the ∆ 2 condition. Let The space A Φ (G) equipped with the above norm and with the pointwise addition and multiplication becomes a commutative Banach algebra [14,Theorem 3.4]. In fact, A Φ (G) is a commutative, regular and semisimple banach algebra with spectrum homeomorphic to G [14, Corollary 3.8]. This Banach algebra A Φ (G) is called as the Orlicz Figà-Talamanca Herz algebra. Let . Then, with the above norm, B Φ (G) becomes a commutative Banach algebra with pointwise addition and multiplication.
Let B(L Φ (G)) be the linear space of all bounded linear operators on L Φ (G) equipped with the operator norm. For a bounded complex Radon measure µ on G and f ∈ L Φ (G), define T µ : ) with respect to the ultraweak topology. It is proved in [14,Theorem 3.5], that for a locally compact group G, the dual of A Φ (G) is isometrically isomorphic to P M Ψ (G). By [14,Theorem 3.6] singletons are sets of spectral synthesis for A Φ (G). Further, every closed subgroup is a set of local synthesis for A Φ (G).
Throughout this paper, G will denote a locally compact group with a fixed left Haar measure dx. Further Φ will always denote a Young function whose complementary Young function is Ψ and the pair (Φ, Ψ) satisfies the ∆ 2 -condition.
In this section, motivated by the results of [10,11], we describe all the non-degenerate * -representations of A Φ (G). Throughout this section and the next, H will denote a Hilbert space.
Proposition 3.1. Let µ be a bounded positive Radon measure on G.
function as cyclic vector.
Proof. (i) and (ii) are just a routine check.
(iii) We show that the constant 1 function is a cyclic vector. Since the measure µ is finite, the conclusion follows from the density of where {µ α } α∈∧ is a summable family of measures with pairwise disjoint support. Now the conclusion follows from (iii) of Proposition 3.1.
In the next result, we characterise all cyclic * -representations. Proof. Let u ∈ A Φ (G). Then, by [15,Pg. 22], it follows that π(u) sp ≤ u sp . By [14,Theorem 3.4], A Φ (G) is a commutative Banach algebra and hence the spectral norm and the operator norm for π(u) coincides. Further, as A Φ (G) is semi-simple and the fact that the spectrum of As a consequence of this inequality and the fact that A Φ (G) is dense in C 0 (G), it follows that π extends to a * -representation of C 0 (G) on H, still denoted as π. Note that π is a cyclic * -representation of the C * -algebra C 0 (G). Let ϕ be the cyclic vector of the It is clear that T ϕ is a positive linear functional on C 0 (G) and hence, by Riesz representation theorem, there exists a bounded positive Radon measure µ such that Let π µ denote the cyclic * -representation of A Φ (G) on L 2 (G, dµ), given by Proposition 3.1.
We now claim that the representations π and π µ of A Φ (G) are unitarily equivalent. Since ϕ is a cyclic vector, in order to prove the above claim, it is enough to show that the correspondence π(u)ϕ → u.1 is an isometry and commutes with π and π µ . Note that the above correspondence is well-defined by (3.1). Let T denote the above well-defined correspondence.
We now show that T is an isometry. Let u ∈ A Φ (G). Then Finally, we show that T intertwines with π and π µ . Let u ∈ A Φ (G). Then, for v ∈ A φ (G), we have, Here is the main result of this section, describing all the non-degenerate Hilbert space representations of A Φ (G). Thus, we are done if we can show that {µ α } α∈∧ is a summable family. Let f : G → C be a continuous function with compact support. Then ⊕ α∈∧ f ∈ ⊕ α∈∧ L 2 (G, µ α ) and hence, The boundedness of sup α∈∧ µ α (G) follows from the uniform boundedness principle and from In this section, we show that the non-degenerate representations described in the previous section can be extended uniquely to B Φ (G). (i) For each u ∈ B Φ (G), there exists a unique operator π(u) ∈ B(H) such that, ∀ v ∈ A Φ (G), (4.1) π(u)π(v) = π(uv) and (4.2) π(v) = π(v).
(ii) The mapping u → π(u) defines a non-degenerate * -representation of B Φ (G) on H.
Proof. (i) Let π be a non-degenerate * -representation of A Φ (G). By [3, Proposition 2.2.7], π is a direct sum of cyclic * -representations, say {π α , H α } α∈∧ . If we can prove (i) for each of these π α 's, then the argument for π is similar to the one given in Corollary 3.4. Thus, in order to prove this, we assume that the representation π is cyclic. Since π is a cyclic * -representation, by Theorem 3.3, π is unitarily equivalent to π µ , for some bounded positive Radon measure µ. So, without loss of generality, let us assume that the non-degenerate * -representation of A Φ (G) is π µ for some bounded positive Radon measure µ.
The following corollary is the converse of the above theorem.
Thus, by Theorem 4.1, it follows that π| AΦ = π. Again by Theorem 4.1, π| AΦ is nondegenerate and hence it follows that the representation π is non-degenerate.

Application to ergodic sequences in
In this section, we discuss an application of ergodic sequences. This section is also motivated from [10] and [11]. Let Before we proceed to the main result of this section, here we give an appropriate definition.
