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One of the central concepts in his study of the coarse geometry of topological groups is the notion of coarsely boundedness (it is called the property (OB) in Rosendal [Ro09]), which is related to the boundedness in the sense of Hejcman [He58]. Here, a Polish group has property (OB) if whenever it acts continuously by isometries on a metric space, all orbits are bounded. The properties (OB), (FH) and (T) are thus closely related, and the distinction of them provides better understanding of non-locally compact Polish groups from the large scale viewpoint. The purpose of this paper is to study some of the above mentioned themes, focusing on the two different topologies on the unitary groups, namely the norm topology and the strong operator topology. We also take advantage of operator algebra theory for the study of Polish groups of unitaries. In order to explain the results of this article, let us introduce some terminology. We say that a Hausdorff topological group G is (i) strongly unitarily representable (SUR) if G is isomorphic as a topological group to a closed subgroup of (H) for some Hilbert space H equipped with the strong operator topology. U (ii) norm unitarily representable (NUR) if G is isomorphic as a topological group to a closed subgroup of (H) for some Hilbert space H equipped with the norm topology. U In 3 we study the unitary representability of some Polish groups consisting of unitary operators. Our work in§ this section is inspired by the works of Megrelishvili [Me08] and Galindo [Ga09] about SUR. If 2 G is a subgroup of (ℓ ), we denote by Gu (resp. Gs) the group G equipped with the norm (resp. the strong) topology. ItU is obvious that if M is a von Neumann algebra, then the unitary group (M) is U s SUR. On the other hand, the group (M)s is NUR if and only if M is finite-dimensional (Corollary 3.12). U ∗ It is also clear that if A is a separable unital C -algebra, then (A)u is NUR. On the other hand, (A)u is SUR if and only if A is finite-dimensional (Theorem 3.3).U These results show the incompatibilityU of the two topologies for the non-locally compact case. Although the result is to be expected from the fact that the only von Neumann algebras which are separable in norm are finite-dimensional ones, the proof is 2 entirely different. Theorem 3.3 is used to prove that the Fredholm unitary group ∞(ℓ ) (Definition 2.2) with the norm topology, which is evidently NUR, is not SUR. More interesting exampleU of Polish groups of unitaries are the so-called Schatten p-unitary groups (ℓ2) (1 p < ) (Definition 2.2). Although Up ≤ ∞ there have been several intensive works on the (projective) unitary representations of p(H) for p =1, 2, 2 2 U not much is known about unitary representability of p(ℓ ). We know that 2(ℓ ) is SUR (see [AM12-2, Theorem 3.2]; in fact it is of finite type, meaning thatU it embeds into the stronglyU closed subgroup of a 2 II1 factor with separable predual), but it is unclear whether 2(ℓ ) is NUR. Also it is unknown whether 2 U p(ℓ ) is SUR or NUR for p = 1, 2, (if it is SUR, then by [AM12-2, Theorem 2.22], it is actually of U 6 ∞ 2 finite type). In contrast, we show in 3.2 that 1(ℓ ) is NUR (Theorem 3.16). We do not know if it is § 2 U SUR. This might suggest that the family p(ℓ ) shares mixed features of both the NUR groups and the SUR groups. U In 4, we study whether the following properties hold for a specific class of Polish groups of unitaries: § ∗ boundedness, properties (FH), (OB) and (T). If A is a unital abelian C -algebra, we show that (A)u is bounded if and only if its spectrum is totally disconnected, which is known to be equivalent to AU having real rank zero. For general A, (A) is bounded if and only if the identity component (A) is bounded U u U0 u and has finite index in (A)u. Thus we focus on 0(A)u. We show that 0(A)u is bounded if and only if it has finite exponentialU length (Definition 4.10) introducedU by RingroseU [Ri91] (Theorem 4.13). Phillips’ survey [Ph93] is a very readable account of this and the related notion of the exponential rank. The notion of exponential length has been studied by experts, and it was one of the important ingredients in Phillips’ proof [Ph00] of the celebrated Kirchberg–Phillips’ classification Theorem for purely infinite simple nuclear C∗-algebras in the UCT class. Thanks to Lin’s result [Li93] that a C∗-algebra A having real rank zero has property weak (FU) of Phillips [Ph91], meaning that any u 0(A) is the norm limit of unitaries with finite spectrum, it follows that such an algebra has finite exponential∈U length. But there exist unital C∗-algebras with finite exponential lengths which do not have real rank zero (see [Ph92] and Remark 4.15). Thus, the real rank zero property is not necessary for 0(A)u to be bounded. In [Pe18, 2 U Example 6.7], Pestov shows that the 2(ℓ ) does not have property (T), and it was left unanswered (see 2 U [Pe18, 3.5]) whether 2(ℓ ) has property (FH). We show that it does not. Actually, we prove a more general§ result that if MU is a properly infinite semifinite von Neumann algebra with a faithful normal semifinite trace τ, then the Lp-unitary group (M, τ) (see Definition 2.2) does not have property (FH) Up for 1 p< . In fact, we prove that any closed subgroup of p(M, τ) which has property (FH) is in fact ≤ ∞ U 2 of finite type (Theorem 4.17, and Theorem 4.28). In particular, any closed subgroup of p(ℓ ) (1 p< ) with property (FH) must be compact (Corollary 4.20 and Corollary 4.33). For technicalU reasons,≤ we need∞ two different proofs depending on whether 1 p 2 or 2

