Pacific Journal of Mathematics REPRESENTATIONS OF LOCALLY COMPACT GROUPS ON QSL p-SPACES AND A p-ANALOG OF THE FOURIER–STIELTJES ALGEBRA VOLKER RUNDE Volume 221 No. 2 October 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 221, No. 2, 2005 REPRESENTATIONS OF LOCALLY COMPACT GROUPS ON QSL p-SPACES AND A p-ANALOG OF THE FOURIER–STIELTJES ALGEBRA VOLKER RUNDE For a locally compact group G and p ∈ (1, ∞), we define Bp(G) to be the space of all coefficient functions of isometric representations of G on quo- tients of subspaces of L p spaces. For p = 2, this is the usual Fourier–Stieltjes algebra. We show that Bp(G) is a commutative Banach algebra that con- tractively (isometrically, if G is amenable) contains the Figà-Talamanca– Herz algebra A p(G). If 2 ≤ q ≤ p or p ≤ q ≤ 2, we have a contractive inclusion Bq (G) ⊂ Bp(G). We also show that Bp(G) embeds contractively into the multiplier algebra of A p(G) and is a dual space. For amenable G, this multiplier algebra and Bp(G) are isometrically isomorphic. Introduction P. Eymard [1964] introduced the Fourier algebra A(G) of a locally compact group G. If G is abelian with dual group 0, the Fourier transform yields an isometric isomorphism of L1(0) and A(G): this motivates (and justifies) the name. For any p ∈ (1, ∞), as usual, we define p0 ∈ (1, ∞) to be such that 1/p+1/p0 = 0 1; we say that p is dual to p. The Figà-Talamanca–Herz algebra A p(G) is defined : → ∞ as the collection of those functions f G ރ such that there are sequences (ξn)n=1 ∞ 0 in L p (G) and (φn)n=1 in L p(G) such that ∞ X (0–1) f (x) = hλp0 (x)ξn, φni (x ∈ G), n=1 where λp0 denotes the regular left representation of G on L p0 (G), and ∞ X (0–2) kξnkkφnk < ∞. n=1 MSC2000: primary 46J99; secondary 22D12, 22D35, 43A07, 43A15, 43A65, 46J99. Keywords: locally compact groups, representations, coefficient functions, QSLp-spaces, Figà-Talamanca–Herz algebras, multiplier algebra, amenability. Research supported by NSERC under grant no. 227043-00. 379 380 VOLKERRUNDE The norm of f ∈ A p(G) is the infimum over all expressions of the form (0–2) satisfying (0–1). These Banach algebras were first considered by C. Herz [1971; 1973]; their study has been an active area of research ever since (see [Cowling 1979; Forrest 1993; 1994; Lambert et al. 2004; Miao 1996], and many more). For p = 2, the algebra A p(G) is nothing but the Fourier algebra A(G). Another algebra introduced in [Eymard 1964] is the Fourier–Stieltjes algebra B(G). For abelian G, it is isometrically isomorphic to M(0) via the Fourier– Stieltjes transform. It consists of all coefficient functions of unitary representations of G on some Hilbert space and contains A(G) as a closed ideal. Is there, for general p ∈ (1, ∞), an analog of B(G) in a p-setting that relates to A p(G) as does B(G) to A(G)? In the literature (see [Cowling 1979; Forrest 1994; Miao 1996; Pier 1984], for instance), sometimes an algebra Bp(G) is considered: it is defined as the multiplier algebra of A p(G). If p = 2 and if G is amenable, we do have B(G) = Bp(G); for nonamenable G, however, B(G) ( B2(G) holds. Hence, the value of Bp(G) as the appropriate replacement for B(G) when dealing with A p(G) is a priori limited to the amenable case. In the present paper, we pursue a novel approach. We define Bp(G) to consist of the coefficient functions of all representations of G on quotients of subspaces of L p0 -spaces, so-called QSLp0 -spaces. This class of spaces is identical with the p0-spaces considered in [Herz 1973] and turns out to be appropriate for our pur- pose (such representations were considered only recently, in a completely different context, in [Jaming and Moran 2000]). We list some properties of our Bp(G): • Under pointwise multiplication, Bp(G) is a commutative Banach algebra with identity. • A p(G) is an ideal of Bp(G), into which it contractively embeds (isometrically if G is amenable). • If 2 ≤ q ≤ p or p ≤ q ≤ 2, we have a contractive inclusion of Bq (G) in Bp(G). • Bp(G) is a dual Banach space. • Bp(G) embeds contractively into the multiplier algebra of A p(G) and is iso- metrically isomorphic to it if G is amenable. This list shows that our Bp(G) relates to A p(G) in a fashion similar to how B(G) relates to A(G) and therefore may be the right