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Home , Locally compact space, Perfect set, Set of uniqueness

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In Section 2, we set up our general framework : on one hand, we set up the modern operator-theoretic framework for sets of uniqueness in general locally compact groups and, on the other hand, we introduce the elementary descrip- tive set-theoretic context on which we will study the complexity of U(G) and U0(G).

In Section 3, we rely heavily on [18] and [19] in order to prove certain preser- vation properties of sets of uniqueness and sets of extended uniquess regarding finite direct products and inverse images. In particular, having in mind our end goal of classifying the complexity of U0(G) for connected abelian Lie groups, we prove new results concerning these preservation properties for sets of extended uniqueness.

In Section 4, we prove that U(G) and U0(G) are coanalytic sets with respect to the Effros Borel space of F(G) under fairly general assumptions on G. Combin- ing results from Section 3 with the more classic previously mentioned study of the complexity of U(T) and U0(T), we establish the complexity of U(G) - for a locally compact connected Lie group G - and of U0(G) - for a connected abelian 1 Lie group G - both as Π1-complete sets of F(G).

2 2 Preliminaries

In section 2.1 we set up a framework to study sets of uniqueness on arbitrary locally compact groups, thus extending the classical definition for T. In section 2.2, we introduce terminology and concepts from classical Descriptive Set The- ory that will be used in further sections concerning the study of the complexity of sets of uniqueness.

2.1 The framework for sets of uniqueness We start by recalling the classical definition of set of uniqueness. A E ⊆ T is said to be a set of uniqueness if the following holds :

∞ inx cne =0, for x∈ / E ⇒ ∀n ∈ Z : cn =0 n=X−∞ where by equality with zero of the trigonometric series with coefficients cn ∈ C N inx we mean convergence of the partial sums SN = n=−N cne , as N → ∞. Recall that whenever such trigonometric seriesP converges to zero on a set of positive , then (cn) ∈ c0(Z) by Cantor-Lebesgue lemma. It is worth to mention that while every set of uniqueness has measure zero (cf. [12], Proposi- tion 1 - I.3), the converse is false. Indeed, there are sets of measure zero which are not sets of uniqueness (cf. [12], Theorem 5 - III.4). The reader may thus find Definition 2.3 natural, as the classic case occurs when G = T (hence, Gˆ = Z). If a subset E ⊆ T is not a set of uniqueness, then it is said to be a set of multi- plicity. The collection of closed sets of uniqueness is denoted by U(T) and the collection of closed sets of multiplicity is denoted by M(T). Both collections will be considered in further sections as subsets of the hyperspace K(T) of closed subsets of the circle endowed with the Vietoris .

Another important class of subsets of T are the so called sets of extended unique- ness. A subset E ⊆ T is said to be a set of extended uniqueness if the following holds : ∞ µˆ(n)einx =0, for x∈ / E ⇒ ∀n ∈ Z :µ ˆ(n)=0 n=X−∞ where µ is a on T andµ ˆ(n) denote its Fourier-Stieltjes coeffi- cients. A set which is not a set of extended uniqueness is said to be a set of restricted multiplicity. The family of closed sets of extended uniqueness is de- noted by U0(T) and we set M0(T)= K(T) \U0(T).

It is clear that U(T) ⊆ U0(T) and in fact, the inclusion is strict (cf. [12], Corollary 17 - VII.3).

We now extend these notions of uniqueness for locally compact groups G.

3 Henceforth, M(G) denotes the of complex Borel measures on G with respect to of measures. We will denote by Lp(G) the Lebesgue space with respect to the on G and we recall that L1(G) is canon- ically identified as a closed ideal of M(G). As usual, λ : G → L(L2(G)) with λs := λ(s) denotes the left of G. We also use λ to denote the representation of M(G) on L2(G) given by :

−1 λ(µ)(ξ)(g)= ξ(s g)dµ(s)= (λsξ)(g)dµ(s) ZG ZG for µ ∈ M(G) and ξ ∈ L2(G). The reduced group C∗-algebra of G is the following C∗-subalgebra :

||.|| ∗ 1 Cr (G)= {λ(f): f ∈ L (G)}

The group of G is denoted by VN(G) and defined as ∗ w ∗ ∗ ∗ VN(G) = Cr (G) . The group C -algebra of G, denoted by C (G), is the 1 ∗ completion of L (G) under the norm ||f||C := supπ{||π(f)||}, where the supre- mum is taken over all unitary equivalence classes of irreducible representations π of G. The Fourier-Stieltjes algebra of G will be denoted by B(G). As a set it consists of all functions u : G → C of the form :

u(s) = (π(s)ξ, η) where π : G → L(H) is a continuous acting on some H and ξ, η ∈ H. The Fourier algebra of G will be denoted by A(G) and as a set it consists of all functions u : G → C of the form :

u(s) = (λsξ, η) for ξ, η ∈ L2(G). Those are algebras with respect to pointwise multiplication. One has that A(G) is a closed ideal of B(G) and that A(G) can be identified with the predual of VN(G) via the pairing :

hu,T i = (Tξ,η) where u ∈ A(G) has the form u(s) = (λsξ, η). If T ∈ VN(G) and u ∈ A(G), the operator u.T ∈ VN(G) is given by the rela- tion hu.T, vi = hT,uvi, for v ∈ A(G).

For further details and properties about these algebras, we refer the interested reader to [18] and [6].

∗ ˆ We out that if G is abelian, then Cr (G) coincides with C0(G) - where Gˆ is the dual group of G - and VN(G) with L∞(Gˆ). Furthermore, in this

4 case the gives an isometric *- between L1(G) and A(Gˆ). In particular, when G is a locally compact abelian group we get that L1(Gˆ) ≈ A(G). For instance, with G = Td we recover the :

A(Td)∗ ≈ ℓ1(Zd)∗ ≈ ℓ∞(Zd) ≈ VN(Td)

Definition 2.1. Let T ∈ VN(G) and g ∈ G. We define supp∗(T ) as the set of elements in G which satisfy one of the following equivalent conditions (cf. Proposition 4.4, in [6]) : (i) For every open neighborhood V of g, there is some element u ∈ A(G) with supp(u)⊆V and such that hu,Si 6= 0. (ii) u.T 6= 0, whenever u ∈ A(G) and u(g) 6= 0. For the sake of completeness, we list some properties of supp∗(T ), for T ∈ VN(G) that will be used in further sections : Lemma 2.2. Let T ∈ VN(G), then supp∗(T ) has the following properties : (i) If T 6= 0, then supp∗(T ) is a non-empty . (ii) supp∗(T ) is the smallest closed set E ⊆ G such that for all u ∈ A(G) ∩ Cc(G) with supp(u) ∩ E = ∅ then hu,T i = 0. (iii) If T = λ(µ) for µ ∈ M(G), then supp∗(T ) = supp(µ). Proof. The reader can find a proof in [6] (respectively, Prop. 4.6, Prop. 4.8 and Remark 4.7). For a closed subset E ⊆ G we will use the following notation :

J(E)⊥ = {T ∈ VN(G): supp∗(T ) ⊆ E}

For a motivation of such notation, we refer the interested reader to [18].

We adopt the definition sets of uniqueness which was introduced in [4] and further generalized in [18] : Definition 2.3. Let E ⊆ G be a closed subset. Then, E is a set of uniqueness ∗ ∗ if S ∈ Cr (G) and supp (S) ⊆ E, imply that S = 0. Any closed set which is not a set of uniqueness, will be called a set of multiplicity (or a M-set). We denote the family of closed sets of uniqueness of G by U(G) and the family of closed sets of multiplicity of G by M(G).

