Contractive projections on Banach algebras

Journal of Functional Analysis 254 (2008) 2513–2533 www.elsevier.com/locate/jfa

Contractive projections on Banach algebras Anthony To-Ming Lau a , Richard J. Loy b,∗ a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada b Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia

Received 8 August 2007; accepted 19 February 2008 Available online 24 March 2008 Communicated by N. Kalton

Abstract In this paper, we explore the properties of projections of norm one on general Banach algebras, in particular the relation with conditional expectations for algebras which arise in harmonic analysis. © 2008 Elsevier Inc. All rights reserved. Keywords: Banach algebra; Conditional expectation; Norm one projection; Locally compact group; Group algebra; Measure algebra; Fourier algebra; Group von Neumann algebra

1. Introduction Let (X, S, μ) be a probability space, and T a σ -subalgebra of S. The conditional expecta tion operator E T : L1 (X, S, μ) → L1 (X, T , μ) is determined by the relation T E T (f ) dμ =  1 T T f dμ for T ∈ T , and all f ∈ L (X, S, μ). Existence and uniqueness of E follow from the T Radon–Nikodym theorem. In particular, uniqueness shows that E is idempotent, and since E T  = 1, E T is a contractive projection. An easy argument with simple functions also shows the fundamental relation E T (f g) = g · E T (f )

  f ∈ L1 (X, S, μ), g ∈ L∞ (X, T , μ) .

(1.1)

Slightly more generally, suppose, inthe above situation that S ∈ T is fixed, and k  0 is  S-measurable and satisfies T k dμ = T χS dμ for T ∈ T , so that E T k = χS . Then again * Corresponding author.

E-mail addresses: [email protected] (A.T.-M. Lau), [email protected] (R.J. Loy). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.02.008

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f → k · E T (f ) is a contractive projection on L1 (X, S, μ) [12, Proposition 1]. In fact the general form of a contractive projection on L1 (X, S, μ) is Uϕ (k · E T )Uϕ + V for some unimodular S-measurable ϕ, where Uϕ (f ) = ϕf and V is a related contraction with V 2 = 0 [12]. Contractive projections have been studied in other situations, see, for example, [1,14,16,33,41]. In many cases it is known that the range of such a projection on certain algebras is often imbued with algebraic structure of its own, and the projection itself is a conditional expectation operator. For example, see [17]. The current paper has the same flavour, though from the different perspective of hypothesising algebraic conditions on the range. Some other recent papers concerned with the conditional expectation property in related settings are [11,13,34]. In this paper we shall study contractive projections whose range is a subalgebra on various classes of Banach algebras, particularly those associated with locally compact groups. 2. Preliminaries In the algebraic setting, motivated by (1.1), an idempotent operator P on an algebra A is a conditional expectation (or quasi-expectation [31,35]) if P (b1 ab2 ) = b1 P (a)b2

  b1 , b2 ∈ P (A), a ∈ A .

(2.1)

Thus (A, P ) is a non-commutative probability space in the sense of [13]. Trivially, any idempotent homomorphism satisfies (2.1). There are several equivalent conditions relevant to our discussion; in particular the form Proposition 2.1(iv) is often more convenient to use. For a ∈ A, let λa : x → ax denote left multiplication by a. Proposition 2.1. Let A be a Banach algebra, P : A → A an idempotent operator. Then each of the following implies the next: (i) (ii) (iii) (iv)

P (b1 a) = b1 P (a) and P (ab2 ) = P (a)b2 for b1 , b2 ∈ P (A), a ∈ A; P λa P = λP (a) P = P λP (a) for a ∈ A; P is a conditional expectation; b1 , b2 ∈ P (A), a ∈ A with P (a) = 0 necessitates P (b1 ab2 ) = 0.

If P (A) is a subalgebra, then (iii) and (iv) are equivalent. Suppose further that P (A) contains an approximate identity for A. Then (i), (ii) and (iii) are equivalent and imply (v) P (A) = {a ∈ A: λa P = P λa }. Consequently, P (A) is a subalgebra, and so (i)–(iv) are equivalent. Proof. This follows from straightforward calculations.

2

Our starting point is a result of Tomiyama [40]. Theorem 2.2. (See [40].) Let A be a unital C ∗ -algebra, P : A → A a contractive projection with P (1) = 1 whose range is a C ∗ -subalgebra. Then P is a conditional expectation.

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Proofs of variants of Theorem 2.2 are given in [37, Theorem III.3.4], which has no identity assumptions, [35] which has an invariant bounded approximate identity assumption, and [6, Corollary 4.2.9] which is for (possibly non-selfadjoint) unital operator algebras and assumes that P is completely contractive. David Bletcher has recently informed the authors that he and Matthew Neal have shown this last result remains true with the unital hypothesis weakened to supposing only that the range subalgebra has a contractive approximate identity. In [5, Question 1] the question is raised as to whether there are forms of Theorem 2.2 for other Banach algebras; it is this question that prompted the current work. Acknowledging this fundamental result of Tomiyama, we make the following definition. Definition 2.3. A Banach algebra will be said to have the Tomiyama property if any contractive projection P : A → A whose range B is a subalgebra satisfies P (b1 ab2 ) = b1 P (a)b2

(b1 , b2 ∈ B, a ∈ A).

(2.2)

Furthermore, we will make use of weaker variants where restrictions are placed on the projections. In particular, we will use the terminology Tomiyama property for unital (respectively positive, L-) projections when (2.2) holds for projections satisfying P (1) = 1 (respectively P  0, P an L-projection). Example 2.4. (See [5].) Define the semigroup S = {1, a, b} by taking product ab = ba = a, a 2 = a, b2 = b, 1 an identity. Set A = 1 (S). Then P (λδ1 + μδa + νδb ) = λδ1 + μδa

(2.3)

defines a norm one projection with P (δ1 ) = δ1 , whose range is a subalgebra, but P is not a conditional expectation. Example 2.5 (David Blecher). Let P : Mn (C) → Mn (C) be the (completely) contractive projection onto the first row. Here the range of P is a subalgebra having no right identity and the first half of Proposition 2.1(i) fails.

Example 2.6. Take a semigroup S with identity such that 1 (S) is Arens regular and c0 (S) fails to be translation invariant. Let Λ : c0 (S) → 1 (S)∗ = ∞ (S) be the natural injection. Then P = Λ∗ : 1 (S)∗∗ → 1 (S) will not in general be a conditional expectation, see Proposition 6.7 below. We are interested in whether Banach algebras in general, and those associated with locally compact groups in particular, have properties akin to Theorem 2.2. Note that for a locally compact group G, ‘averaging’ over a compact group of automorphisms of G gives a contractive projection from the group algebra or group C ∗ -algebra onto a subalgebra which is a positive conditional expectation [30, §1]. Our interest is in whether the range being a subalgebra is in itself sufficient for certain contractive projections to be conditional expectations. There are two special (and related) classes of contractive projections which have received major attention over the years, cf. [19].