substitute for B(G) when working with Figà-Talamanca–Herz algebras. The main challenge when defining Bp(G) and trying to establish its properties is that the powerful methods from C∗- and von Neumann algebras are no longer at one’s disposal for p 6= 2, so that one has to look for appropriate substitutes. REPRESENTATIONS OF LOCALLY COMPACT GROUPS ON QSL p -SPACES 381 1. Group representations and QSL p-spaces We begin with defining what we mean by a representation of a locally compact group on a Banach space: Definition 1.1. A representation of a locally compact group G (on a Banach space) is a pair (π, E) where E is a Banach space and π is a group homomorphism from G into the invertible isometries on E which is continuous with respect to the given topology on G and the strong operator topology on Ꮾ(E). Remarks. 1. Our definition is more restrictive than the usual definition of a representation, which does not require the range of π to consist of isome- tries. Since we will not encounter any other representations, however, we feel justified to use the general term “representation” in the sense just defined. 2. Any representation (π, E) of a locally compact group G induces a represen- tation of the group algebra L1(G) on E, i.e. a contractive algebra homomor- phism L1(G) to Ꮾ(E) — which we shall denote likewise by π — through Z (1–1) π( f ) := f (x)π(x) dx ( f ∈ L1(G)), G where the integral (1–1) converges with respect to the strong operator topol- ogy. 3. Instead of requiring π to be continuous with respect to the strong operator topology on Ꮾ(E), we could have demanded that π be continuous with respect to the weak operator topology on Ꮾ(E): both definitions are equivalent by [de Leeuw and Glicksberg 1965]. Definition 1.2. Let G be a locally compact group, and let (π, E) and (ρ, F) be representations of G. Then: (a) (π, E) and (ρ, F) are said to be equivalent if there is an invertible isometry V : E → F such that V π(x)V −1 = ρ(x)(x ∈ G). (b) (ρ, F) is called a subrepresentation of (π, E) if F is a closed subspace of E such that ρ(x) = π(x)|F (x ∈ G). (c) (ρ, F) is said to be contained in (π, E) — in symbols: (ρ, F) ⊂ (π, E) — if (ρ, F) is equivalent to a subrepresentation of (π, E). Throughout, we shall often not tell a particular representation apart from its equivalence class. This should, however, not be a source of confusions. In this paper, we are interested in representations of locally compact groups on rather particular Banach spaces: 382 VOLKERRUNDE Definition 1.3. Let p ∈ (1, ∞). (a) A Banach space is called an L p-space if it is of the form L p(X) for some measure space X. (b) A Banach space is called a QSLp-space if it is isometrically isomorphic to a quotient of a subspace of an L p-space. Remarks. 1. Equivalently, a Banach space is a QSLp-space if and only if it is a subspace of a quotient of an L p-space. 2. Trivially, the class of QSLp-spaces is closed under taking subspaces and quo- tients. L 3. If (Eα)α is a family of QSLp-spaces, its `p-direct sum `p- α Eα is again a QSLp-space. 0 ∗ 4. If E is a QSLp-space and if p ∈ (1, ∞) is dual to p, the dual space E is an QSLp0 -space. In particular, every QSLp-space is reflexive. 5. By [Kwapien´ 1972, §4, Theorem 2], the QSLp-spaces are precisely the p- spaces in the sense of [Herz 1971], i.e. those Banach spaces E such that for any two measure spaces X and Y the amplification map Ꮾ(L p(X), L p(Y )) → Ꮾ(L p(X, E), L p(Y, E)), T 7→ T ⊗ idE is an isometry. In particular, an Lq -space is a QSLp-space if and only if 2 ≤ q ≤ p or p ≤ q ≤ 2. Consequently, if 2 ≤ q ≤ p or p ≤ q ≤ 2, then every QSLq -space is a QSLp-space. 6. All ᏸp,1-spaces in the sense of [Lindenstrauss and Rosenthal 1969] — and, g more generally, all Lp,1-spaces in the sense of [Defant and Floret 1993] — are QSLp-spaces. 7. Since the class of L p-space is stable under forming ultrapowers ([Heinrich 1980]), so is the class of QSLp-spaces (this immediately yields that QSLp- spaces are not only reflexive, but actually superreflexive). In the case where X = Y = ރ, the QSLp-spaces are therefore precisely those that occur in [Le Merdy 1996, Theorem 4.1] and play the rôle played by Hilbert spaces in Ruan’s representation theorem for operator spaces ([Effros and Ruan 2000, Theorem 2.3.5]).