For a closed subset E ⊆ G, we let M(E) ⊆ M(G) be the subset of measures with contained in E and we adopt the definition of sets of extended uniqueness given in [18] :

5 Definition 2.4. Let E ⊆ G be a closed subspace. Then, E is a set of extended ∗ uniqueness if λ(M(E)) ∩ Cr (G) = {0}. Any closed set which is not a set of extended uniqueness, will be called a set of restricted multiplicity (or a M0-set). We denote the family of closed sets of extended uniqueness of G by U0(G) and the family of closed sets of restricted multiplicity of G by M0(G). Remark 2.5. Indeed, Definitions 2.3 and 2.4 generalize the classical notions : (i) Let PM(T) be the space of pseudomeasures and P F (T) be the space of pseudofunctions. Moreover, for a closed subset E ⊆ T, let PM(E) denote the subset of pseudomeasures supported on E. It is known that PM(E)= J(E)⊥ (see [8]). (ii) If G is abelian or compact then the class of measures µ ∈ M(G) which ∗ satisfy λ(µ) ∈ Cr (G) coincide with the class of Rajchman measures, i.e. measures whose Fourier-Stieltjes coefficients vanish at infinity (see [2]). (iii) One can reformulate the classical definition for elements in M(T) and M0(T) as follows (cf. [12], Theorem 1 - II.4 and Propostion 6 - II.5): E ∈M(T) ⇔ PM(E) ∩ P F (T) 6= {0}

E ∈M0(T) ⇔ M(E) ∩ P F (T) 6= {0} ∗ ˆ Recalling that if G is abelian one has that Cr (G) ≈ C0(G), the conclusion follows from (i) and (ii).

2.2 Borel hierarchy and analytic sets Definition 2.6. A is a completely metrizable separable . A measurable space (X, Σ) is a standard Borel space if there is some Polish topology T on X such that its Borel algebra - i.e. the smallest σ-algebra containing T - coincides with Σ. In other words, if B(T ) = Σ. The study of regularity and definability properties on Polish spaces plays a cen- tral role in classical . For completeness, we mention some examples of Polish spaces which will appear in further sections :

• A topological group1 is called a Polish group if it is Polish as a topologi- cal space. Any which is second countable is Polish. In particular, every connected (hence second countable) locally compact Lie group and every connected abelian Lie group (being isomorphic to Rn × Tm for some positive integers n,m) is Polish.

• If X is a topological space, let K(X) be the collection of its compact sub- sets. The Vietoris topology on K(X) is the topology generated by subsets of the following form : {K ∈ K(X): K ⊆ U} and {K ∈ K(X): K ∩ U= 6 ∅}

1We assume that topological groups are Hausdorff.

6 for an U ⊆ X. The following is a well known result : Theorem 2.7. Let X be a Polish space. Then, K(X) endowed with the Vietoris topology is a Polish space. Proof. The reader can find a proof in [11] (Theorem 4.25). • If X is a topological space, let F(X) be the collection of its closed subsets. The collection F(X) endowed with the σ-algebra generated by sets of the following form : {F ∈F(X): F ∩ U= 6 ∅}, for open sets U ⊆ X is called the Effros space on F(X), which we will denote by E(X). The relevance of this example emerges from the next two well known results : Theorem 2.8. Let X be a Polish space. Then, F(X) is a standard Borel space. Proof. The reader can find a proof in [11] (Theorem 12.6). Theorem 2.9. Let X be a compact Polish space. Then, E(X) on F(X)= K(X) coincides with the Borel algebra generated by the Vietoris topology. Proof. The reader can find a proof in [11] (Example 12.7). If furthermore X is locally compact, E(X) is induced by the Fell topology (which is a compact metrizable topology that coincides with the Vietoris topology when- ever X is compact) on F(X) which has as a basis sets of the form :

{F ∈F(X): F ∩ K = ∅, F ∩U1 6= ∅, ..., F ∩Un 6= ∅} with K varying over K(X) and each Ui ⊆ X an open set.

For a Polish space X, its Borel algebra B(X) is stratified in the so called Borel hierarchy. Given a family of subsets F⊆P(X) we adopt the usual notations :

Fσ = { An : An ∈F} n[∈ω ¬F = {¬A : A ∈F} We define the first level of the Borel hierarchy as follows :

0 0 0 Σ1(X)= {U ⊆ X : U is open} and Π1(X)= ¬Σ1(X)

We proceed by transfinite recursion, for 1 <α<ω1 as follows :

0 0 0 0 Σα(X) = ( Πβ(X))σ and Πα(X)= ¬Σα(X) β<α[

0 0 It is straightforward to verify that if β < α, then Σβ(X) ⊆ Σα(X) and that any Borel set of X lies in some level of the Borel hierarchy :

0 0 B(X)= Σα(X)= Πα(X) α<ω[ 1 α<ω[ 1

7 1 The class of analytic sets of X, denoted by Σ1(X), is defined as follows : 1 Σ1(X)= {A ⊂ X : A = π(B),B ∈B(X × Y ) for some Polish space Y } where π denotes the projection map. The class of coanalytic sets of X, denoted 1 by Π1(X), is defined as 1 1 Π1(X)= ¬Σ1(X) An equivalent and often convenient definition of an analytic set is as follows :

Let X be Polish and A ⊆ X. Then, A is analytic if and only if there is a Polish space P and a continuous map f : P → X such that f(P )= A.

It turns out that the class of Borel sets of a Polish space coincides with the class of those sets which are simultaneously analytic and coanalytic : 1 1 Theorem 2.10. Let X be a Polish space. Then, B(X)=Σ1(X) ∩ Π1(X). Proof. The reader can find a proof in [11] (Theorem 14.11).

1 1 Furthermore, if X is uncountable then Σ1(X) and Π1(X) extend B(X). This was famously proved by Suslin after detecting a mistake in Lebesgue’s work.2 1 Theorem 2.11. Let X be an uncountable Polish space. Then, B(X) ( Π1(X). Proof. The reader can find a proof in [11] (Theorem 14.12). One can argue through different lenses that sets which are (co)analytic and not Borel are of high complexity, thus their detection being a topic of interest. A commonly used technique to identify these sets involves the concept of Γ- 1 1 hardness. In what follows, we consider Γ(X) to be either Σ1(X) or Π1(X). Definition 2.12. Let X be a Polish space and A ⊆ X. Then, A is said to be Γ(X)-hard if for every zero-dimensional Polish space Y and B ∈ Γ(Y ), there is a continuous map f : Y → X such that B = f −1(A). If, furthermore A ∈ Γ(X), then A is said to be Γ(X)-complete. We note that if A ⊆ X is a Γ(X)-complete set, then A ∈ Γ(X) \B(X). Indeed, by Theorem 2.11 there exists a set B ∈ Γ(ωω) \B(ωω). If A is Γ(X)-hard, then there is a continuous map f : ωω → X such that B = f −1(A). Since continu- ous preimages preserve the property of being Borel, we conclude that A/∈B(X).

1 We finish the section with the following example of a Π1(K([0, 1]))-complete set, which shall be used further : ′ 1 Theorem 2.13. Let Q = Q ∩ [0, 1]. Then, the following set is Π1(K([0, 1]))- complete (wrt Vietoris topology) : K(Q′)= {K ∈ K([0, 1]) : K ⊆ Q′} Proof. The reader can find a proof in [12] (Theorem IV.2.1). 2Lebesgue claimed in [14] that the projection of a of the plane onto the is again a Borel set.