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Definition 2.7. Let P be a projection on a Banach space X. Then P is an (i) L-projection if x = P x + (I − P )x for x ∈ X; (ii) M-projection if x = max{P x, (I − P )x} for x ∈ X. P is proper if P = 0, P = 1. Clearly for any L- or M-projection P , both P and 1 − P are contractive projections. In certain cases a characterization of such projections is well known. In particular, for L1 spaces there are only the “obvious” ones: Theorem 2.8. Let (X, S, μ) be a measure space such that L1 (μ)∗ = L∞ (μ). Then L-projections on L1 (μ) are exactly the characteristic projections PS : f → 1S ·f for some S ∈ S. Unless L1 (μ) is two-dimensional, it has no non-trivial M-projections. For the proof for L-projections, see [36, Corollary 2], [19, Example 1.6(a)], or [2, Proposition 4.9]. The restriction on (X, S, μ) is fairly weak, in its absence the result is the same except that more general S need to be considered, see [2]. The statement about M-projections follows from [19, Theorem 1.8]. Let A be a Banach algebra. Recall that for f ∈ A∗ , ϕ ∈ A, f · ϕ, ϕ · f ∈ A∗ are defined by f · ϕ, ψ = f, ϕψ ,

ϕ · f, ψ = f, ψϕ (ψ ∈ A).

A subspace X of A∗ is A-left invariant (respectively A-right invariant) if X · ϕ ⊆ X (respectively ϕ · X ⊆ X) for all ϕ ∈ A. Now suppose X ⊆ A∗ is A-left invariant. For m ∈ X ∗ , f ∈ X, define m · f ∈ A∗ by m · f, ϕ = m, f · ϕ (ϕ ∈ A). The subspace X is A-left-introverted if X ∗ · X ⊆ X. Finally, for an A-left-introverted subspace of A∗ , define a product on X ∗ by mn, f = m, n · f (m, n ∈ X ∗ , f ∈ X). Throughout this paper, G will always denote a locally compact group with fixed left Haar measure. Given a function f : G → C, the left (right) translation of f by x ∈ G is defined by (x f )(y) = f (xy) ((rx f )(y) = f (yx)). The standard Lebesgue spaces with respect to left Haar measure will be denoted Lp (G), 1  1  ∞; CB(G) will denote the space of all bounded continuous complex-valued functions on G with the supremum norm; C0 (G) will denote the closed subspace of CB(G) of functions vanishing at infinity, C00 (G) will denote the dense subspace of C0 (G) consisting of the functions with compact support. Note that CB(G) is isometrically isomorphic to a closed translation-invariant subspace of L∞ (G). The dual of C0 (G) is the space M(G) of bounded regular Borel measures on G, and we have the decomposition M(G) = L1 (G) ⊕ Md (G) ⊕ Ms (G) where Md (G) and Ms (G) are the discrete and singular measures, respectively [20, §19]. ˜ Here Δ is the modular For f ∈ L∞ (G), and ϕ ∈ L1 (G), f · ϕ = Δ1 ϕ˜ ∗ f and ϕ · f = f ∗ ϕ. function of G, and ϕ(x) ˜ = ϕ(x −1 ) for x ∈ G. Let LUC(G) denote the space of those f ∈ CB(G) such that the map G → (CB(G),  · ) : x → x f is continuous. This in fact coincides with the space of those f ∈ CB(G) such that x → x f : G → (CB(G), weak) is continuous. Thus LUC(G) is L1 (G)-invariant; it is in fact the maximal 1 (G)-left-introverted subspace of ∞ (G) contained in CB(G) [4, Theorem 5.7]. Note

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that LUC(G) is exactly the space of right uniformly continuous functions of CB(G) as defined in [20]. If f ∈ LUC(G), n ∈ LUC(G)∗ , then for x ∈ G, (n · f )(x) = n, x f [22, Lemma 3]. The right orbit of a function f ∈ CB(G) is given by RO(f ) = {rx f : x ∈ G}. Recall that a function f ∈ CB(G) is almost periodic (weakly almost periodic) if RO(f ) is precompact in the norm topology (weak topology) of CB(G). We shall denote the spaces of such functions by AP(G) and W (G), respectively; they are also left-introverted-subspaces of L∞ (G) [43, Lemma 6.4]. If X is a weak∗ -closed subspace of L∞ (G), then X is L1 (G)-invariant if and only if X is left and right translation invariant [43, Lemma 6.3]. If X is a norm-closed subspace of W (G), then X is L1 (G)-invariant if and only if X is left and right translation-invariant. We shall repeatedly need the following simple observation. Proposition 2.9. Let G be a locally compact group, K a subgroup of positive measure. Then (i) K is a clopen set; (ii) the projection PK : f → 1K · f maps onto a subalgebra and is a conditional expectation. Proof. Clause (i) follows from [20, Corollary 20.17]. Clause (ii) is a direct calculation that ker P is invariant under the action of B, so PK is a conditional expectation by Proposition 2.1(iv). 2 3. Locally compact groups Let G be a locally compact group, P (G) ⊆ CB(G) be the set of continuous positive definite functions on G, B(G) its linear span. Then B(G) can be identified with the dual of the group C ∗ -algebra C ∗ (G), this latter being the completion of L1 (G) under the norm    f  = sup π(f ): π a continuous unitary representation of G . Indeed the action is given by  ϕ, f =

ϕ(t)f (t) dλ

  ϕ ∈ B(G), f ∈ L1 (G) .

G

With pointwise multiplication and the dual norm, B(G) is a commutative Banach algebra, the Fourier–Stieltjes algebra of G, see [15]. Let Pρ (G) be the closure of P (G) ∩ C00 (G) under the compact open topology, Bρ (G) its linear span. Then Bρ (G) is a closed ideal in B(G), and is the dual of the reduced C ∗ -algebra Cρ∗ (G). This latter is the norm closure in B(L2 (G)) of the operators {ρ(f ): f ∈ L1 (G)}, where ρ(f )(h) = f ∗ h, h ∈ L2 (G). It is well known that Bρ (G) = B(G) if and only if G is amenable. Finally, the Fourier algebra A(G) is the norm closure of the linear span of P (G) ∩ C00 (G). It is a closed ideal of B(G) contained in Bρ (G). An alternate description is that ϕ ∈ A(G) has the form ϕ(x) = ρ(x)h, k for h, k ∈ L2 (G), where ρ(x)(h)(y) = h(x −1 y), (x, y ∈ G) is the left regular representation of G on L2 (G). As a consequence, A(G) may be regarded as the set of ultraweakly continuous functionals on VN(G), the von Neumann algebra generated by ρ in B(L2 (G)), and so A(G) is the unique predual of VN(G). For a Hilbert space H , and X a subset of B(H ), we denote by X wot the closure of X in the weak operator topology of B(H ).