2 2 for 1 p 2, there is a well-defined affine isometric action of p(M, τ) on L (M, τ) (regarded as a real Hilbert≤ space≤ by taking the real part of the inner product) givenU by

u x = ux + (u 1), u (M, τ), x L2(M, τ). · − ∈Up ∈

It might be interesting to note that the absence of property (FH) for p(M, τ) is valid even for M of the 2 U form N B(ℓ ), where N is a type II1 factor with property (T) in the sense of [CJ85]. Finally, we show ⊗ ∗ that if A is a unital separable C -algebra admitting a representation π which generates a type II1 factor with separable predual having property Gamma, then (A)u does not have property (T) (Theorem 4.34). This is an immediate consequence of the definition itself,U but shows that there exist many C∗-algebras whose unitary groups do not have property (T), while at the same time many of them do have property ∞ (FH) (or even (OB)). In particular, the unitary group of the UHF algebra M2∞ of type 2 does not have property (T) but has property (OB).

2 Preliminaries

We consider complex Hilbert spaces unless explicitly stated otherwise. For a Hilbert space H, we denote by (H) (resp. K(H)) the multiplicative (resp. additive) group of unitary operators (resp. compact operators)U of H. σ(a) denotes the spectrum of a closed, densely defined operator a on H. For 1 p< , ≤ ∞ Sp(H) is the additive group of Schatten p-class operators on H. We use the following abbreviations for the operator topologies: SOT for the strong operator topology, SRT for the strong resolvent topology (see e.g., [AM15, 2]). We sometimes use to denote the operator norm, if it is appropriate to distinguish § k·k∞ it from other norms, such as the Schatten p-norm p. For a topological group G, we denote by N (G) the set of all open neighborhoods of the identity ink·kG. If G is a subgroup of (H), we often denote by G U u or (G, ∞) (resp. Gs or (G, SOT)) the group G equipped with the norm (resp. the strong operator) topology.k·k For a unital C∗-algebra A, its unitary group (resp. the set of self-adjoint elements in A) is denoted by (A) (resp. A ). (A) is the identity component of (A) in the norm topology. U sa U0 U Definition 2.1. Let M be a semifinite von Neuman algebra with separable predual and τ a normal faithful semifinite trace on M. The noncommutative Lp-space associated with M is denoted by Lp(M, τ). We denote by p the p-norm induced by τ. If x is a closed, densely defined operator affiliated with M and x = ∞ λk·kde(λ) is the spectral resolution of x , then | | 0 | | 1 R p x = λp dτ(e(λ)) . k kp Z[0,∞) ! Details about non-commutative integration theory can be found e.g., in [Tak, Chapter IX.2]. Definition 2.2 ([AM12-2]). Let (M, τ) be as in Definition 2.1 and 1 p < . The p-unitary group of ≤ ∞ p M, denoted by p(M, τ) is the group of all unitaries u (M) such that u 1 L (M, τ). The group is endowed withU the metric topology given by the following∈ U bi-invariant metric− ∈

d(u, v)= u v , u,v (M). k − kp ∈Up In the case M = B(H) for a Hilbert space H and τ is the usual trace, the group is called the Shcatten p-unitary group and is denoted by (H). The group (H)= u (ℓ2) u 1 K(H) endowed with Up U∞ { ∈U | − ∈ } the norm topology is sometimes called the Fredholm unitary group. If M is finite, then p(M, τ)= (M)s is independent of the choice of τ and p. U U Definition 2.3. Let G be a topological group, π : G (H) be a strongly continuous unitary represen- tation. →U (i) For ( =)Q G and ε> 0, a vector ξ H is (Q,ε)-invariant if the following inequality holds: ∅ 6 ⊂ ∈ sup π(x)ξ ξ <ε ξ . x∈Q k − k k k

(ii) We say that π almost has invariant vectors, if it has a (Q,ε)-invariant vector for every compact subset Q of G and every ε> 0. We say that

3 (iii) G has Property (T) if every strongly continuous unitary representation of G which almost has invariant vectors admits a nonzero invariant vector. (iv) G has Property (FH) if every continuous affine isometric action of G on a real Hilbert space has a fixed point. (v) G has Property (OB) if whenever G acts continuously on a metric space by isometries, all orbits are bounded. (vi) G is bounded if for every V N (G), there exists n N and a finite set F G such that G = V nF . ∈ ∈ ⊂ The following results help us understand the implications among the above properties. Theorem 2.4 ([Ro13, Theorem 1.11, Proposition 1.14]). Let G be a Polish group. Then (i) The following three conditions are equivalent. (a) G has property (OB). (b) For every symmetric V N (G), there exist a finite subset F G and k N such that G = (F V )k, . ∈ ⊂ ∈ (c) Every compatible left-invariant metric on G is bounded. (ii) The following three conditions are equivalent. (a) G is bounded. (b) Every left uniformly continuous function on G is bounded. (c) G has property (OB) and every open subgroup is of finite index. Here, a map f from a topological group G to a metric space (X, d) is called left uniformly continuous, N if for every ε > 0, there exists U (G) such that supx∈G d(f(ux),f(x)) < ε holds (this is sometimes called the right uniform continuity∈ in the literature). Also, a topological group G has property (FH) if and only if for every continuous affine isometric action of G on a real Hilbert space, there is a bounded orbit (see e.g., [BHV08, Proposition 2.2.9]). Therefore, the following implications hold:

bounded (OB) (FH) ⇒ ⇒ It is known that none of the above two implications can be reversed in general. Also, (T) (FH) hold for any topological group [De77] (see also [BHV08, Theorem 2.12.4]). In general, there is no⇒ implication between (T) and (OB). As pointed out in [Pe18], any countable discrete group with property (T) does 2 not have property (OB) (consider the isometric action on its Cayley graph). On the other hand, ∞(ℓ ) is bounded [At89] hence has property (FH), but it fails to have property (T) [Pe18, Example 6.6]U (see also Remark 4.36). For more details about the above mentioned properties, we refer the reader to [BHV08] for property (T) and (FH), [Ro09, Ro13] for property (OB) and boundedness (see also [He58]).

3 Unitary Representability

In this section we study a class of Polish groups which arise as closed subgroups of the unitary group. Definition 3.1. We say that a Hausdorff topological group G is (i) strongly unitarily representable (SUR) if G is isomorphic as a topological group to a closed sub- group of (H) for some Hilbert space H equipped with the strong operator topology. U (ii) norm unitarily representable (NUR) if G is isomorphic as a topological group to a closed subgroup of (H) for some Hilbert space H equipped with the norm topology. U (iii) of finite type, if G is isomorphic as a topological group to a closed subgroup of (M)s for a finite type von Neumann algebra M. U (iv) SIN (Small Invariant Neighborhoods) if for every U N (G), there exists V N (G) such that V U and gVg−1 = V for every g G. ∈ ∈ ⊂ ∈

4 If G is a metrizable topological group, then G is SIN, if and only if G admits a compatible two-sided invariant metric. Thus, every NUR group is SIN (the metric is given by the norm). It is also clear that a finite type Polish group is SIN and SUR. The converse need not be true [AMTT17]. However, we do not know whether there exists a connected SUR SIN Polish group which is not of finite type. Recall also that a topological group G has small subgroups if for every neighborhood U of the identity in G, there exists a non-trivial subgroup of G contained in it. Remark 3.2. Every locally compact Polish groups is SUR via the left regular representation (and it is also NUR if the group is discrete). The solution to the Hilbert’s 5th problem (see e.g., [Tao]) asserts that a compact Polish group does not have small subgroups if and only if it is a compact Lie group, in which case the group can be realized as a closed subgroup of (n) for some n N, which is NUR. Thus for compact Polish groups, NUR is equivalent to the absenceU of small subgroups.∈ If a locally compact Polish group is totally disconnected but perfect, then by van Dantzig’s Theorem (see. e.g., [Tao]), it has small subgroups and therefore is not NUR (see Proposition 3.11 below).

3.1 Strong Unitary Representability (SUR) We start by showing the next theorem. Theorem 3.3. Let A be a unital C∗-algebra. Then the following conditions are equivalent. (i) The additive group A with the norm topology is SUR.

(ii) The additive group Asa with the norm topology is SUR. (iii) (A) is SUR. U u (iv) A is finite-dimensional. The proof is inspired by the works of Megrelishvili [Me08] and Galindo [Ga09]. Recall the notion of cotype. We refer the reader to [AK06] for details.

Definition 3.4. A Banach space X has cotype q [2, ) if there exists a constant Cq > 0 such that for every finite set of vectors x ,...,x X the following∈ ∞ inequality holds: 1 n ∈ 1 1 n q n q q xi Cq E εixi , (1) k k ! ≤ i=1 i=1 X X ∞ where (εn)n=1 is a Rademacher sequence, i.e., a sequence of i.i.d.’s on a probability space with P(εn = 1 1) = P(εn = 1) = 2 for all n N. If X has cotype q for some q [2, ), we say that X has nontrivial cotype. − ∈ ∈ ∞ We will also use the following characterization of the SUR, which has been known to experts (we follow [Ga05, Theorem 2.1]).