8 3 Sets of operator multiplicity and sets of mul- tiplicity

In section 3.1 we introduce two notions of sets of operator multiplicity and relate these with sets of multiplicity as introduced in section 2.1. In section 3.2, we prove some preservation properties concerning direct products and inverse images.

3.1 Definitions

The concepts of operator M-sets and operator M0-sets were introduced in [17] and are of special importance as they allow the use of operator theoretic meth- ods in the study of questions emerging from . We refer the reader to [18],[19],[15] and [20] for further details. In this section, we state the relevant definitions and results needed in further sections, mainly following [18].

A measure space (X,µ) will be called a standard measure space if µ is a with respect to some completely metrizable separable and locally com- pact topology on X. For standard measure spaces (X,µ) and (Y,ν), a subset R ⊆ X × Y is said to be a rectangle if it is of the form R = A × B for A and B measurable. Endowing the product X × Y with the product measure :

(i) E ⊆ X × Y is said to be marginally null if E ⊆ (X0 × Y ) ∪ (X × Y0) for µ(X0) = 0 and ν(Y0) = 0. (ii) E, F ⊆ X × Y are said to be marginally equivalent if E∆F is marginally null. We denote this by E ∼ F . (iii) A subset E ⊆ X × Y is said to be ω-open if it is marginally equivalent to a countable of rectangles. The of a ω-open set will be called ω-closed.

Given Hilbert spaces H1 and H2, we denote as usual the space of compact op- 2 erators in L(H1, H2), by K(H1, H2). Henceforth, H1 = L (X,µ) and H2 = 2 L (Y,ν). The space C1(H2, H1) of nuclear operators is identified with the Ba- nach space dual of K(H1, H2) via hT,Si = tr(TS). Moreover, one can identify C1(H2, H1) with the space Γ(X, Y ) of all (marginal equivalence classes of) fuc- tions h : X × Y → C which admit a representation :

∞ h(x, y)= fi(x)gi(y) Xi=1

∞ 2 ∞ 2 with fi ∈ H1 and gi ∈ H2 such that i=1 ||fi||2 < ∞ and i=1 ||gi||2 < ∞. The duality between L(H1, H2) and Γ(PX, Y ) is given by : P

2 2 hT,f ⊗ gi = (T f, g), for T ∈ L(H1, H2) and f ∈ L (X,µ),g ∈ L (Y,ν)

9 ∞ If f ∈ L (X,µ), let Mf ∈ B(H1) be the operator of multiplication by f. The collection {Mf }f∈L∞(X,µ) is a maximal abelian selfadjoint algebra (masa). If

A ⊆ X is measurable, we write P (A)= MχA where χA is the characteristic map of A. A subspace W ⊆ B(H1, H2) is called a masa-bimodule if MΨTMϕ ∈ W for all Ψ ∈ L∞(X,µ), T ∈ W and ϕ ∈ L∞(Y,ν).

Now let E ⊆ X × Y be a ω-closed set and T ∈ B(H1, H2). We say that E supports T (or that T is supported on E) if P (B)MP (A) = 0 whenever A × B ∩ E ∼ ∅. For a subset M⊆B(H1, H2), there exists a smallest (up to marginal equivalence) ω-closed set which support every operator T ∈ M, that we denote by supp(M). On the other hand, it is known that for every ω-closed set E there exists the smallest and the largest weak∗ closed masa-bimodule - respectively Mmin(E) and Mmax(E) - with support E.

We are now in position to introduce operator M-sets : Definition 3.1. Let (X,µ) and (Y,ν) be standard measure spaces and κ ⊆ X × Y be a ω-closed set. Then, κ is an operator M-set if :

K(H1, H2) ∩ Mmax(κ) 6= {0}

Otherwise, κ is said to be an operator U-set.

In order to define operator M0-sets, we need some additional terminology. We follow [1] and consider σ to be a complex measure of finite total variation defined on the product σ-algebra of X × Y and let |σ| denote the variation of σ and |σ|X , |σ|Y be the marginal measures of |σ|. Such measure σ is said to be an Arveson measure if there is a constant c> 0 such that the following hold :

|σ|X ≤ cµ and |σ|Y ≤ cν

The set of all Arveson measures on X × Y is denoted by A(X, Y ) and for some σ ∈ A(X, Y ), we denote the smallest constant satisfying its defining inequalities by ||σ||A. For a σ-closed subset F ⊆ X × Y , we denote by A(F ) the set of all Arveson measures σ in X × Y such that supp(σ) ⊆ F .

An Arveson measure σ ∈ A(Y,X) defines an operator Tσ : H1 → H2 which will be called pseudointegral. These operators were introduced in [1]. Indeed, for σ ∈ A(Y,X) one can consider the sesquilinear form φ : H1 × H2 → C given by : φ(f,g)= f(x)g(y)dσ(y, x) ZY ×X By Riesz Representation Theorem, it follows that there is an unique operator Tσ : H1 → H2 such that (Tσf,g)= φ(f,g).

For a given ω-closed subset κ ⊆ X × Y we letκ ˆ = {(y, x) : (x, y) ∈ κ}. We can now state the following :

10 Theorem 3.2. Let σ ∈ A(Y,X). There exists an unique Tσ : H1 → H2 such that :

(Tσf,g)= f(x)g(y)dσ(y, x), for f ∈ H1,g ∈ H2 ZY ×X

Moreover, ||Tσ|| ≤ ||σ||A and for a given ω-closed subset κ ⊆ X ×Y the operator Tσ is supported on κ if and only if supp(σ) ⊆ κˆ. Proof. The reader can find a proof in [18] (Theorem 3.2).

We can finally define operator M0-sets : Definition 3.3. Let (X,µ) and (Y,ν) be standard measure spaces and κ ⊆ X × Y be a ω-closed set. Then, κ is an operator M0-set if :

There is a non-zero measure σ ∈ A(ˆκ) such that Tσ ∈ K(H1, H2)

Otherwise, κ is said to be an operator U0-set. We finish this section with the bridge between sets of multiplicity and their operator theoretic counterpart. Let G be a group and E ⊆ G. We define :

E∗ = {(s,t) ∈ G × G : ts−1 ∈ E}

We note that if G is second countable and E is closed, then E∗ is ω-closed. We have the following central results : Theorem 3.4. Let G be a locally compact second countable group and E ⊆ G be a closed subset. Then, the following are equivalent : (i) E is a M-set. (ii) E∗ is an operator M-set. Proof. The reader can find a proof in [18] (Theorem 4.9). Theorem 3.5. Let G be a locally compact second countable group and E ⊆ G be a closed subset. Then, the following are equivalent :

(i) E is a M0-set.

∗ (ii) E is an operator M0-set. Proof. The reader can find a proof in [18] (Theorem 4.12).

3.2 Preservation properties In this section we prove some preservation properties involving direct products and inverse images. We divide the section into two subsections. In section 3.2.1 we essentially follow [18] and deal with M-sets. In section 3.2.2, we establish some new results concerning the aforementioned preservation properties for M0- sets.

11 3.2.1 Operator M-sets and M-sets

Let (Xi,µi) and (Yi,νi) be standard measure spaces. We define the following map : ρ : (X1 × Y1) × (X2 × Y2) → (X1 × X2) × (Y1 × Y2)

((x1,y1), (x2,y2)) 7→ ((x1, x2), (y1,y2)) We note that the following useful identity holds :

∗ ∗ ∗ ρ(E1 × E2 ) = (E1 × E2) for Ei ⊆ Gi where G1 and G2 are groups. Since we already know how to relate operator M-sets with M-sets (section 3.1), the key element to prove preservation of M-sets under finite direct products is the following result :

Theorem 3.6. Let (Xi,µi) and (Yi,νi) be standard measure spaces and Ei ⊆ Xi × Yi be ω-closed sets. The set ρ(E1 × E2) is an operator M-set if and only if both E1 and E2 are operator M-sets. Proof. The reader can find a proof in [18] (Theorem 5.11).