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Theorem 3.1. Let G be a locally compact group. Then L1 (G) has the Tomiyama property for L-projections onto ∗ -subalgebras. Indeed, given a non-zero L-projection P of L1 (G) onto a ∗ -subalgebra, there is measurable subset S ⊆ G and a unique clopen subgroup H ⊆ G such that: (i) (ii) (iii) (iv) (v) (vi)

P = PS : f → 1S · f , f ∈ L1 (G); VN(S) = {ρ(f ): f ∈ L1 (S)}−wot = VN(H ); S is dense in H ; H is the subgroup generated by S; in the case that G is σ -compact S can be taken to equal H ; P is a conditional expectation.

Proof. (i) Theorem 2.8 gives that P = PX : f → 1X · f for some measurable X ⊂ G, and λ(X) > 0 since P is non-zero. Take E ⊂ X of non-zero finite measure. Then 1E ∈ L1 (X), whence so is 1E −1 (and E −1 \ X is null). The function x → λ(E ∩ xE) = (1E  1E −1 )(x) is continuous and non-zero at x = e and lies in L1 (X). Thus   U = x ∈ G: 1E  1E −1 (x) > 0 is an open set containing e, and U \ X is null. By a suitable modification to X, we obtain a set S which contains U , and PX = PS . (ii) We first note that VN(S) is A(G)-invariant and set   H = x ∈ G: ρ(x) ∈ VN(S) . Then by [38, Theorems 6 and 8], H is a closed subgroup of G and  VN(S) = ρ(h): h ∈ H }−wot = VN(H ). Now let U ⊆ S be the open neighbourhood of e given above. For x0 ∈ U , take {Vα } to be a basis of compact neighbourhoods of x0 all lying inside U , directed by reverse set inclusion. For each α, let fα = 1Vα /λ(Vα ) ∈ L1 (S). For ϕ ∈ A(G),



1 ρ(fα ), ϕ − ϕ(x0 ) = λ(Vα )



 ϕ(t) − ϕ(x0 ) dt.

Vα

Taking the limit over α, continuity of ϕ shows that limα ρ(fα ), ϕ = ϕ(x0 ). That is, ρ(fα ) → ρ(x0 ) weak∗ , so that ρ(x0 ) ∈ VN(S). It follows that U ⊆ H , whence H is a clopen subgroup. Uniqueness follows by [39]. (iii) Now suppose K ⊂ S \ H is compact, λ(K) > 0. Then 1K ∈ 1S · L1 (G) so T = ρ(1K ) ∈ VN(S). Now, for h0 ∈ H , take ϕ ∈ A(G) with ϕ(h0 ) = 0, supp(ϕ) ⊂ H . Then ϕ · T = 0, yet / VN(H ) = VN(S), ϕ(h0 ) = 0, so that h0 = supp(T ). It follows that supp(T ) ∩ H = ∅, so that T ∈ contradiction. Thus S \ H is null. By discarding the null set outside H , we may thus suppose that S ⊆ H ; note that this does not effect PS nor VN(S).

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Finally, suppose that x0 ∈ H \ S. By definition of H , there is {fα } ∈ L1 (G) such that w∗ −→ ρ(x0 ). But by regularity, there is ϕ ∈ A(G) with ϕ(x0 ) = 1, ϕ|S = 0, so that ρ(1S · fα ) −



0 = ρ(1S · fα ), ϕ → ρ(x0 ), ϕ = 1, which is impossible. Thus S ⊇ H , and hence equality holds. (iv) The subgroup generated by S lies in K, it is open and hence closed, so equals H by (iii). (v) Suppose that G is σ -compact. For each K ⊂ S of finite measure, K −1 \ S is null since −1 −1 1K −1 ∈ L1 (S).

It follows that S \ S is null, so we may suppose that S = S . Take S = En where 0 < λ(En ) < ∞ for each n. Take U as before (instead of using E this could be done for each En then the union taken, but this is not necessary). Replacing S by U ∪ S we may suppose that S = S −1 and U ⊆ S. Let z ∈ U , and take n ∈ N. Let V0 be an open neighbourhood of e such that V0 z ⊂ U , and set V = {V ⊆ V0 : V an open neighbourhood of e}. Then for V ∈ V, we have 1En  1V  δz ∈ L1 (S) since L1 (S) is a subalgebra. Noting that (λ(V )−1 1V )V ∈V is an approximate identity for L1 (G), it follows that 1En  δz = 1En z ∈ L1 (S). Thus En z \ E is null. It follows that Sz \ S is null, and hence that Sz  S is null. Now suppose that H \ S has positive measure. Take x0−1 ∈ H , and a set V of finite positive measure such that e ∈ V ⊆ U and x0−1 V ⊆ H \ S. Set h = 1x −1 V . Since ρ(x0 ) lies in the 0

wot wot closure of ρ(1S · L1 (G)), there is a net (fα ) ⊂ L1 (G) such that ρ(1S · fα ) − −→ ρ(x0 ). In particular



 1V

     (1S · fα )(x)h x −1 y dλ(x) − h x0−1 y dλ(y) → 0.