Theorem 3.5 (Folklore). Let G be a Polish group with the identity 1G. The following conditions are equivalent. (i) G is SUR. (ii) G is isomorphic as a topological group to a closed subgroup of (ℓ2) . U s (iii) There exists a family of continuous positive definite functions on G which generates a neigh- borhood basis of 1 in theF sense that for every V N (G), there exist f ,...,f and open sets G ∈ 1 n ∈F O1,...,On in C such that n 1 f −1(O ) V. G ∈ i i ⊂ i=1 \ (iv) There exists a family of continuous positive definite functions on G which separates 1 and F G closed sets not containing 1G in the sense that whenever A is a closed set in G with 1G / A, there exists an f such that the following inequality holds: ∈ ∈F

sup f(g)

5 We prepare several lemmas. Lemma 3.6. If a Banach space embeds as a topological vector space into L0(Ω,µ) equipped with the measure topology for some probability space (Ω,µ), then it has cotype 2. Proof. See [BL00, Corollary 8.17]. Lemma 3.7. Let X be a separable real Banach space. If X as an additive topological group is SUR, then there exists a probability space (Ω,µ) such that X embeds isomorphically into L0(Ω,µ) equipped with the topology of convergence in measure. Proof. Let π be a topological group isomorhism of X onto an SOT-closed subgroup of (ℓ2). The image π(X) generates an abelian von Neumann algebra A with separable predual. Then weU may identify A with L∞(Ω,µ) where (Ω,µ) is a probability space. For each a X, the map R t π(ta) (A) is a strongly continuous one-parameter unitary group. By Stone The∈ orem, there exists∋ 7→ a unique∈ self-adjoint U operator T (a) affiliated with A such that eitT (a) = π(ta) (t R). Since X is an abelian group, it is easy to see that T : X L0(Ω,µ) is a real linear map. We show that∈ T is a topological group isomorphism from → 0 ∞ X onto its image. Since X,L (Ω,µ) are both metrizable, it suffices to show that for a sequence (an)n=1 n→∞ n→∞ in X, an 0 if and only if T (an) T (0) = 0 in measure. Now we use [RS81, Theorem VIII.21] and thek factk that→ the convergence in measure→ in L0(Ω,µ) is the same as the strong resolvent convergence by [AM12-1, Lemma 3.12]. Then

n→∞ n→∞ a 0 t R [π(ta )= eitT (an) 1 (SOT)] k nk → ⇔ ∀ ∈ n → T (a ) n→∞ 0 (SRT) ⇔ n → T (a ) n→∞ 0 (in measure). ⇔ n → This finishes the proof, because T (X) is then a Polish subgroup of the Polish group L0(Ω,µ), which therefore must be closed in L0(Ω,µ).

∞ Lemma 3.8. Let (an)n=1 be a sequence of bounded self-adjoint operators on H converging to 0 in the norm resolvent topology. Then lim a =0. n→∞ k nk Proof. Let a = λ dE (λ) be the spectral resolution of a (n N). First, we show that sup N a < n σ(a) an n ∈ n∈ k nk . Assume by contradiction that (a )∞ is unbounded. Then for each k N, there exists n N ∞ R n n=1 ∈ k ∈ with n1 < n2 < < nk, such that Ea (R [ k, k]) = 0. Let ξk be a unit vector contained in ··· nk \ − 6 ran Ea (R [ k, k]). Then nk \ − 2 −1 −1 2 λ 2 (an i) ξk (0 i) ξk = d Ea (λ)ξk k k − − − k λ2 +1 k nk k Z|λ|>k 2 k→∞ k 0, ≥ k2+1 6→ which contradicts the assumption. Therefore C = sup N a < , and n∈ k nk ∞ a = (a i)[(0 i)−1 (a i)−1](0 i) (C + 1) (a i)−1 (0 i)−1 n→∞ 0. k nk k n − − − n − − k≤ k n − − − k →

Proposition 3.9. Let A be a separable unital abelian C∗-algebra such that (A) is SUR. Then the U u additive group Asa with the norm topology is SUR.

Proof. Since (A)u is an SUR Polish group, there exists a topological group isomorphism π of (A)u onto an SOT-closedU subgroup of (H) for some separable Hilbert space H. Define for each n U N a continuous function ψ : A C Uby ∈ n sa → ∞ ψ (a)= e−t π(e−iat)ξ , ξ dt, a A, n h n ni ∈ Z0 ∞ where ξn n=1 is a dense subset of the unit ball of H. Since A is abelian, the map Asa a π(e−iat{)ξ }, ξ C is a continuous positive definite function for all t R. This implies that∋ ψ 7→is h n ni ∈ ∈ n