Corollary 3.7. Let G1 and G2 be locally compact second countable groups and let E1 and E2 be closed sets. Then, the following holds :

(i) If E1 or E2 are sets of uniqueness, then E1 × E2 is a set of uniqueness

(ii) If E1 and E2 are sets of multiplicity, then E1 × E2 is a set of multiplicity Proof. We only prove (i), since the argument for (ii) is entirely analogous (see [18], Corollary 5.12). Suppose that E1 is a set of uniqueness. By Theorem ∗ ∗ ∗ ∗ 3.4, E1 is an operator U-set. Since ρ(E1 × E2 ) = (E1 × E2) , it follows from ∗ Theorem 3.6 that (E1 × E2) is an operator U-set and thus, by Theorem 3.4, we conclude that E1 × E2 is a set of uniqueness. Concerning inverse images, the following two results will be of crucial impor- tance in further sections :

Theorem 3.8. Let (X,µ), (Y,ν), (X1,µ1), (Y1,ν1) be standard Borel spaces and suppose that ϕ : X → X1 and Ψ: Y → Y1 are measurable maps. Let E ⊆ X1×Y1 and F = {(x, y) ∈ X × Y : (ϕ(x), Ψ(y)) ∈ E}. If ϕ∗µ and Ψ∗ν are equivalent, respectively, to µ1 and ν1, then E is an operator M-set if and only if F is an operator M-set. Proof. The result follows from Theorem 5.5 and Remark 5.6 in [18]. Corollary 3.9. Let G and H be locally compact second countable groups with Haar measures mG and mH respectively and E ⊆ H be a closed set. If ϕ : G → H is a continuous isomorphism, then E is a M-set if and only if ϕ−1(E) is a M-set. Proof. Since ϕ is a homomorphism, ϕ−1(E)∗ = (ϕ−1 × ϕ−1)(E∗) and since it is an isomorphism, ϕ∗mG and mH are equivalent. Thus, the result follows from Theorem 3.4 and Theorem 3.8.

12 3.2.2 Operator M0-sets and M0-sets

In this section we study preservation properties of M0-sets under inverse images and finite direct products, towards a classification of the complexity of U0(G), for G connected abelian Lie group. Every such group is isomorphic, for some non-negative integers m,n, to Rn ×Tm. Consequently, we are mostly concerned with inverse images of the quotient map q : Rn × Tm → Tn+m and finite prod- ucts of T. Nevertheless, we establish some preservation results with greater generality.

We recall that if G is either compact or abelian, the class of measures µ ∈ M(G) ∗ satisfying λ(µ) ∈ Cr (G) coincides with the class of Rajchman measures. For the case q : R → T, suppose that E ⊆ T is a closed set and that q−1(E) is −1 a M0-set. Thus, there is a Rajchman measure ν 6= 0 supported on q (E). Consider the pushforward measure µ := q∗ν on T, which is a non-zero measure on T supported on E. Furthermore, µ is Rajchman. Indeed, this follows from the observation that if f is a function on T and g is its 2π-periodic extension as a function on R, then R g(x)dν(x) = T f(t)dµ(t). In particular,ν ˆ(n)=ˆµ(n). Hence, sinceν ˆ ∈ C0(RR) it follows thatR lim|n|→∞ µˆ(n) = 0 and µ is Rajchman. It follows that E is a M0-set. In order to use this idea within a more general setting, we establish some notation.

Let G be locally compact and H ⊆ G be a closed normal subgroup. Given f ∈ Cc(G), we define : φ(f)([x]) := f(xh)dh ZH

One can prove that φ(f) ∈ Cc(G/H) and that in fact, φ : Cc(G) → Cc(G/H) is surjective (cf. Proposition 1.3.7, in [10]). The Haar measures on G and on H can be normalized in such way that the Weil’s formula holds for all f ∈ Cc(G):

f(x)dx = ( f(xh)dh)dµ[x] ZG ZG/H ZH where µ is a G-invariant measure on G/H (since H is closed normal, in this case µ is a Haar measure on G/H). One can verify that given f ∈ L1(G), Weil’s formula still holds and that ||φ(f)||1 ≤ ||f||1 and thus, φ is actually a ∗-homomorphism from L1(G) onto L1(G/H).

Every unitary representation of G determines a non degenerate ∗-representation π˜ of L1(G) as follows :

hπ˜(f)ξ, ηi = f(x)hπ(x)ξ, ηidx, for f ∈ L1(G) and ξ, η ∈ H(π) ZG For any representation π of G/H and ξ, η ∈ H(π), one shows that :

hπ ◦ q(f)ξ, ηi = hπ(φ(f))ξ, ηi, for f ∈ L1(G)

13 where q : G → G/H is the quotient map. This implies that ||φ(f)||C∗(G/H) ≤ ∗ ∗ ||f||C∗(G) and thus, φ can be extended to a -homomorphism from C (G) onto C∗(G/H). For more details, the interested reader is referred to [10]. Theorem 3.10. Let G be a locally compact second countable abelian (or com- pact) group, H ⊆ G a normal closed subgroup and E ⊆ G/H be a closed set. −1 Then, if q (E) is a M0-set, so is E. −1 −1 Proof. Since q (E) is a M0-set, there is a measure µ ∈ M(q (E)) such that ∗ ∗ λG(µ) 6= 0 and λG(µ) ∈ C (G). Recall that if λG(µ) ∈ C (G), then µ is Rajchman. Since λG is faithful, it follows that µ is a non-zero Rajchman measure and we can, if needed after considering |µ|, assume that µ is a non-zero positive 3 Rajchman measure. Hence, it follows that the pushforward measure q∗µ ∈ M(E) is non-zero and thus, λG/H (q∗µ) 6= 0. It then suffices to prove that ∗ λG/H (q∗µ) ∈ C (G/H) and we can conclude that E is a M0-set. Indeed :

φ(λG(µ))λG/H (φ(f)) = φ(λG(µ)λG(f)) = φ(λG(µ ∗ f))

= λG/H (φ(µ ∗ f)) = (λG/H ◦ q)(µ ∗ f)

= (λG/H ◦ q)(µ)(λG/H ◦ q)(f)

Since π ◦ q(µ)= π(q∗µ) for any representation π we conclude that :

∗ ∗ 4 φ(λG(µ)) = λG/H ◦ q(µ)= λG/H (q∗µ) ∈ C (G/H)= Cr (G/H)

In order to prove a converse to Theorem 3.10 we rely on a certain operator ∗ ∗ that allows going from Cr (G/H) to Cr (G) while respecting the support in a convenient way.

Let G be a locally compact group, θ ∈ A(G) ∩ Cc(G) and T ∈ VN(G/H), where H is a closed normal subgroup of G. Then, one can show that the fol- lowing functional is bounded (Theorem 3.7 in [19]) :

A(G) ∋ u 7→ hT, φ(θu)i

Thus, there is an operator Φθ(T ) ∈ VN(G) such that :

hΦθ(T ),ui = hT, φ(θu)i ,u ∈ A(G)

Lemma 3.11. Let θ ∈ A(G) ∩ Cc(G). Then, the following holds : ∗ ∗ (i) Φθ maps Cr (G/H) into Cr (G). ∗ ⊥ ∗ −1 ⊥ (ii) If T ∈ Cr (G/H) ∩ J(E) , then Φθ(T ) ∈ Cr (G) ∩ J(q (E)) , where q : G → G/H is the quotient map. 3Let G be a locally compact group. Then, if µ ∈ M(G) is Rajchman, so is |µ| (cf. Corollary 12 in [5]). 4Recall that the quotient of an by a normal subgroup, is an amenable group again.