(3.1)

For x ∈ / S, (1S · fα )(x) = 0 and for each fixed y ∈ V , if x ∈ S, then x −1 y ∈ S −1 y = Sy ⊆ S ∪ (Sy \ S), so that h(x −1 y) = 0 for almost all such x. It follows that 

  (1S · fα )(x)h x −1 y dλ(x) = 0,

whence (3.1) has (fixed) value −λ(V ) = 0, contradiction. Thus we replace S by S ∪ (H \ S) = H . (vi) Note that an integrable function on G vanishes outside a σ -finite set. By regularity it thus vanishes almost everywhere outside a σ -compact set. By Proposition 2.1, it suffices to show that for non-zero f1 , f2 ∈ 1S · L1 (G), g ∈ 1G\S · L1 (G), we have P (f1 gf2 ) = 0. Take a σ -compact set K outside of which f1 , f2 and g are almost everywhere zero. Set S1 to be the subgroup generated by K. Since f1 , f2 and g are non-zero, λ(K) > 0, whence λ(S1 ) > 0 so that S1 is clopen. Then   1S1 ∩S · L1 (G) = 1S · L1 (S1 ) = 1S · L1 (G) ∩ L1 (S1 ) is a ∗ -subalgebra, and hence P1 : f → 1S1 ∩S · f from L1 (S1 ) → 1S · L1 (S1 ) satisfies the conditions of (v), so there is a clopen subgroup H1 with (S ∩ S1 )  H1 null. Thus P1 is a conditional expectation by Proposition 2.9. But then P (f1 gf2 ) = P1 (f1 gf2 ) = 0 by Proposition 2.1, as required. 2

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Remark. The hypothesis of the range being a ∗ -subalgebra is necessary to get the group H , as is shown by the restriction map 1 (Z) → 1 (N). It is used in the first paragraph of the proof to get 1E −1 ∈ L1 (X). Note that for invariant subalgebras the following results are known. Theorem 3.2. (See [3, Theorem 3.1].) Let G be a locally compact group. The non-zero weak∗ closed C0 (G)-invariant ∗ -subalgebras A of M(G) are exactly the subspaces M(H ) for some closed subgroup H of G. In particular, A is the range of the L-projection μ → 1H · μ on M(G), where H = {x ∈ G: δx ∈ A}. The proof of this result in [3] gives the following. Proposition 3.3. Let G be a locally compact group. The non-zero weak∗ -closed C0 (G)-invariant subspaces A of M(G) are exactly the subspaces M(H ) for some closed subset H of G. In particular, A is the range of the L-projection μ → 1H · μ on M(G), where H = {x ∈ G: δx ∈ A}. Theorem 3.4. (See [3, Corollary 3.2].) Let G be a locally compact group. The non-zero closed L∞ (G)-invariant ∗ -subalgebras A of L1 (G) are exactly the subspaces L1 (H ) for some clopen 1 subgroup H of G. In

particular, A is the range of the L-projection μ → 1H · μ on L (G), where H is the closure of {x ∈ G: x ∈ supp(f ): f  0, f ∈ A}. In the discrete case Theorem 3.1 shows that S = H . A more direct proof is given in the following, which answers the semigroup question stated at Section 9(i) below in the discrete case. Theorem 3.5. Let G be a discrete group. Then (i) a projection on 1 (G) is an L-projection if and only if it is of the form f → 1S · f for some subset S; (ii) a projection on 1 (G) is an L-projection with range a subalgebra if and only if it is of the form f → 1S · f for some subsemigroup S; (iii) a projection on 1 (G) is an L-projection with range a ∗ -subalgebra if and only if it is of the form f → 1S · f for some subgroup S. Proof. In each case the ‘if’ is clear. (i) Let P be an L-projection of 1 (G). As above, P = PS for some subset S of G. But this is easily shown directly in the current situation. For each x ∈ G, provided P (δx ) = δx and P (δx ) = 0, we have   P (δx )  (I − P )(δx )  δx = P (δx ) + (I − P )(δx ) = P (δx ) + (I − P )(δx ) , P (δx ) (I − P )(δx ) where P (δx ) + (I − P )(δx ) = 1 since P is an L-projection. But δx is extreme in the unit ball. It follows that either P (δx ) = δx or P (δx ) = 0. Set S = {x ∈ G: P (δx ) = δx }. Clearly 1 (S) ⊆ B, and 1 (G \ S) ⊆ ker P . Thus B = 1 (S) and ker P = 1 (G \ S), so that P = P |S .

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(ii) When B is a subalgebra, that S is a semigroup is immediate. (iii) Since δx∗ = δx −1 this is clear. 2 Theorem 3.6. Let G be a locally compact group. The following are equivalent: (i) G is a discrete torsion group; (ii) M(G) has the Tomiyama property for L-projections; (iii) M(G) has the Tomiyama property for unital L-projections. Proof. (i) ⇒ (ii). Let P be a L-projection onto a subalgebra of 1 (G). From Theorem 3.5(ii) P = PS where S is a semigroup, and hence a subgroup since every element has finite order. (In particular e ∈ S so that P (δe ) = δe ∈ B.) Thus G \ S is the union of the non-trivial cosets of S. Thus G \ S is invariant under the action of S, and so (ii) holds. (ii) ⇒ (iii) is trivial. (iii) ⇒ (i). Supposing G is not discrete, write M(G) = L1 (G) ⊕ Md (G) ⊕ Ms (G) where each summand is non-trivial [20, (19.26)]. Let P be the projection onto the first two summands. Clearly, P (δe ) = δe , and P is an L-projection. But Ms (G) is not invariant under the action of B = L1 (G) ⊕ Md (G). Indeed, L1 (G) has a bounded approximate identity (eα ), and for all μ ∈ M(G), eα  μ → μ weak∗ . So for μ ∈ Ms (G), μ = 0, there is α such that eα  μ ∈ L1 (G) \ {0}. But this means that M(G) fails (iii). Thus M(G) = 1 (G). Suppose that a ∈ G has infinite order. Then the semigroups + Sa = {e, a, a 2 , . . .} and Sa− = {a −1 , a −2 , . . .} are disjoint. Thus        1 (G) = 1 Sa+ ⊕ 1 Sa− ⊕ 1 G \ Sa+ ∪ Sa− .

(3.2)

Let P be the L-projection onto the first summand. Clearly ker P is not invariant under the action of 1 (Sa+ ). 2 Proposition 3.7. Let G be a locally compact group. If LUC(G)∗ has the Tomiyama property then G is a discrete torsion group. Proof. By [18], LUC(G)∗ = M(G) ⊕ C0 (G)⊥

(3.3)

with 1 -norm between summands. Now use the same argument as in Theorem 3.6 (iii) ⇒ (i).

