aa r X i v : . [ m a t h . OA ] M a y A TORSION-FREE ALGEBRAICALLY C*-UNIQUE GROUP
EDUARDO SCARPARO
Abstract.
Let p and q be multiplicatively independent integers. We showthat the complex group ring of Z [ pq ] ⋊Z admits a unique C ∗ -norm. The proofuses a characterization, due to Furstenberg, of closed × p − and × q − invariantsubsets of T . Introduction
Given a group G , there are in general many C ∗ -norms on its complex groupring C [ G ]. For example, if G = Z , identifying C [ Z ] with complex polynomials p ( z ) = P α n z n defined on T , we have that any infinite closed subset F ⊂ T givesrise to a distinct C ∗ -norm k·k on C [ Z ], defined by k p k := sup z ∈ F | p ( z ) | , for p ∈ C [ Z ](this was noted in [6, Chapter 19] and [8]). .Following [2], we say that a group G is algebraically C ∗ -unique if C [ G ] admits aunique C ∗ -norm.Clearly, any algebraically C ∗ -unique group must be amenable. In [8], Grigorchuk,Musat and Rørdam proved that any locally finite group is algebraically C ∗ -uniqueand asked whether the converse holds. Furthermore, Alekseev and Kyed obtainedin [2] large classes of amenable groups which are not algebraically C ∗ -unique.In [4], Caspers and Skalski studied the question of C ∗ -uniqueness in the contextof discrete quantum groups, and showed that there exists a C ∗ -unique discretequantum group which is not locally finite.As told by Alekseev in his report for the 2019 workshop “C ∗ -algebras” at Ober-wolfach [1], Ozawa pointed out, during the meeting, that the lamplighter group( L Z Z ) ⋊ Z is algebraically C ∗ -unique and not locally finite, thus answering inthe negative the question in [8]. Alekseev asked then in [1] whether there exists a torsion-free group which is algebraically C ∗ -unique.In this paper, by adapting Ozawa’s argument, we provide an example of a torsion-free algebraically C ∗ -unique group G . The proof that G is algebraically C ∗ -uniqueuses a certain result by Furstenberg ([7]) on diophantine approximation.2. A torsion-free algebraically C ∗ -unique group Fix p and q integer such that p, q ≥ r, s ∈ N suchthat p r = q s (this means that p and q are multiplicatively independent ).Let Z [ pq ] be the additive group { a ( pq ) n ∈ Q : a ∈ Z , n ∈ N } , and α the actionof Z on Z [ pq ] given by α ( n,m ) ( x ) := p n q m x , for n, m ∈ Z and x ∈ Z [ pq ]. Wewill show that the trosion-free group Z [ pq ] ⋊ Z is algebraically C ∗ -unique. By [2, Mathematics Subject Classification.
Lemma 2.2], this is equivalent to showing that, given an ideal I E C ∗ ( Z [ pq ] ⋊ Z )such that I ∩ C [ Z [ pq ] ⋊ Z ] = { } , we have that I = { } . Remark 2.1.
In [9], Huang and Wu studied unitary representations of Z [ pq ] ⋊ Z in connection with Furstenberg’s conjecture on × p − and × q − invariant probabilitymeasures on T .Notice that, because p and q are multiplicatively independent, we have that, if n, m ∈ Z are not both zero, then p n q m = 1. This implies that the action of Z on Z [ pq ] is faithful.Recall that an action of a group G on a locally compact Hausdorff space X issaid to be topologically free if, for each g ∈ G \ { e } , the set of points of X fixed by g has empty interior.The following lemma follows easily from [5, Lemma 2.1]. For the sake of com-pleteness, we include a proof. Lemma 2.2.