14 Proof. The reader can find a proof in [19] (Theorem 3.7). Motivated by the definition of property (l) in [19], we introduce the following property : Definition 3.12. Let G be a locally compact group and H ⊆ G be a closed normal subgroup. For any θ ∈ Cc(G), let |θ(x)| := |θ(x)|. We say that H has the property (|l|2) if for every proper compact K ⊆ G/H there exists θ ∈ 2 A(G) ∩ Cc(G) such that φ(|θ|) = c1 and φ(θ )= c2, for positive real constants c1,c2 on a neighborhood of K. Remark 3.13. As an important example, Z ⊆ R has the property (|l|2). In- deed, after identifying T with [0, 1) let’s fix a proper compact K ⊆ T. We may assume that 0 ∈/ K - otherwise replacing K with a suitable translation. Using the regularity of A(R), let θ ∈ A(R) such that θ(x) = 1 on a neighborhood U of K and θ(x) = 0 whenever x∈ / (0, 1). If x ∈U, note that :

φ(|θ|)([x]) = |θ(x + h)| = θ(x + h)θ(x + h)= φ(θ2)([x]) = 1 hX∈Z hX∈Z An entirely analogous argument shows that Zn ⊆ Rn and Zn ×{1}m ⊆ Rn ×Tm also have the property (|l|2). Theorem 3.14. Let G be a locally compact group, H ⊆ G a closed normal subgroup with property (|l|2) and E ⊆ G/H a compact set. Furthermore, assume that G/H is abelian and second countable. Let q : G → G/H denote the quotient −1 map and suppose that E is a M0-set. Then, q (E) is also a M0-set.

Proof. Since E ⊆ G/H is a M0-set, there is a non-zero Rajchman measure µ′ ∈ M(E). Moreover, since supp(|µ′|) ⊆ E and |µ′| is also Rajchman, we may assume that there is a non-zero positive Rajchman measure µ supported by E. Since G/H is second countable and µ 6= 0 one has that µ(E) > 0. ∗ For any θ ∈ A(G) ∩ Cc(G), Φθ(λ(µ)) ∈ Cr (G) by Lemma 3.11. Thus, if one −1 shows that Φθ(λ(µ)) = λ(ν) for some ν ∈ M(q (E)) such that λ(ν) 6= 0, it −1 follows that q (E) is a M0-set. In order to obtain such measure ν, appealing to Riesz-Markov-Kakutani Rep- resentation Theorem it is enough to choose θ such that the following linear functional Ψ is bounded :

C0(G) ∋ u 7→ ( θ(xh)u(xh)dh)dµ(x) ZG/H ZH

Indeed, if Ψ is bounded then there is some measure ν ∈ M(G) such that :

Ψ(u)= u(s)dν(s) ZG By construction, for u ∈ A(G) we have that :

hΦθ(λ(µ)),ui = hλ(µ), φ(θu)i = Ψ(u)= hλ(ν),ui

15 and we can conclude that Φθ(λ(µ)) = λ(ν). Thus, it remains to choose an appropriate θ. Since E is compact and H ⊆ G has the property (|l|2), let 2 θ ∈ A(G) ∩ Cc(G) be such that φ(|θ|) = c1 > 0 and φ(θ ) = c2 > 0 on a neighborhood of E. It follows that :

| ( θ(xh)u(xh)dh)dµ(x)|≤ | θ(xh)u(xh)dh|dµ(x) ZG/H ZH ZG/H ZH

≤ ||u||∞ |φ(θ)(x)|dµ(x) ZG/H

≤ ||u||∞c1µ(E)

Hence, Ψ is bounded. Moreover, it follows from Lemma 3.11 that ν ∈ M(q−1(E)). It remains to check that indeed λ(ν) 6= 0. For this purpose, it suffices to choose u ∈ A(G) such that Ψ(u) 6= 0. We let u = θ and note that :

2 ( θ(xh)θ(xh)dh)dµ(x) = φ(θ )(x)dµ(x)= c2µ(E) > 0 ZG/H ZH ZG/H

Corollary 3.15. Let q : Rn × Tm → Tn+m be the quotient map for integers n+m n,m ≥ 0 and E ⊆ T be a closed subset. Then, E is a M0-set if and only if −1 q (E) is a M0-set. Proof. The result follows from Theorem 3.10 and Theorem 3.14.

We finish the section with results concerning the preservation of operator M0- sets and M0-sets under finite direct products. It will be useful to consider the ∗ left and right slice maps. Given ω ∈ (K(H2,K2)) = C1(K2,H2), the left slice map Lω : K(H1 ⊗ H2,K1 ⊗ K2) → K(H1,K1) is defined on elementary tensors by : Lω(A ⊗ B)= ω(B)A

A useful property of Lω is that if T ∈ K(H1 ⊗ H2,K1 ⊗ K2) is supported on ρ(κ1 × κ2) then, supp(Lω(T )) ⊆ κ1 (cf. [18]). The right slice operator Rω : K(H1 ⊗ H2,K1 ⊗ K2) → K(H2,K2) is defined analogously.

Theorem 3.16. Let (Xi,νi), (Yi,µi) be standard measure spaces and κi ⊆ Xi × Yi be ω-closed sets, for i =1, 2. The set ρ(κ1 × κ2) is an operator M0-set if and only if both κ1 and κ2 are operator M0-sets.

Proof. Let κ1 and κ2 be operator M0-sets, so that there are non zero measures

σ1 ∈ A(ˆκ1) and σ2 ∈ A(ˆκ2) such that Tσ1 ∈ K(H1,K1) and Tσ2 ∈ K(H2,K2).

Following the proof of Theorem 3.8 in [18], we conclude that Tσ := Tσ1 ⊗ Tσ2 ∈ 2 2 K(L (X1 × X2),L (Y1 × Y2)) for a non zero Arveson measure σ supported in \ ρ(κ1 × κ2). Hence, ρ(κ1 × κ2) is an operator M0-set.

Conversely, suppose that ρ(κ1 × κ2) is an operator M0-set so that there is a

16 \ non zero measure σ ∈ A(ρ(κ1 × κ2)) and Tσ ∈ K(H1 ⊗ H2,K1 ⊗ K2). Note that Lω(Tσ) ∈ K(H1,K1) and supp(Lω(Tσ)) ⊆ κ1. Thus, it suffices to prove that for some ω, Lω(Tσ) is a pseudo- operator - say Tγ - for a non-zero Arveson measure γ. It will follow from Theorem 3.2 that γ ∈ A(ˆκ1) and thus, κ1 is an operator M0-set. The case for κ2 is entirely analogous, using the right slice operator Rω instead.