2

Proposition 3.8. Let G be a locally compact group. Suppose that every positive, contractive projection P : B(G) → B(G) with range a ∗ -subalgebra is a conditional expectation. Then G is compact. Proof. Suppose not, and set W ∗ (G) = B(G)∗ . Note that W ∗ (G), being the second dual of C ∗ (G), is the von Neumann algebra generated by {w(g): g ∈ G}, where {w, Hw } is the universal continuous unitary representation of G. Now W ∗ (G) acts on A(G) by x · ϕ, y = ϕ, yx , ϕ · x, y = ϕ, xy , x, y ∈ W ∗ (G), ϕ ∈ A(G). Note that for ϕ ∈ B(G), g ∈ G, ϕ(g) = ϕ, w(g) . Since A(G) is translation invariant, we have w(g) · ϕ, ϕ · w(g) ∈ W ∗ (G) for each g ∈ G and ϕ ∈ A(G). Thus by [37, Theorem III.2.7(i)] A(G) is invariant under this action of W ∗ (G). So,

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by [37, Theorem III.2.7(iii)], there is a central projection z ∈ W ∗ (G) such that A(G) = z · B(G). Since G is not compact, z = 1. Then P : ϕ → z · ϕ is a projection of B(G) onto A(G), P  = 1. Further, if ϕ ∈ B(G), and ϕ  0, that is x ∗ x, ϕ  0 for all x ∈ W ∗ (G), then since z is a central projection,

x ∗ x, z · ϕ = zx ∗ x, ϕ = (zx)∗ zx, ϕ  0. Thus P is positive. But since A(G) is an ideal in B(G), ker P cannot be invariant, that is, Proposition 2.1(4) fails. 2 The following can be shown by the same method. Proposition 3.9. Let G be a locally compact group. Suppose that every positive contractive projection P : Bρ (G) → Bρ (G) with range a ∗ -subalgebra is a conditional expectation. Then G is compact.  denote the norm closure of A(G) · VN(G) in VN(G). Here ϕ · T , ψ = T , ϕψ , Let UC(G)  is a C ∗ -subalgebra of VN(G). (When G is abelian, ϕ, ψ ∈ A(G), T ∈ VN(G). Then UC(G)  is just the subspace of bounded uniformly continuous functions on G.)  Now UC(G)  is UC(G)  ∗ becomes a Banach algebra under the an introverted subspace of VN(G) = A(G)∗ , so UC(G)  ∗. first Arens product. Also, there is an isometric embedding of Bρ (G) into UC(G) Proposition 3.10. Let G be a locally compact group. Suppose that every positive contractive  ∗ → UC(G)  ∗ with range a subalgebra is a conditional expectation. Then projection P : UC(G) G is compact.  ∗  Bρ (G) ⊕ Cρ∗ (G)⊥ . Proof. By [27, Lemma 5.2], there is an isometric isomorphism UC(G) Let P be the projection onto the first summand. Then P is positive and contractive because  ∗ , [23], and Cρ∗ (G)∗ = Bρ (G). Cρ∗ (G) ⊆ UC(G) Let Q be the projection of Bρ (G) onto A(G) given by Qϕ = z · ϕ where z is the central projection in Wρ∗ (G) = Cρ∗ (G)∗∗ such that A(G) = zBρ (G). Then Q ◦ P is a positive contractive  ∗ onto A(G) with kernel K = (1 − z)Bρ (G) ⊕ Cρ∗ (G)⊥ . Since A(G) is projection from UC(G) an ideal in Bρ (G), K cannot be invariant. 2 These three results reduce the respective questions to the compact case. Proposition 3.11. Let G be a compact abelian group. Then A(G) has the Tomiyama property for L-projections if and only if G is 0-dimensional.  having Proof. A(G) having the Tomiyama property for L-projections is the same as 1 (G)  the property, so by Theorem 3.6 we have G is a torsion group, so G is 0-dimensional [20,  is a discrete torsion group by [20, CorolTheorem 24.21]. Conversely, if G is 0-dimensional, G lary 24.18]. So L-projections onto subalgebras are conditional expectations by Theorem 3.6. 2 Corollary 3.12. Let G be an abelian locally compact group. Then B(G) satisfies the Tomiyama property for L-projections if and only if G is compact and 0-dimensional.

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Proof. Suppose B(G) satisfies the Tomiyama property for L-projections. The projection in Proposition 3.8 is an L-projection by [37, Theorem III.2.14], so it follows that G is compact. The rest is Theorem 3.11. 2 4. Compact groups In the case of compact groups, we have a stronger result than Theorem 3.1 in that the range subalgebra is not assumed to be ∗ -closed, this is part of the conclusion. Moreover the result is obtained by simple measure theoretic arguments. Theorem 4.1. Let G be a compact group. Then L1 (G) has the Tomiyama property for L-projections. Indeed, a non-zero L-projection P of L1 (G) onto a subalgebra B has the form P = PK : f → 1K · f for a unique clopen subgroup K, and B = L1 (K). In particular, P is a conditional expectation onto a ∗ -subalgebra. Proof. By Theorem 2.8, P : f → 1E · f for some measurable non-null subset E of G. We need to show that E differs from a clopen subgroup by a null set. Consider the function      (4.1) 1E  1E (x) = 1E (y)1E y −1 x dy = λ E ∩ xE −1 . G

(Note the difference from Theorem 3.1.) This is a continuous function with  1E  1E (x) dλ(x) = λ(E)2 > 0.

(4.2)

G

Define U = {x ∈ G: 1E 1E (x) > 0}. Then U is an open set in G, and it is non-empty by (4.2). Also, U \ E is null since 1E  1E ∈ B. Let z ∈ U . Then as in Theorem 3.1 we have that 1E  δz = 1Ez ∈ B. Thus Ez \ E is null, so that λ(Ez ∩ E) = λ(Ez). But λ(Ez) = λ(E) so that E \ Ez is also null, and hence Ez  E is null. Thus E  Ez−1 is null, and so E −1  zE −1 is null. But then from (4.1)     1E  1E (z) = λ E ∩ zE −1 = λ E ∩ E −1 . This holding for every u ∈ U shows that   1E  1E = λ E ∩ E −1 1U ,

(4.3)

since both sides vanish outside U . In particular, U is a clopen set. Integrating (4.3) gives   λ(E)2 = λ E ∩ E −1 λ(U ). Since λ(U ) = λ(U \ E) + λ(U ∩ E)  λ(E), we have λ(U ) = λ(E) = λ(E ∩ E −1 ). The first equality gives E  U , and hence E −1  U −1 , are null. The second gives E \ E −1 , and so E −1 \ E, are null, whence E  E −1 is null.