6 a continuous positive definite function as well. By Stone Theorem, for each a Asa there exists a self- ita itT (a) ∈ adjoint operator T (a) on H such that π(e )= e (t R) holds. Note that ψn(a) is then expressed as ∈ ψ (a)= i (T (a) i)−1ξ , ξ . (2) n − h − n ni ∞ We show that the family ψn n=1 generates a neighborhood basis of 0 in Asa. Let τ be the norm topology ′ { } ′ on Asa and τ be the weakest topology on Asa making the functions ψn (n N) continuous. Then τ has a countable basis ∈ J = ψ−1(U ) J N, n ,m N (1 j J) , B  nj mj ∈ j j ∈ ≤ ≤  j=1  \  where U ∞ is a countable basis for R. Thus both τ and τ ′ are second countable, and by definition m m=1   τ ′ τ {holds.} Note that because A is abelian, it holds that for a net (a ) in A and a A , ⊂ j j∈J sa ∈ sa j→∞ j→∞ a a (in τ ′) n N ψ (a ) ψ (a) j → ⇔ ∀ ∈ n j → n j→∞ n N (T (a ) i)−1ξ , ξ (a i)−1ξ , ξ ⇔ ∀ ∈ h j − n ni → h − n ni (∗) j→∞ ξ, η N (T (a ) i)−1ξ, η (a i)−1ξ, η ⇔ ∀ ∈ h j − i → h − i (∗∗) T (a ) SRT T (a) ⇔ j → ∞ Here, we used in ( ) the density of ξn n=1 and the polarization identity, in ( ) the fact that the weak resolvent convergence∗ implies the{ strong} resolvent convergence for self-adjoint∗∗ operators. Therefore, thanks to the fact that SRT is a vector space topology on the set T (Asa) of strongly commuting self- ′ adjoint operators [AM12-1, Lemma 3.12], it follows that τ is a vector space topology on Asa. We then show that τ ′ τ, thus τ ′ = τ. By the first countability of both τ and τ ′, it suffices to show that ⊃ ∞ the notion of sequential convergence to 0 in both topologies agree. Let (an)n=1 be a sequence in Asa ′ n→∞ converging to 0 in τ . Then T (an) T (0) (SRT), whence by [RS81, Theorem VIII.21], we have n→∞ → t R [eitT (a) eitT (0) (SOT)], so that because π is a topological embedding, the last condition is ∀ ∈ → itan n→∞ −1 n→∞ −1 equivalent to t R [ e 1 0], which implies that (an 0) (0 i) in norm. Indeed, ∀ ∈ k − k → itan − → − because each an is bounded, the map t e is norm-continuous. Thanks to the dominated convergence theorem, it then follows that 7→

∞ −1 −1 −t −itan (an i) (0 i) = i e (e 1) dt k − − − k 0 − ∞Z n→∞ e−t e−itan 1 dt 0. ≤ k − k → Z0 ′ Then by Lemma 3.8, lim an = 0 holds. Thus τ = τ , and Asa is SUR by Theorem 3.5. n→∞ k k

Proof of Theorem 3.3. Since A is isomorphic as an additive group to Asa Asa, (i) (ii) holds. (ii) (iv): Assume that dim A = . Let B A be a maximal abelian× -subalgebra⇔ of A. Then by Proposition⇒ 4.40, dim B = holds.∞ Then B ⊂as a Banach space does not have∗ cotype 2 by Lemma 4.38, ∞ sa so that Bsa is not SUR by Lemma 3.6. Therefore Asa( Bsa) is not SUR either. (iii) (iv) We show the contrapositive. Assume that⊃ dim A = . Let B be a maximal abelian C∗- subalgebra⇒ of A. Then dim B = by Proposition 4.40. By∞ Lemma 4.39, B contains a separable infinite-dimensional unital C∗-subalgebra∞ C. If (C) were SUR, then so would be C by Proposition U u sa 3.9. By Lemma 3.7, it follows that Csa would be isomorphic as a topological vector space to a closed 0 subspace of L (Ω,µ), which implies that Csa has cotype 2 by Lemma 3.6. However, this contradicts Lemma 4.38. Therefore (C)u, hence (A)u is not SUR. (iv) (iii): By dim A< U , (A) = U(A) is compact hence SUR. ⇒ ∞ U u U s (iv) (ii) Since Asa is a finite-dimensional Banach space, it is isomorphic as an additive Polish group to some⇒Rn (n R), which is SUR. ∈ In [Pe18, The paragraph after Lemma 5.1] (see also 3.4 of the cited paper), Pestov implicitly men- 2 § tioned that the Fredholm unitary group ∞(ℓ ) (equipped with the norm topology) is SUR (recall by Theorem 3.5 that SUR is equivalent to havingU a family of continuous positive definite functions separating points and closed sets). We remark that this is not the case. We show that it is not the case. More generally, we show the following (take A = B(ℓ2), I = K(ℓ2)).

7 Corollary 3.10. Let A be a unital C∗-algebra, I be a separable two-sided closed ideal of A. Then the group I (A) = u (A) u 1 I equipped with the norm topology is SUR if and only if I is finite-dimensional.U { ∈ U | − ∈ }

Proof. If 1 I, then (A) = (I), which is SUR if and only if dim I < by Theorem 3.3. Thus we ∈ UI U ∞ assume that I is proper and (A) is SUR. We will show that dim I < . Let I be the unitization of I, UI ∞ which can be identified with I + C1 A. First, we observe that the map π : (A) u [u] (I)/T A ⊂ UI ∋ 7→ ∈U is an isomorphism of topological groups, where [u] is the class of u I (A) in thee quotient group. Clearly ∈U ∗ ∗ π is continuous. Let u = u0 + z1 (I), where u0 I and z C. Since u u = (u0 + z1)(u0 + z1)e = (u∗u + zu∗ + zu )+ z 21 = 1, z∈= U 1 holds. Therefore∈ zu ∈(A) and zu 1 = zu I. This shows 0 0 0 0 | | | | ∈ U − 0 ∈ that zu I (A) and π(zu) = [u], whencee π is surjective. For u I (A), if [u] = 1, then there exists w T such∈ U that wu = 1, while u 1= w 1 I. Since 1 / I, this∈ U shows that w = 1 and u = 1 holds. Therefore∈ π is injective. To see that− π is open,− ∈ it suffices (since∈ all the topological groups involved are ∞ n→∞ n→∞ Polish) to show that whenever a sequence (un)n=1 in I (A) satisfies [un] [1], we have un 1. ∞ U → → Fix such (un)n=1 and write un = vn + 1 (vn I, n N). Then for each n N, there exists zn T such that lim z u 1 = lim z v (1 z )∈ = 0.∈ If z 1 does not tend∈ to 0 as n , then∈ there n k n n − k n k n n − − n k n − → ∞ exists a subsequence (z )∞ and z = 1 such that z z k→∞ 0. For each k,ℓ N, nk k=1 0 6 nk − 0 → ∈ z v z v z v (1 z ) + (1 z ) (1 z ) + (1 z ) z v k nk nk − nℓ nℓ k≤k nk nk − − nk k k − nk − − nℓ k k − nℓ − nℓ nℓ k k,ℓ→∞ 0. → Therefore the limit v = lim z v I exists. This implies that k nk nk ∈