We let Ω = Y1 × Y2 × X1 × X2, π, πXi , πYi be respectively the projections of Ω 2 2 onto Y1 × X1, Xi and Yi and fi ∈ L (Xi,νi), gi ∈ L (Yi,µi), for i = 1, 2 with ω = f2 ⊗ g2. Note that if σ is Arveson (in particular with finite total variation), then so is \ |σ|. Consequently, we can assume that σ ∈ A(ρ(κ1 × κ2)) is finite, non-zero and non-negative and as Ω is completely metrizable and separable, it follows that σ is Radon.5 Since σ 6= 0 and is Radon, there is a compact K ⊆ Ω with σ(K) > 0. ′ Note that πX2 (K) := C ⊆ X2 and πY2 (K) := C ⊆ Y2 are compact sets and let ′ U, V be open sets such that C ⊆ U and C ⊆ V. Since X2 and Y2 are normal, there are continuous functions f2 : X2 → [0, 1] and g2 : Y2 → [0, 1] such that χC ≤ f2 ≤ χU and χC′ ≤ g2 ≤ χV . Since ν2 is Radon, then :

inf{ν2(W): C ⊆W and W is open} = ν2(C) < ∞ and we can choose V such that ν2(V) < ∞. Similarly, we can choose W such that µ2(W) < ∞. Consequently :

2 f2 (x2)dν2 ≤ χU dν2 = ν2(U) < ∞ ZX2 ZX2

2 and we conclude that f2 ∈ L (X2,ν2). Similarly, we conclude that g2 ∈ 2 L (Y2,µ2). This is our choice of f2 and g2, defining ω. Now note that :

(Lω(Tσ)f1, g1)= hLω(Tσ),f1 ⊗ g1i = hTσ, (f1 ⊗ g1) ⊗ ωi = (Tσ(f1 ⊗ f2), g1 ⊗ g2) Since σ is an Arveson measure on Ω we have :

(Lω(Tσ)f1,g1)= f1(x1)f2(x2)g1(y1)g2(y2)dσ((y1,y2), (x1, x2)) ZΩ

Letting y = (y1,y2) and x = (x1, x2), we define a measure γ on Y1 × X1 as follows :

γ(E) := χπ−1(E)((y, x))f2(x2)g2(y2)dσ(y, x) ZΩ Therefore, we may conclude that :

(Lω(Tσ)f1,g1)= f1(x1)g1(y1)dγ(y1, x1) ZY1×X1

5Let X be a completely metrizable and and µ a (positive) Borel measure on X. Then, X is Radon (cf. [3], Theorem 7.1.7). In fact, if w(X) is the weight of X, then there is a non-tight finite (positive) Borel measure on X if and only if w(X) is a measurable cardinal (cf. [7], Theorem 438H). Existence of measurable cardinals is not provable within ZFC.

17 If one proves that γ an Arveson measure, we may conclude that Lω(Tσ) is pseudo-integral. It will then be enough to check that γ is non-zero and we are done. On one hand :

γ(Y1 × X1)= χΩ(y, x)f2(x2)g2(y2)dσ(y, x) ≤ χΩ(y, x)dσ < ∞ ZΩ ZΩ and thus, γ has finite total variation. On the other hand, let E = α × X1, with a measurable set α ⊆ Y1. Then, we have that :

|γ|(α × X1) ≤ χα×Y2×X1×X2 ((y, x))f2(y2)g2(x2)d|σ|((y, x)) ZΩ

= χα(y1)χY2×X1×X2 (y2, x1, x2)f2(y2)g2(x2)d|σ|((y, x)) ZΩ 1 1 1 1 2 2 2 2 2 2 ≤ ( f2 g2d|σ|) ( χαχY2×X1×X2 d|σ|) := A B ZΩ ZΩ ≤ kν1(α), for some finite positive k. In fact : Since σ is Arveson :

1 1 1 2 2 2 2 2 A ≤ c( f2 (x2)dν2) ( g2(y2)dµ2) = c||f2||||g2|| < ∞ ZX2 ZY2

Furthermore, note that given measurable spaces (Xi,νi) and (Yi,µi), a measure σ on (X1 × X2) × (Y1 × Y2) can be identified with a measureσ ˜ on X1 × (X2 × Y1 ×Y2) as the product σ-algebras can be canonically identified. In this way, the marginal measures |σ˜|(α) and |σ|(α × X2) - for α ⊆ X1 measurable - coincide. If σ is Arveson, it then follows that there is some positive finite constant d such 1 1 2 2 2 2 that |σ˜(α)|≤ da1(α). Thus, B ≤ (dν1 (α)) . It remains to check that γ is non-zero : let E = πY1 (K) × πX1 (K) which is compact, hence closed and thus Borel subset of Y1 × X1. Then :

γ(E)= χπ−1(E)(y, x)f2(x2)g2(y2)dσ(y, x) ≥ σ(K) > 0 ZΩ since if (y1,y2, x1, x2) ∈ K, then χπ−1(E) = 1 and f2(x2) = g2(y2) = 1, by construction.

We can thus establish the corresponding fact for M0-sets :

Corollary 3.17. Let G1 and G2 be locally compact second countable groups and E1 ⊆ G1 and E2 ⊆ G2 be closed sets. Then :

(i) If E1 and E2 are sets of restricted multiplicity, then E1 × E2 is a set of restricted multiplicity.

(ii) If E1 and E2 are sets of extended uniqueness, then E1 × E2 is a set of extended uniqueness. ∗ ∗ Proof. Let E1 and E2 be M0-sets. By Theorem 3.5, E1 and E2 are operator M0- ∗ ∗ ∗ sets. Since ρ(E1 × E2 ) = (E1 × E2) it follows by Theorem 3.16 and Theorem 3.5 that E1 × E2 is a M0-set. The proof is entirely analogous for the case of sets of extended uniqueness.

18 4 Complexity of sets of uniqueness and extended uniqueness

1 In section 4.1 we follow [12] and establish the Π1-hardness of U(T) and U0(T). In section 4.2, we prove the coanalycity of the collection of closed sets of unique- ness and of the collection of closed sets of extended uniqueness (with respect to the Effros Borel space) in greater generality. Combining these results with the 1 Π1-hardness of U(T) (and U0(T)) and using the preservation properties proven in section 3.2, we locate the complexity of the following sets :

• U(G) and U0(G) are coanalytic if G is locally compact, Hausdorff and second countable.

1 • U(G) is Π1-complete for any connected and locally compact Lie group G.

1 • U0(G) is Π1-complete if G is a connected abelian Lie group.

4.1 The case of T 1 In order to prove the Π1-hardness of U(T) we use the Salem-Zygmund Theorem, which is a remarkable fact relating sets of uniqueness of the circle with Pisot numbers.

Consider an interval [a,b] with length l = b − a and a sequence of positive real parameters : 0= η0 < η1 < ... < ηk < ηk+1 =1 such that for i < k, ξ := 1 − ηk < ηi+1 − ηi. Moreover, let E be the following disjoint union of closed sets :

k [a + lηi,a + lηi + lξ] i[=0

We say that E was obtained from [a,b] by a dissection of type (ξ, η1, ..., ηk). We now start from [a,b] = [0, 2π] and apply iteratively this operation of dissec- tion. As result, we have a decreasing sequence of closed sets E1 ⊇ ... ⊇ En ⊇ En+1 ⊇ ... and define : E(ξ, η1, ..., ηk)= En n\∈ω It is clear that E is a non-empty . For instance, the usual 1 2 in this notation is simply the set E( 3 , 3 ). The Salem-Zygmund Theorem gives us a full characterization of these type of sets in terms of sets of uniqueness :

19 1 Theorem 4.1. E(ξ, η1, ..., ηk) is a set of uniqueness if and only if ξ is a Pisot 6 1 number and each ηi ∈ Q( ξ ). Proof. The reader can find a proof in [12] (Theorem III.3.1). Another remarkable fact about sets of uniqueness of T which is needed is Bary’s Theorem, which states an important closure property of U(K(T)) : Theorem 4.2. The union of countably many closed sets of uniqueness of T is also a set of uniqueness. Proof. The reader can find a proof in [12] (Theorem I.5.1).7