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But then U  U −1 is null. Both of the sets here being clopen, it follows that U = U −1 . For z ∈ U we have Ez  E is null, whence we have U z  U null. Again, both of the sets here are clopen, so that U z = U . Hence U is a subgroup. Since E  U is null we have the desired clopen subgroup. If K  is any clopen subgroup with B = 1K  · L1 (G). Then 1K = 1K  almost everywhere, whence everywhere by continuity. Thus K = K  and uniqueness follows. The final statement is Proposition 2.9(ii). 2 Corollary 4.2. Let G be a compact group, P a non-zero L-projection of L1 (G) onto a subalgebra B. Then B is closed under involution, and P is a conditional expectation. Proof. Since P = PK for a subgroup K by Theorem 4.1, closure under involution is clear. The other statement follows as in the finite case above. 2 Corollary 4.3. Let G be a compact connected group. Then there is no proper L-projection of L1 (G) whose range is a subalgebra. For a general contractive projection, we have the following partial result. Proposition 4.4. Let P be a contractive projection of L1 (G), G a compact group, onto a subalgebra B. Then there is a measurable set S such that supp f \ S is null for each f ∈ B. Suppose that 1S ∈ B. Then S is equivalent to a clopen subgroup, and B = P (L1 (S)) ⊆ L1 (S). Proof. The existence of S is shown in [12, Proposition 2]. Now use the argument of Theorem 4.1. 2  Note that for any compact group G, the conditional expectation P : f → ( f (x) dx)1G is a contractive, multiplicative projection whose range B is a subalgebra. Here the set S of Proposition 4.4 is all of G, yet B has dimensional one. Again, in Example 8.2, S = G, yet B has dimension two. We have been unable to determine whether or not the contractive projection P in Proposition 4.4 must be a conditional expectation, we suspect not. Note there is always f ∈ B with supp f  S null, but the argument of Theorem 4.1 does not work unless f is a characteristic function. Theorem 4.1 also shows that in general the range of an L-projection on L1 (G), G compact, will not be a subalgebra, for if P is an L-projection then so is I − P , and clearly both cannot satisfy the conclusions of Theorem 4.1. 5. The algebras p (Λ) Theorem 5.1. For 1  p < ∞, the algebra p (Λ) has the Tomiyama property. Proof. First suppose that Λ is countable, so set Λ = N. Let P be a contractive projection onto a subalgebra B. By [28, Theorem 2.a.4] or [32, Corollary 3.2], there are norm-one elements yn ∈ B spanning B, and norm-one elements yn∗ ∈ q (N), such that

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(i) supp(yn ) ∩ supp(ym ) = ∅ if n = m, (ii) for x ∈ p (N), P (x) =



yn∗ , x yn .

n

As was remarked in [32, Corollary 3.2], the following is a consequence of (i) and (ii): (iii) supp(yn ) = supp(yn∗ ). This latter fact is crucial for our argument, so we give a proof. Since P yn = yn , it follows that yn∗ , ym = 0 for n = m. We have yn p = yn∗ q = 1 and      

 ∗ y ∗ (j ) · yn (j )  y ∗  · yn p = 1. yn (j )yn (j )  1 = yn∗ , yn = n n q j

j

It follows that  ∗  y (j ) = 1 n

for all j ∈ supp(yn ).

(5.1)

Since yn∗  = 1, it follows that yn∗ and yn have singleton support and that these are equal. Now B is a subalgebra, so (i) and (iii) imply that some multiple of yn is a non-zero idempotent en , and en em = 0 for n = m. (ii) implies that for each x, and n = m, yn∗ , xem = 0, and yn∗ , xen = yn∗ , x . Thus for x ∈ ker P , for each n and m, 0 = yn∗ , x = yn∗ , xem so that xem ∈ ker P . It follows that ker P is invariant under multiplication by elements of B. That is, Proposition 2.1(2) holds. Now consider the case of a general set Λ. Suppose P is a contractive projection on p (Λ) with range a subalgebra B. Let b1 , b2 ∈ B, a ∈ p (Λ). Define S1 = supp(b1 ) ∪ supp(b2 ) ∪ supp(a),     (n  2), supp P (δx ) : x ∈ Sn−1 Sn =  S= Sn . n1

Then S is countable, and P maps 1 (S) into itself. For g ∈ B ∩ p (S), clearly P (g) = g. And for g ∈ p (S), certainly P (g) ∈ B, and further P (g) ∈ p (S) by construction of S. Thus P (p (S)) = B ∩ p (S), which is a subalgebra. Since S is countable the first part shows that P |p (S) is a conditional expectation. Thus P (b1 ab2 ) = b1 P (a)b2 as required. 2 6. Dual algebras Let X be a left-introverted subspace of L∞ (G) which is contained in CB(G). Given μ ∈ M(G), define τ (μ) ∈ X ∗ by 

τ (μ), f = f (x) dμ(x) (f ∈ X).

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Provided C0 (G) ⊆ X, τ (μ) is the unique norm preserving extension of μ ∈ C0 (G)∗ to X, see [26, Lemma 1], so that τ is a linear isometry of M(G) into X ∗ . The following is an obvious extension of Proposition 3.7. Proposition 6.1. Let G be a locally compact group, X a left-introverted subspace of L∞ (G), C0 (G) ⊆ X ⊆ CB(G). If X ∗ has the Tomiyama property then G is a discrete torsion group. Proof. By [24, Lemma 4.1], τ above is an isometric algebraic isomorphism, and there is an isometric direct sum decomposition   X ∗ = τ M(G) ⊕ C0 (G)⊥ , where C0 (G)⊥ is a weak∗ -closed ideal of X ∗ . Now proceed as before.

2

 If J is a weak∗ -closed right ideal Lemma 6.2. Let X be a left-introverted subspace of UC(G). of X ∗ , then J is also a left ideal. Proof. Similar to [27, Lemma 5.1].

2

 be a left-introverted subspace of VN(G) containing Cρ∗ (G), and Lemma 6.3. Let X ⊂ UC(G) set   J = m ∈ X ∗ : m, T = 0 for all T ∈ Cρ∗ (G) . Then J is a weak∗ -closed ideal in X ∗ , and X ∗ = Bρ (G) ⊕ J isometrically. Proof. Similar to [27, Lemma 5.2].

2

 be a left-introverted C ∗ -subalgebra of VN(G). SupProposition 6.4. Let Cρ∗ (G) ⊆ X ⊆ UC(G) ∗ pose that X has the Tomiyama property for positive projections. Then G is compact. Proof. We have X ∗ = Bρ (G) ⊕ J , now use the argument of Proposition 3.10.

2

Remark. If X is a left-introverted subspace of VN(G) containing Cρ∗ (G), then Theorem 6.4 remains true under the stronger hypothesis with “positive” omitted. Define   WAP(A∗ ) = f ∈ A∗ : the map A → A∗ : ϕ → ϕ . f is weakly compact . Proposition 6.5. Let A be a unital Banach algebra, X ⊆ A∗ an A-left-introverted subspace, Λ : X → X ∗∗ the natural inclusion. Take the Arens product on X ∗∗∗ as the second dual of X ∗ . Then Λ∗ : X ∗∗∗ → X ∗ is a conditional expectation with range X ∗ if and only if X ⊆ WAP(A∗ ).