0 = lim znk vnk (1 znk ) = v (1 z0) . k→∞ k − − k k − − k

Then because v I and 1 z0 C1, we have z0 = 1, a contradiction. Therefore π is open and consequently, (I∈)/T is SUR.− Since∈ the map (I) u = u + z1 ([u],z) (I)/T T is an embedding U U ∋ 0 7→ ∈U × of the Polish group (I) onto a closed subgroup of (I)/T T, which is SUR, it follows that (I) is U U × U SUR, whence I ise finite-dimensional by Theoreme 3.3. e e e e 3.2 Norme Unitary Representability (NUR) Next, we turn our attention to NUR groups. First, the following result follows from a well-known fact that Banach Lie groups do not have small subgroups (see e.g., [Om97, Theorem III.4.1]). We include a simple proof for the reader’s convenience. Proposition 3.11. Let H be a Hilbert space and G be a topological group. If G has small subgroups, then there is no norm-continuous injective homomorphism from G into (H). U Proof. Assume by contradiction that π : G (H) is a norm-continuous injective homomorphism. Since G has small subgroups, for each U N (→UG), there exists a subgroup G = 1 contained in U. In ∈ U 6 { G} particular, there exists an element gU GU 1G . Since π is injective, its spectrum σ(π(gU )) contains an ∈ \{ } kU iθU element λU T 1 . Then there exists kU N such that Re(λU ) 0 (we may assume that λU = e ∈ \{π } ∈ π ≤ π with 0 < θU < . Then we can choose kU N such that kU ). Since GU is a subgroup | | 2 ∈ 2|θU | ≤ ≤ |θU | kU kU kU of G, we have gU GU U. This shows that lim gU = 1G, whence lim π(gU ) 1 = 0. ∈ ⊂ U∈N (G) U∈N (G) k − k However, for each U N (G), we have ∈ π(gkU ) 1 λkU 1 1, k U − k ≥ | U − |≥ which is a contradiction.

∞ The above proposition shows in particular that the Polish groups of the form n=1 Gn, where each Gn is a non-trivial SUR Polish group, are not NUR, although they are SUR (since SUR passes to countable products). Another immediate consequence is that for an infinite-dimensional vonQ Neumann algebra M, the group (M) is never NUR. U s Corollary 3.12. For a von Neumann algebra M with separable predual, the Polish group (M)s is NUR, if and only if M is finite-dimensional. U

8 Proof. If dim M < , then (M)s = (M)u is compact, so it is NUR. Conversely, if M is infinite- dimensional, let A be∞ a maximalU abelianU subalgebra of M which is an infinite-dimensional abelian von Neumann algebra with separable predual (cf. Proposition 4.40). Then A has a direct summand A0 ∞ ∞ −n −n+1 ∞ isomorphic to either ℓ or L ([0, 1]). Let Jn = (2 , 2 ] (n N). Then the set of all f L ([0, 1]) ∈ ∈ ∞ which are essentially constant on each Jn (n N) forms a von Neumann subalgebra isomorphic to ℓ ∞ N ∈ and (ℓ )s = T has small subgroups. Thus, in both cases the group (A0)s, hence (M)s, has small subgroups.U The conclusion now follows from Proposition 3.11. U U Proposition 3.13. Let C([0, 1], R) be the Banach space of all continuous real valued functions on [0, 1]. Then C([0, 1], R) as an additive Polish group is NUR. Proof. Let G = C([0, 1], R). Let m: G B(L2([0, 1])) be the map given by [m(f)g](x)= f(x)g(x), f 2 → ∞ ∞ 2 ∈ C([0, 1], R), g L ([0, 1]) and x [0, 1]. Fix an enumeration (tn)n=1 of Q [ 1, 1]. Let H = n=1 L ([0, 1]) ∈ ∈ 2πit∩nm−(f) ∞ and π : G (H) be the unitary representation given by π(f) = (e )n=1, f G. Then π is a → U ∈ L ∞ topological group isomorphism of G onto π(G) endowed with the norm topology. To see this, let (fn)n=1 be a sequence in G. If f n→∞ 0, then k nk∞ →