1 We can now establish Π1-hardness for U(T): 1 Theorem 4.3. The set of closed sets of uniqueness of T is Π1(K(T))-hard. Proof. By Theorem 2.13 it is enough to define a continuous map F : K([0, 1]) → K(T) such that K(Q′) = F −1(U(T)). We start by defining a continuous map f : [0, 1] → K(T) as follows : 1 3 x 3 x 7→ E( , + , ) 4 8 9 4 It follows by Theorem 4.1 that Q = f −1(U(K(T))). Now, we define a map F : K([0, 1]) → K(T) as follows :

K 7→ f(x) x[∈K

By the properties of the Vietoris topology, F is continuous. Moreover, by The- orem 4.2, F −1(U(T)) = K(Q′) as we wanted. Remark 4.4. The proof of Theorem 4.1 given in [12] actually shows a stronger 1 1 statement : If ξ is not a Pisot number or some ηi ∈/ Q( ξ ), then E(ξ, η1, ..., ηk) is a set of restricted multiplicity (Theorem III.4.4, in [12]). Since sets of uniqueness are also sets of extended uniqueness, the proof of Theorem 4.3 also shows the following :

1 Theorem 4.5. The set of closed sets of extended uniqueness of T is Π1(K(T))- hard. 6An algebraic integer x is called a Pisot number if x> 1 and all its conjugates have absolute value smaller than 1. 7Let κ < 2ℵ0 be a cardinal. Under the assumption MA(κ)+ ¬CH one can prove that in fact the union of κ many closed sets of uniqueness of T is also a set of uniqueness. Since MA(ℵ0) holds in ZFC, this includes the case of Theorem 4.2. The reader can find a proof in [16] (Theorem 4.86).

20 4.2 General case We start this last section by identifying on which Polish spaces we will work on.

• For a locally X, we will consider the Fell topology on F(X), which is compact metrizable, induces the Effros Borel space and coincides with the Vietoris topology whenever X is compact.

• Let X be a locally compact, Hausdorff and second countable space. Then, 8 ∗ ∗ C0(X) is separable and since M(X)= C0(X) , it follows that B1(M(G)) is w - metrizable for a locally compact and second countable group G (we adopt the convention that topological groups are Hausdorff). Furthermore, by Banach- Alaoglu Theorem, it follows that :

∗ G locally compact second countable group, then (B1(M(G)), w ) is Polish

• Let G be locally compact and second countable. Any element of u ∈ A(G) is of the form u = f ∗ gˇ for f,g ∈ L2(G) (cf. Theorem 2.4.3, in [10]) and with norm ||u|| = inf{||f||2||g||2} with infimum taken over all representations of the form u = f ∗ gˇ. Since G is second countable, then L2(G) is separable and consequently, A(G) is also separable. Thus, and since VN(G) = A(G)∗, ∗ B1(VN(G)) is w -metrizable. Similarly as before, we conclude that :

∗ G locally compact second countable group, then (B1(VN(G)), w ) is Polish

• Let G be locally compact and second countable. It follows that L1(G) is ∗ 1 separable. Let C (G) be the completion of L (G) under the norm ||.||C∗(G). It ∗ is well known that ||f||C∗(G) ≤ ||f||1 and thus, C (G) is separable. Since it is defined as a completion, it follows that C∗(G) is Polish.

• We recall that if G is locally compact, for any ϕ ∈ A(G)∗ there is an unique operator Tϕ ∈ VN(G) such that :

2 hTϕ(f),gi2 = hϕ, g ∗ fˇi = hϕ, f ˇ∗ g˜i, for f,g ∈ L (G)

∗ The assignment ϕ 7→ Tϕ is then a surjective linear isometry from A(G) to VN(G) and a if one considers the w∗ topology on both the ∗ domain and codomain. Furthermore, if ϕ ∈ A(G) is given by hϕµ,ui =

G u(s)dµ(s) for some µ ∈ M(G), we have that Tϕµ actually coincides with Rλ(µ). For a proof of this collection of facts, the reader is referred to [10] (The- orem 2.3.9). In the remaining of this section, we consider the representation λ of M(G) restricted to the unit ball and with the following :

∗ ∗ λ : (B1(M(G)), w ) → (VN(G), w )

8Since X is second countable, so it is its one-point compactification X˜. Thus, C(X˜) is separable as X˜ is compact, Hausdorff and second countable. Since subspaces of separable metric spaces are separable, we are done

21 It follows from previous observations that λ is continuous. Indeed, since B1(M(G)) ∗ is w -metrizable, it is sufficient to verify sequential continuity : if µn → µ, it follows by definition that ϕµn → ϕµ and thus, λ(µn) → λ(µ). Note that in fact, the definition of the pairing A(G)∗ = VN(G) entails the continuity of λ even if one considers the WOT topology in the codomain.

4.2.1 M-sets Theorem 4.6. Let G be a locally compact and second countable group. Then, U(G) is coanalytic.

∗ ∗ Proof. Step 1 : Consider C (G) and Cr (G) respectively as the completions of 1 L (G) under the norm ||.||∗ and ||.||λ (as usual, ||f||λ := ||λ(f)||). Since ||.||λ ≤ ∗ ∗ 2 ||.||∗ this entails a continuous surjective map, Υ : C (G) → Cr (G) ⊆ L(L (G)), taking the operator norm in the codomain. Note that C := B1(VN(G)) ∩ ∗ Cr (G) is closed in the operator norm and thus, since Υ is continuous, P := Υ−1(C) is also closed. As a closed subspace of a Polish space, it follows that P is also Polish. Let’s denote the restriction of Υ to P still as Υ. Then, ∗ Υ: P → B1(VN(G)) is a function whose image is B1(VN(G)) ∩ Cr (G), as Υ is surjective. Since Υ was continuous with respect to the operator norm, it will remain continuous with respect to the w∗-topology in the range. Thus :

∗ ∗ Γ1 := Cr (G) ∩ B1(VN(G)) ⊆ (B1(VN(G)), w ) is analytic

∗ Step 2 Considering F(G) with the Fell topology and (B1(VN(G)), w ), the following set is Borel (even closed) :

∗ Γ2 = {(S, E) ∈ B1(VN(G)) ×F(G): supp (S) ⊆ E}

Let {(Sn, En)} ⊆ Γ2 such that (Sn, En) → (S, E). It is enough to prove that supp∗(S) ⊆ E. We use the characterization of supp∗(T ) given in Lemma 2.2 for T ∈ VN(G). Let u ∈ A(G) ∩ Cc(G) such that supp(u) ∩ E = ∅ and W = {F ∈ F(G): F ∩ supp(u) = ∅} which is an open neighborhood of E (in the Fell topology). Since En → E, then Ek ⊆W for all sufficiently large k. Since ∗ supp (Sk) ⊆ Ek, then for all sufficiently large k one has that hu,Ski = 0. Onthe ∗ w ∗ other hand, since Sn −−→ S, it follows that hu,Si = 0 and thus supp (S) ⊆ E as we wanted.

Step 3: Define P ⊆ B1(VN(G)) ×F(G) as the following subset :

−1 P = {(S, E) : (S, E) ∈ Γ2,S 6=0} ∩ π1 (Γ1) where πi (i = 1, 2) denotes the projection onto the factors of the product −1 B1(VN(G)) ×F(G). By Step 1, Γ1 is analytic and thus so is π1 (Γ1) since 9 π1 is Borel. By Step 2, and since the condition S 6= 0 is Borel and the inter- section of analytic sets is analytic, P is analytic. Again, since π2 is Borel, it 9If X,Y are Polish, f : X → Y a Borel function, A ⊆ X and B ⊆ Y analytic, then f(A) and f −1(B) are also analytic (cf [12], Theorem 14.4).