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Proof. The argument of [24, Lemma 1.4] shows that multiplication in X ∗ is separately weak∗ continuous if and only if X ⊆ WAP(A∗ ). Now apply [5, Proposition 2.2]. 2 Corollary 6.6. Let S be a semigroup with identity, X a left-introverted subspace of ∞ (S) and Λ : X → X ∗∗ the natural injection. Then Λ∗ is a conditional expectation if and only if X ⊂ WAP(S). Recall that a semigroup S is weakly cancellative if the equations xt = u, tx = u have only finitely many solutions x ∈ S for each t, u ∈ S. Proposition 6.7. Let S be a semigroup with identity, Λ : c0 (S) → ∞ (S) the inclusion map. Suppose that 1 (S) is Arens regular. Then the projection Λ∗ : ∞ (S)∗ → 1 (S) ⊂ ∞ (S)∗ is a conditional expectation if and only if S is weakly cancellative. Proof. [10, Theorem 7.13] shows that S is weakly cancellative if and only if 1 (S) is a dual Banach algebra, that is, if and only if the multiplication in 1 (S) = c0 (S)∗ is separately weak∗ continuous. The result now follows from [5, Proposition 2.2]. 2 For Example 2.5, note that c0 (S) translation invariant necessitates S weakly cancellative. 7. Compact right topological groups Let G be a compact right topological group with a strong normal system of closed subgroups. Consequently G has a Haar measure λ which is the unique right invariant Borel probability measure on G; λ is also left invariant under translation by elements of the topological centre. See [25] for details and notation. Definition 7.1.   M1 (G) = μ ∈ M(G): μ · f is defined and in L1 (G) for all f ∈ L1 (G) .  Here (μ · f )(x) = μ, fx = f (yx) dμ(y). In the case that G is a topological group,  (μ ∗ f )(x) =

  f y −1 x dμ(y) =

 f (yx) d μ(y) ˇ = (μˇ · f )(x)

so that μˇ · f ∈ L1 (G) whenever f ∈ L1 (G). It follows that M1 (G) = M(G). Remark 7.2. (i) When μ = δz for z ∈ Λ(G), then μ ∈ M1 (G). For given integrable f1 ∼ f2 , there is a Borel set E such that λ(E) = 0 and f1 = f2 off E. Then ( z f1 )(x) = f1 (zx) = f2 (zx) = ( z f2 )(x) whenever zx ∈ / E, that is, x ∈ / z−1 E. But λ(z−1 E) = 0 since z ∈ Λ(G). Further,  (μ · f )(x) =

f (yx) dδz (y) = ( z f )(x).

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But, again since z ∈ Λ(G), 

  f (zx) dλ(x) =



  f (x) dλ(x) < ∞.

(ii) Suppose μ ∈ L1 (G), and f1 , f2 are integrable with f1 = f2 off E, E Borel with λ(E) = 0. Then for any x ∈ G, (f1 )x = (f2 )x off Ex −1 . Then μ(Ex −1 ) = λ(Ex −1 ) = 0, so μ · f1 = μ · f2 . (iii) Recall from [25],   D(G) = g ∈ C(G): ( y g)ˇ∈ C(G) for all y ∈ G . Then D(G) ⊂ M1 (G). Let μ ∈ M1 (G) and define πμ : L1 (G) → L1 (G), f → μ · f . Consider the adjoint operator πμ∗ : L∞ (G) → L∞ (G):

πμ∗ (h) . f = h, μ · f



  h ∈ L∞ (G), f ∈ L1 (G) .

Define J μ to be the closed linear span of {f − μ · f , f ∈ L1 (G)}. Then   Jμ⊥ = h ∈ L∞ (G): πμ∗ (f ) = f . Remark 7.3. In the case that G is a compact group, Jμ is a closed right ideal in L1 (G), see [8,42]. Here μ · f = μˇ ∗ f , f ∈ L1 (G), and it follows that πμ∗ (h) = μˇ ∗ h, h ∈ L∞ (G). Definition 7.4. A Borel function h : G → C is μ-harmonic if h ∈ Jμ⊥ . In the case that μ(G) = 1, the constant function 1 is μ-harmonic. Proposition 7.5. Suppose μ ∈ M1 (G) with μ  1. Then there is a contractive projection Pμ : L∞ (G) → Jμ⊥ commuting with πμ∗ . Proof. For each n ∈ N, define a map on L∞ (G) by Λn (h) = μn · h. Each Λn is weak∗ to weak∗ continuous and contractive. Let B(L∞ (G)) be the locally convex space of all bounded linear maps on L∞ (G) with the weak∗ -operator topology. Let K be the closed convex hull of {Λn : n ∈ N} in B(L∞ (G)), so K is convex and compact. The map Φ : K → K : Λ → πu∗ ◦ Λ is affine and continuous. So by Markov–Kakutani it has a fixed point Pμ . Such Pμ has the desired properties. 2 Remark 7.6. Since πμ commutes with right translation it easily follows that Pμ also commutes with right translation. Remark 7.7. There is a net (Λα ) in {(πμ∗ )n : n ∈ N} such that for each h ∈ L∞ (G), Λα (h) → Pμ (h) weak∗ . By Mackey–Arens the net can be chosen to converge uniformly on compact subsets of L1 (G).

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Remark 7.8. We have μ · 1 = Pμ (1) = μ(G)1, and on the other hand μ · Pμ (1) = Pμ (1), whence Pμ (1) = 1 if μ(G) = 1, but Pμ (1) = 0 if μ(G) = 1. Corollary 7.9. Suppose that μ ∈ M1 (G) with μ(G) = 1. Then Jμ⊥ is linearly isometric to an abelian von Neumann algebra. Proof. We have 1 ∈ Jμ⊥ and Jμ⊥ is weak∗ -closed. Thus its closed unit ball has an extreme point. Thus by [29], P (A) is linearly isometric to a unital commutative C ∗ -algebra. 2 In fact [16] show that result remains true without the assumption of A having an identity, and explicitly construct the C ∗ -structure as follows. Let u be an extreme point of the closed unit ball of P (A), and define operations on P (A) by f ×u g = P (f u∗ g) and f  = P (uf ∗ u). Then (P (A), ×u , ) is a commutative C ∗ -algebra with identity u, and original norm. Remark 7.10. In the case that μ  0, μ(G) = 1 is equivalent to μ = 1, and Pμ is a positive contractive projection from L∞ (G) onto Jμ⊥ with Pμ 1 = 1. By Tomiyama [40], Pμ is a conditional expectation. Definition 7.11. For each 1  p  ∞, set   Mp (G) = μ ∈ M(G): μ · f is defined and in Lp (G) for all f ∈ Lp (G) . Each Mp (G) is a closed subspace of M(G). Definition 7.12. For each 1  p  ∞ with conjugate index q, set  D (G) = ϕ ∈ Lq (G): for each Borel subset E ⊆ G, p

 x →

   ϕ yx −1 dλ(y) is measurable .