2πitnm(fk) 2πitfk (x) k→∞ π(fk) 1 = sup e 1 = sup max e 1 0. k − k n∈N k − k |t|≤1 x∈[0,1] | − | →

k→∞ k→∞ Also if π(f ) 1 0, then for each t [ 1, 1], one has e2πitm(fk) 1 0, and by the group k k − k → ∈ − k − k → property, the same convergence holds for all t R. Moreover, because the map t e−te−itm(fk ) is norm-continuous, the dominated convergence theorem∈ implies that 7→

∞ k→∞ (m(f ) i)−1 (0 i)−1 = i e−t(e−itm(fk ) 1) dt 0 (norm). k − − − − → Z0

Then by Lemma 3.8, we have lim m(fk) = lim fk ∞ = 0. Therefore π is a topological group k→∞ k k k→∞ k k isomorphism of the Polish group G onto π(G) endowed with the norm topology. In particular, π(G) is norm-separable whence π(G) is a Polish group in the norm topology, of which π(G) is a Polish subgroup. Therefore π(G) is norm-closed. Corollary 3.14. Every separable Banach space is NUR. Proof. Since C([0, 1]) is isomorphic as a topological group to C([0, 1], R) C([0, 1], R), C([0, 1]) is NUR. Since every separable Banach space admits a linear isometric embedding into× C([0, 1]) by Banach–Mazur Theorem (see e.g., [AK06, Theorem 1.4.3]), the claim follows from Proposition 3.13. Remark 3.15. Corollary 3.14 implies that there exist non-locally compact Polish groups which are both SUR and NUR. Namely, ℓp (1 p 2) as an additive group is such an example (the fact that they are SUR is a classical result of Shoenberg≤ ≤ [Sc38]). We do not know if there exists a Polsih group which is both NUR and SUR but fails to be of finite type (cf. [AMTT17]).

2 Next, we show a more interesting example: 1(ℓ ) is NUR. To state the theorem precisely, we need the theory of Fermion Fock space. We briefly recallU some basic notations and facts about it and refer the reader to [Ar18, Chapter 6] for details. Let H be a Hilbert space. For each natural number p 1, let Sp be the symmetric group of degree p 1, which acts on the p-fold tensor product Hilbert space≥ pH by ≥ ⊗ U(σ)ξ ξ = ξ ξ , σ S , ξ , , ξ H. 1 ⊗···⊗ p σ(1) ⊗···⊗ σ(p) ∈ p 1 ··· p ∈ Set 1 A = sgn(σ)U(σ). p p! σX∈Sp p p Then Ap is an orthogonal projection, and its range H = Ap H is called the p-fold anti-symmetric tensor product of H. Put 0H = C. The Fermion Fock∧ space over⊗ H is defined by ∧ ∞ F = F (H)= p(H). f ∧ p=0 M The vector Ω = (1, 0, 0, ) F is called the Fock vacuum. ··· ∈

9 We next define the annihilation and creation operators. For each vector ξ H, we define the creation operator a†(ξ) as a bounded linear operator on F so that ∈

a†(ξ)ψ = p +1A (ξ ψ), p 0, ψ pH. p+1 ⊗ ≥ ∈⊗ The annihilation operator a(ξ) is definedp by a(ξ) = a†(ξ)∗, and hence a(ξ)∗ = a†(ξ). They satisfy the canonical anti-commutation relations (CARs):

a(ξ),a(η)∗ = ξ, η , a(ξ),a(η) = a(ξ)∗,a(η)∗ =0 { } h i { } { } for all ξ, η H. Here, for two bounded linear operators A, B, the anti-commutator A, B = AB + BA ∈ 2 ∗ 2 { } is used. It is known that a(ξ) = (a(ξ) ) = 0 and a(ξ) ∞ = ξ H . Moreover, if H is separable infinite-dimensional and ξ ∞ is an orthogonal basisk for Hk, then k k { n}n=1 a(ξ )∗ a(ξ )∗Ω ℓ 0, n

Γ(u)a(ξ)Γ(u)∗ = a(uξ), u (H), ξ H. ∈U ∈ We can now state the theorem.

2 2 Theorem 3.16. 1(ℓ ) is isomorphic to a closed subgroup of (CAR(ℓ ))u via the second quantization map Γ. In particular,U (ℓ2) is NUR. U U1 We need the following three lemmas.

∞ ∞ Lemma 3.17. If a sequence xn n=1 of bounded operators on H satisfies n=1 xn ∞ < , then the limit { } k k ∞ N P X = lim (1 + xn) = lim (1 + x1)(1 + x2) (1 + xN ) N→∞ N→∞ ··· n=1 Y exists in the operator norm topology and that the inequality

∞ X 1 exp x 1 k − k∞ ≤ k nk∞ − n=1 ! X holds. Proof. For natural numbers M >N, we have

M N N M (1 + xn) (1 + xn) (1 + xn) (1 + xn) 1 − ≤ − n=1 n=1 ∞ n=1 ∞ n=N+1 ∞ Y Y Y Y N M (1 + x ) (1 + x ) 1 ≤ k nk∞ k nk∞ − (n=1 )( ) Y n=YN+1 0 (M,N ). → → ∞