22 follows that π2(P ) is analytic. Note that :

π2(P )= M(G) and we conclude that U(G) is coanalytic. Given a class Γ of sets in a standard Borel space X and A ⊆ X, we say that A is Borel Γ-hard if for every standard Borel space Y and B ∈ Γ(Y ), there is a Borel function f : Y → X such that B = f −1(A). This definition is very similar to the one of Γ-hardness, except that we use Borel reductions instead of 1 continuous reductions. In fact, if X is Polish then A ⊆ X is Σ1-hard if and only 1 1 if A is Borel Σ1-hard (cf. [12], pp. 207). Evidently, the same holds for Π1. By 1 Theorem 4.3, U(T) ⊆F(T) is Π1-hard and thus, since F(T) is Polish, it is also 1 Borel Π1-hard. By Theorem 4.5, the same can be said about U0(T). Now let X be locally compact and A ⊆ X such that there is a Borel function −1 −1 f : F(T) →F(X) such that f (A)= U(T) or f (A)= U0(T). It follows that 1 1 A is Borel Π1-hard and thus, Π1-hard. Theorem 4.7. Let G be a connected locally compact Lie group. Then, U(G) is 1 Π1-complete. 1 Proof. By Theorem 4.6, it is enough to prove that U(G) is Π1-hard. We divide the analysis in two cases :

Step 1: Suppose that G is abelian and thus, G ≈ Rn × Tm. By Corollary n m 1 3.9 it suffices to show that U(R × T ) is Π1-hard. Firstly, note that for any positive integer k, the function g : F(T) → F(Tk) such that E 7→ Ek is measurable and, by Theorem 3.7, is such that U(T) = −1 k k 1 g (U(T )). Thus, U(T ) is Π1-hard. Now, let q : Rn × Tm → Tn+m be the quotient map and f : F(Tn+m) → F(Rn × Tm) be given by E 7→ q−1(E). Note that this map is measurable. Indeed, if A = {F ∈F(Rn×Tm): F ∩U= 6 ∅}, then f −1(A) = {F ∈ F(Tn+m): F ∩ q(U) 6= ∅}. Since q is the quotient map by an action of a subgroup, q(U) is an open for each open set U and we may conclude that f is measurable. Furthermore, by Theorem 3.8 combined with Theorem 3.4 one has that E ∈ U(Tn+m) if and only if q−1(E) ∈ U(Rn × Tm) and we are done.

Step 2: To consider the general case, note that G/[G, G] is a connected abelian 1 Lie group. By Step 1, U(G/[G, G]) is Π1-hard and we use the same argument as before with the quotient map q : G → G/[G, G].

4.2.2 M0-sets Theorem 4.8. Let G be a locally compact and second countable group. Then, U0(G) is coanalytic.

23 ∗ Proof. Step 1 : Since the ball B1(VN(G)) is w -closed by Banach-Alaoglu −1 ∗ Theorem, it follows that P := λ (B1(VN(G))) ⊆ (B1(M(G)), w ) is closed ∗ and thus, Polish. Consequently, λ(P ) ⊆ (B1(VN(G)), w ) is analytic. By the ∗ proof of Theorem 4.6, Cr (G) ∩ B1(VN(G)) is also analytic. Since intersection ∗ of analytic sets is analytic, then the following is analytic in (B1(VN(G)), w ):

∗ A := λ(P ) ∩ Cr (G) ∩ B1(VN(G)) \{0}

∗ It follows that the following is analytic in (B1(M(G)), w ):

−1 ∗ Λ1 := λ (A)= {µ ∈ B1(M(G)) : λ(µ) ∈ Cr (G) ∩ B1(VN(G)) \{0}}

∗ Step 2 : Consider (B1(M(G)), w ) and F(G) with the Fell topology and let :

∗ Λ2 = {(µ, E): supp (λ(µ)) ⊆ E} where πi (for i =1, 2) is the projection onto the factors of B1(M(G)) ×F(G). The proof that Λ2 is closed, hence analytic, goes exactly as in the proof that ∗ w Γ2 is closed in Theorem 4.6. Indeed, as pointed out earlier, µn −−→ µ implies ∗ w that λ(µn) −−→ λ(µ) and we can reproduce the argument used in the proof of Theorem 4.6.

Step 3 : Define P ⊆ B1(M(G)) ×F(G) as the following subset :

∗ P = {(µ, E): µ ∈ M(E), λ(µ) ∈ Cr (G) ∩ B1(VN(G)), λ(µ) 6=0}

−1 Since P = π1 (Λ1) ∩ Λ2 it follows by Step 1) and Step 2) that P is an- alytic. Thus, to prove that U0(G) is coanalytic it is enough to show that M0(G)= π2(P ): On one hand, if E ∈ π2(P ), then there is a measure µ ∈ B1(M(G)) such that ∗ (µ, E) ∈ P . By definition of P , this means that λ(M(E)) ∩ Cr (G) 6= {0} and thus, E is a M0-set. Conversely, suppose that E ⊆ G is a M0-set and thus there is some µ ∈ M(E) ∗ µ such that λ(µ) ∈ Cr (G) \{0}. In particular, µ 6= 0. Then,µ ˜ = ||µ|| ∈ ∗ B1(M(G)) ∩ M(E) and T := λ(˜µ) ∈ Cr (G) \{0}. If ||T || ≤ 1, then λ(˜µ) ∈ ∗ ′ µ˜ ′ Cr (G) ∩ B1(VN(G)). Otherwise, we consider µ = ||T || and λ(µ ). In any case, ∗ there is a measure µ ∈ B1(M(G))∩M(E) such that λ(µ) ∈ Cr (G)∩B1(VN(G)) and λ(µ) 6= 0. Thus, E ∈ π2(P )

1 Theorem 4.9. Let G be a connected abelian Lie group. Then, U0(G) is Π1- complete.

1 Proof. By Theorem 4.8 it is enough to prove that U0(G) is Π1-hard.

Step 1: By Theorem 3.17 and considering the map h : F(T) 7→ F(Tk) given k k 1 by E 7→ E we conclude that U0(T ) is Π1-hard.

24 Step 2 : Let G1 and G2 be locally compact abelian groups, ϕ : G1 → G2 be an isomorphism (of topological groups) and µ ∈ M(G1). This induces a pushforward measure ϕ∗µ on G2 and an isomorphism ϕ∗ : Gˆ1 → Gˆ2 given by χ 7→ χ ◦ ϕ−1 such that :

µˆ(χ)= χ(x)dµ(x)= ϕ∗(χ)(y)dϕ∗µ(y)= ϕ∗µ(ϕ∗(χ)) ZG1 ZG2 d Hence, ifµ ˆ ∈ C0(Gˆ1) it follows that ϕ∗µ ∈ C0(Gˆ2). If E ⊆ G1 is a M0-set there is some µ ∈ M(E) such that µ 6= 0 is Rajchman and thus, ϕ∗(µ) ∈ M(ϕ(E)) and d is a non zero Rajchman measure implying that ϕ(E) is also a M0-set. Analo- gously, we may conclude that E ⊆ G1 is a M0-set if and only if ϕ(E) is a M0-set.

Step 3 : Any such G is isomorphic to Rn × Tm for some integers n,m ≥ 0 n m 1 and consequently, by Step 2, it suffices to prove that U0(R × T ) is Π1-hard. In order to do so, we appeal to Corollary 3.15 and consider the Borel reduction f : F(Tn+m) →F(Rn ×Tm) given by E 7→ q−1(E), where q : Rn ×Tm → Tn+m is the quotient map.

5 Acknowledgements

The author would like to express gratitude to L. Turowska for her extremely helpful suggestions.

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