E

Lemma 7.13. For every 1  p  ∞, Dp (G) ⊂ Mp (G). Proof. Let ϕ ∈ Dp (G). Then ϕ ∈ Lq (G), and a simple approximation argument shows that x → f (y)ϕ(yx −1 ) dλ(y) is measurable for each f ∈ Lp (G). Now since G is compact, ϕ ∈ L1 (G), so it may be considered as an element of M(G). Further, for f ∈ Lp (G),     (ϕ · f )(x) = ϕ, fx = ϕ(y)f (yx) dλ(y) = ϕ yx −1 f (y) dλ(y) since λ is right invariant. This function is measurable, and   (ϕ · f )(x)  ϕq f p , whence ϕ · f ∈ L∞ (G) ⊆ Lp (G).

2

We have the following analogue to Proposition 7.5, see [9].

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Proposition 7.14. Let 1  p < ∞, μ ∈ Mp (G) with μ = 1, πμ : Lp (G) → Lp (G) given by πμ (f ) = μ · f . Then there is a contractive projection Pμ : Lq (G) → Jμ⊥ commuting with πμ∗ . In particular, Jμ⊥ is isometrically isomorphic to Lq (Ω, Σ, λ) for some measure space (Ω, Σ, λ). For a similar result for locally compact groups, see [7]. The final statement follows from [41, Theorem 6] or [21, Theorem 3, §17]. 8. Finite groups We conclude with a brief look at the special case of finite groups. Example 8.1. Let G be a finite group, and consider the projection P = PH on 1 (G) given by f → 1H · f for some subset H of G. This is an L-projection and is positive in the pointwise sense, that is, f  0 implies that P (f )  0. Now 1 (G) is an involutive algebra under convolution  and involution ∗ : f → f ˇ where f ˇ(x) = f (x −1 ). For the range to be a subalgebra, H must be a semigroup, and hence a group since all elements have finite order. In particular e ∈ H and so P (δe ) = δe . In fact P is positive in the convolution sense, that is, P takes the positive cone generated by f  f ∗ , f ∈ 1 (G), to the positive cone in 1 (H ). For take f ∈ 1 (G) and x ∈ G. Then (f  f ∗ )(x) = x f, f , so that f  f ∗ is a positive definite function on G. But then (f  f ∗ )|H is also a positive definite function. So there exists g ∈ 2 (H ) such that (f  f ∗ )|H = g  g ∗ . Thus, with g˜ the extension of g to all of G, setting it to be zero outside H , P (f  f ∗ ) = 1H · (f  f ∗ ) = g˜  g˜ ∗ and so is positive. By Proposition 2.9, P is a conditional expectation. Example 8.2. Here we give a simple example of a positive conditional expectation on a group algebra which is not given by restriction, nor by averaging.  Let |G| be the order of the finite group G, and set m = |G|−1 x∈G δx . Define  Qf (x) =

f (e), 1 |G|−1

x = e,



y=e f (y),

x = e.

Then Q is a contractive projection onto the unital subalgebra B spanned by {δe , m}. Here       ker Q = f : f (e) = 0 and f (y) = 0 = f : f (e) = 0 and f (y) = 0 . y=e

y

 Since f  m = m  f = ( x∈G f (x))m, we have ker Q invariant under B. Hence Q is a conditional expectation. Since Q(1) = 1 as well as Q(δe ) = δe , Q is positive in the pointwise sense. To see that Q is positive in the convolution sense, note firstly that for f ∈ 1 (G),  ∗

Q(f  f )(x) =

2 y∈G |f (y)| ,

1 |G|−1



z=e



y∈G f (y)f (z

x = e, −1 y),

x = e.

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Set α=

  f (y)2 ,

β=

   1 f (y)f z−1 y , |G| − 1 z=e y∈G

y∈G

and note that β=

2        1   − f (y)2 . f (y)  |G| − 1  y∈G

y∈G

It follows that Q(f ∗ f ∗ ) is a positive definite function for each f ∈ 1 (G). To see this, let x1 , . . . , xn be distinct elements in G, λ1 , . . . , λn ∈ C. Then      −1  ∗ 2 |λi | α + β λi λj Q(f ∗ f ) xi xj = λi λj . (8.1) i=j

Here α  β by Hölder’s inequality, and distinctness of x1 , . . . , xn requires n  |G|. By elementary induction one verifies that the n × n matrix with diagonal entries equal to α, and all off-diagonal entries equal to β, has eigenvalues α − β  0 (of multiplicity n − 1), and eigenvalue 2   2  n − 1   |G| − n    f (y) + f (y)  0 α + (n − 1)β =  |G| − 1 |G| − 1 y∈G

y∈G

whence (8.1) is non-negative. Remark 8.3. The role of P  = 1 in our considerations is crucial—the only finite group such that every projection onto a unital subalgebra is a conditional expectation is Z2 . For let a ∈ G be an element of order at least 3, S the (abelian) subgroup generated by a. For i = 1, 2, 3, let χi be distinct characters on S, and set Mi = {f : χi , f = 0}. Take f ∈ M2 with χ1 , f = χ3 , f = 1. Then  1 (G) = 1 (S) ⊕ 1 (G \ S) = (M2 ∩ M3 ) ⊕ Cδe ⊕ Cf ⊕ 1 (G \ S). The projection onto the subalgebra (M2 ∩ M3 ) ⊕ Cδe fails to be a conditional expectation since M2 ∩ M3 is an ideal, so that f (M2 ∩ M3 ) ⊆ M2 ∩ M3 and there is g ∈ f (M2 ∩ M3 ) with χ1 , g = 0 so that f (M2 ∩ M3 ) = {0}. So for the property to hold, all elements of G have order 2. So G is abelian (for a, b ∈ G, ab = (ab)−1 = b−1 a −1 = ba), and if |G| > 2, the same argument with characters, but on 1 (G) rather than 1 (S), gives a projection which is not a conditional expectation. 9. Open questions • Let G be a locally compact group. (i) Suppose that S is a measurable subset of G such that 1S · L1 (G) is a subalgebra, that is, is closed under convolution. Does it follow that S differs from a subsemigroup of G by a null set? In particular, is H \ S always null in Theorem 3.1?

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