Let A be a torsion-free discrete abelian group and β : G y A a faithfulaction. Then b β : G y b A is topologically free.Proof. Take g ∈ G such that the set F g of points of b A fixed by b β g has non-emptyinterior; we will show that g = e .Since F g is a subgroup of b A , the fact that F g has non-empty interior implies that F g is open. Moreover, since b β g is continuous, we also have that F g is closed.From the fact that A is torsion-free, we obtain that b A is connected, hence F g = b A .Since β is a faithful action, we conclude that g = e . (cid:3) From the lemma above, we obtain that the action b α : Z y [Z [ pq ] is topologicallyfree. Hence, given a non-zero ideal I E C ∗ ( Z [ pq ] ⋊ Z ) ≃ C ( [Z [ pq ]) ⋊ Z , we havethat I ∩ C ∗ ( Z [ pq ]) = { } (for a proof of this general fact about topologically freeactions, see, for instance, [6, Theorem 29.5]).Let ϕ : T → T z z pq and X := lim ←− ( T , ϕ ) = { ( x n ) n ∈ N ∈ Q n ∈ N T : ∀ n ∈ N , x n = ϕ ( x n +1 ) } . Lemma 2.3.
There is an isomorphism ˜ ψ : C ∗ ( Z [ pq ]) → C ( X ) such that ˜ ψ ( δ a ( pq ) m )( x ) = ( x m ) a , for a ∈ Z , m ∈ N and x = ( x n ) n ∈ N ∈ X .Proof. Let Ev : C ∗ (cid:18) Z (cid:20) pq (cid:21)(cid:19) → C \Z (cid:20) pq (cid:21)! be the isomorphism given by point-evaluation, i.e., given u ∈ Z [ pq ] and τ ∈ [Z [ pq ],we have that Ev( δ u )( τ ) = τ ( u ). TORSION-FREE ALGEBRAICALLY C*-UNIQUE GROUP 3
Let H : [Z [ pq ] → X be the continuous map given by H ( τ ) := ( τ ( pq ) n )) n ∈ N , for τ ∈ [Z [ pq ]. Also let ˜ H : C ( X ) → C ( [Z [ pq ]) be the homomorphism induced by H .Let ψ : Z [ pq ] → C ( X ) be given by ψ ( a ( pq ) m )( x ) = ( x m ) a , for a ∈ Z , m ∈ N and x = ( x n ) n ∈ N ∈ X . It is straightforward to check that ψ is a well-defined unitaryrepresentation of Z [ pq ]. Let ˜ ψ : C ∗ ( Z [ pq ]) → C ( X ) be the canonical extension of ψ . An application of the Stone-Weierstrass theorem shows that ˜ ψ is surjective.Furthermore, it can be readily checked that ˜ H ◦ ˜ ψ = Ev. Since we know that Evis an isomorphism, we conclude that ˜ ψ is also an isomorphism. (cid:3) We denote by ˜ α the action of Z on C ∗ ( Z [ pq ]) induced by α . There is an action β : Z y X such that, for f ∈ C ( X ) and ( r, s ) ∈ Z , we have that ˜ ψ ◦ ˜ α ( r,s ) ◦ ˜ ψ − ( f ) = f ◦ β − r,s ) . One can readily check that, for x ∈ X and r, s ∈ Z non-negative integers, it holds that β − r,s ) ( x ) = x p r q s . Furthermore, β (1 , is the leftshift on X .We will need the following result of Furstenberg ([7]), whose precise formulationwe take from [3, Theorem 1.2]: Theorem 2.4. If B ⊂ T is an infinite closed set which is × p − and × q − invariant,then B = T . Theorem 2.5.
The group Z [ pq ] ⋊ Z is algebraically C ∗ -unique.Proof. Let I E C ( X ) ⋊Z be an ideal and suppose that, under the identification givenin Lemma 2.3, we have that I ∩ C [ Z [ pq ]] = { } . We will show that I ∩ C ( X ) = { } ,and therefore I = { } .Notice that C ( X ) ∩ I is a Z -invariant ideal of C ( X ), hence there is F ⊂ X a Z -invariant closed set such that C ( X ) ∩ I = C ( F c ).For n ∈ N , let π n : X → T be the canonical projection. Let B := π ( F ). Thefact that F is Z -invariant implies that B = π n ( F ) for n ∈ N and that B = { z p : z ∈ B } = { z q : z ∈ B } . Since I ∩ C [ Z [ pq ]] = { } , we have that B contains infinitely many points, forotherwise there would be a non-zero polynomial vanishing on B .Using Theorem 2.4, we conclude that B = T , hence F = X and I = { } . (cid:3) References [1]
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Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway
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