B is said to be strongly (resp. weakly) ergodic if for any non-degenerate * -representation {π, H π } of A Φ (G) the sequence { π(u n )η} converges strongly (resp. weakly) to an element of H f , for every η ∈ H, where (ii) ⇒ (iii). Fix x ∈ G with x = e. Define π : A Φ (G) → C as π(u) = u(x). Then π defines a non-degenerate * -representation of A Φ (G) on C. By Theorem 4.1, the representation {π, C} can be extended uniquely to a non-degenerate * -representation π of B Φ (G) on C such that π(u)z = u(x)z for all u ∈ B Φ (G). Since {u n } is weakly ergodic the set {C f } is non-empty. In order to prove (iii) it is enough to show that the set C f consists only of the zero vector. Suppose to the contrary that there exists 0 = z ∈ C f . Since G is Hausdorff, there exists an open set U containing e but not x. Let v denote the function given by [14,Proposition 5.5], corresponding the open set U. Then v ∈ S Φ A and v(x)z = 0, which is a contradiction. Thus the set C f consists only of the zero vector. Hence (iii).
(iii) ⇒ (i). Let π be a non-degenerate * -representation of A Φ (G). By Corollary 3.4, π is unitarily equivalent to the representation {π µ , L 2 (G, dµ)} for some positive measure µ defined on G. So, without loss of generality, let us assume that π is of the form π µ for some positive measure µ on G. Let π µ denote the extension of π µ from A Φ (G) to B Φ (G) as in Theorem 4.1. Let f ∈ L 2 (G, dµ). We now claim that the sequence { π µ (u n )(f )} converges strongly. As L 2 (G, dµ) is complete, in order to prove the claim, it is enough to show that the sequence { π µ (u n )(f )} is Cauchy. Note that, for any n, m ∈ N, Thus, by dominated convergence theorem and by (iv), we have, Let g ∈ L 2 (G, dµ) denote the limit of the sequence { π µ (u n )(f )}. Our next claim is that g is a fixed point of π µ (u) for each u ∈ S Φ A . Again, this is a consequence of the dominated convergence theorem.

Spectral subspaces
In this section, we study the spectral subspaces associated to a non-degenerate Banach space representation of A Φ (G). Our main aim in this section is to prove Corollary 6.9. Most of the ideas of this section are taken from [12]. Let X be a Banach space and let π be an algebra representation of A Φ (G) on X. For ϕ ∈ X and x * ∈ X * , define T x * ,ϕ : We say that the representation π is continuous if T x * ,ϕ is a continuous linear functional on A Φ (G) for each ϕ ∈ X and x * ∈ X * . It follows from uniform boundedness principle that the linear map π : A Φ (G) → B(X) is norm continuous.
From now onwards, X will denote a Banach space and π an algebra representation of A Φ (G) on X.
Let E be a closed subset of G. Define 3. An immediate consequence of the above definition is that, if E = G then X E = X.
Lemma 6.4. The set X E is a closed π-invariant subspace of X.
Proof. Note that for any x * ∈ X * , ϕ 1 , ϕ 2 ∈ X E and α ∈ C, we have Thus, it follows from (i) and (iii) of Lemma 6.2 that X E is a linear space. Further, closedness of X E is an immediate consequence of (iv) from Lemma 6.2. Again, note that, for any u ∈ A Φ (G), ϕ ∈ X and x * ∈ X * , we have T x * ,π(u)ϕ = u.T x * ,ϕ and hence the invariance of X E under π follows from (ii) of Lemma 6.2.
The subspace X E is called as the spectral subspace associated with the representation π and the closed set E. Lemma 6.5. Let π be a non-degenerate representation of A Φ (G).
(i) The space Proof. (i) is an easy consequence of the non-degeneracy of π, while (ii) is trivial.
The following is an immediate corollary of Remark 6.3 and Lemma 6.5.
Corollary 6.6. There exists a smallest closed non-empty set E of G such that X E = X.
Proposition 6.7. Let K 1 and K 2 be disjoint compact subsets of G.
Proof. The proof of this follows exactly as given in [12, Proposition 2 (iii)].
Theorem 6.8. Let π be a non-degenerate representation of A Φ (G) such that the only spectral subspaces are the trivial subspaces. Then there exists x ∈ G such that X {x} = X.
Proof. Choose a smallest non-empty closed set E such that X E = X, which is possible by Corollary 6.6. Suppose there exists x, y ∈ E such that x = y. As G is locally compact and Hausdorff, there exists an open set U and a compact set K such that x ∈ U ⊂ K and y / ∈ K. Since A Φ (G) is regular, there exists u ∈ A Φ (G) such that u = 1 on U and supp(u) ⊂ K.
Let v ∈ A Φ (G) be arbitrary. Let v 1 = v − uv and v 2 = uv so that v = v 1 + v 2 . Let V = {z ∈ G : v 1 (z) = 0}. The choice of u tells us that x / ∈ V . Again, using the regularity of A Φ (G), choose a function w ∈ A Φ (G) such that w = 1 on some open set W containing x and supp(w) ∩ V = ∅. Further, it is clear that v 1 w = 0.
We now claim that π(v) = 0. Let ϕ ∈ X and x * ∈ X * . If z ∈ W, then w(z) = 1 and hence T x * ,π(v1w)ϕ = 0 as T x * ,π(v1w)ϕ = w.T x * ,π(v1)ϕ . Thus supp(T x * ,π(v1)ϕ ) ⊂ W c . Therefore, using the non-degeneracy of π, it follows that, if π(v 1 )ϕ = 0 then X W c = X and hence, by the choice of the set E, it follows that E is a subset of W c . On the other hand, x / ∈ W c and x ∈ E and hence E is not a subset of W c . Therefore, π(v 1 ) = 0. Similarly, one can show that π(v 2 ) = 0. Thus π(v) = 0. Since v is arbitrary, it follows that π(v) = 0 for all v ∈ A Φ (G), which is a contradiction. Thus the set E is a singleton. Corollary 6.9. Let π be a non-degenerate representation of A Φ (G) such that the only spectral subspaces are the trivial subspaces. Then π is a character.