10 N Hence the limit X = limN→∞ n=1(1 + xn) exists in the operator norm topology. Similarly, we obtain ∞ Q ∞ ∞ X 1 (1 + x ) 1 exp( x ) 1 exp x 1 k − k∞ ≤ k nk∞ − ≤ k nk∞ − ≤ k nk∞ − n=1 n=1 n=1 ! Y Y X This completes the proof. Lemma 3.18. For any u, v (ℓ2), we have u v Γ(u) Γ(v) . ∈U k − k∞ ≤k − k∞ Proof. A direct computation shows that

∗ u v ∞ = sup uξ vξ ℓ2 = sup Γ(u) Γ(v) a(ξ) Ω F Γ(u) Γ(v) ∞, k − k kξk=1 k − k kξk=1 k{ − } k ≤k − k which is the desired result.

2 2 Before going to the last lemma, recall that ∞(ℓ ) is a closed subgroup of (ℓ ) with respect to the norm topology. U U

2 1/2 2 Lemma 3.19. Let u ∞(ℓ ), and let 0 <ε<π/2. If Γ(u) 1 ∞ (2 2cos ε) , then u 1(ℓ ) and u 1 2ε hold.∈U k − k ≤ − ∈U k − k1 ≤ Proof. Let

∞ u = λ ξ , ξ , λ =1, ξ ∞ is an orthonormal basis for ℓ2 nh n ·i n | n| { n}n=1 n=1 X be the spectral resolution of u (ℓ2). By Lemma 3.18, we have u 1 (2 2cos ε)1/2, whence ∈ U∞ k − k∞ ≤ − we can write λ as λ = eiθn with some real number θ ε. Put D = n N θ 0 and n n | n| ≤ + { ∈ | n ≥ } D− = n N θn < 0 . { ∈ | } ∞ We claim that θn ε. To show the claim, we may assume that ♯D+ = , and let θn n∈D+ ≤ ∞ { ℓ }ℓ=1 be the subsequence of θ ∞ consisting of those θ for which n D holds. We show, by induction on P { n}n=1 n ∈ + L N, that for every L N, the inequality L θ ε holds. Once we prove this, the claim follows. ∈ ∈ ℓ=1 nℓ ≤ The case L = 1 follows from the definition of θn. We next assume that the inequality holds for some L N. Since P ∈ 1 ∗ ∗ (2 2cos ε) 2 Γ(u) 1 (Γ(u) 1)a(ξ ) a(ξ ) Ω − ≥k − k∞ ≥ − n1 ··· nL+1 F ∗ ∗ ∗ ∗ = a(uξn1 ) a(uξ nL+1 ) Ω a(ξn1 ) a(ξnL+1 ) Ω F ··· − ··· ∗ ∗ ∗ ∗ = a(exp (iθn1 )ξn1 ) a(exp (iθnL+1 )ξnL+1 ) Ω a(ξ n1 ) a(ξnL+1 ) Ω F ··· − ··· L+1

= exp i θnℓ 1 , ! − ℓ=1 X we get L+1 2πm ε θ 2πm + ε − ≤ nℓ ≤ Xℓ=1 for some integer m. But we know that

L π 0 θ ε, 0 θ ε, 0 <ε< . ≤ nℓ ≤ ≤ nL+1 ≤ 2 Xℓ=1 L+1 Hence ℓ=1 θnℓ ε holds, which is the case L + 1. This finishes the proof of the claim. ≤ ∞ Similarly, we can show that θn ε. This together with the claim implies that θn 2ε. P n∈D− ≤ n=1 | |≤ Recall the elementary inequality 0 2 2cos x x2 (x R). Then we obtain P ≤ − ≤ ∈ P ∞ ∞ ∞ iθ 1 u 1 = e n 1 = (2 2cos θ ) 2 θ 2ε, k − k1 | − | − n ≤ | n|≤ n=1 n=1 n=1 X X X so that u (ℓ2) and u 1 2ε hold. ∈U1 k − k1 ≤

11 Proof of Theorem 3.16. We split the proof into two steps. 2 2 ku−1k1 Step 1. For any u 1(ℓ ), we have Γ(u) CAR(ℓ ) and Γ(u) 1 ∞ e 1 holds. In particular, the map ∈ U ∈ k − k ≤ − (ℓ2), (CAR(ℓ2)), , u Γ(u) U1 k·k1 → U k·k∞ 7→ is continuous.     To see this, let

∞ u = λ ξ , ξ , λ =1, ξ ∞ is an orthonormal basis for ℓ2 nh n ·i n | n| { n}n=1 n=1 X be the spectral resolution of u (ℓ2). Since ∈U1 ∞ ∞ (λ 1)a(ξ )∗a(ξ ) λ 1 = u 1 < , k n − n n k∞ ≤ | n − | k − k1 ∞ n=1 n=1 X X ∗ ∞ F we can apply Lemma 3.17 to the sequence ((λn 1)a(ξn) a(ξn))n=1 of bounded operators on to get that the limit − N ∗ 2 Xu = lim 1 + (λn 1)a(ξn) a(ξn) CAR(ℓ ) N→∞ { − } ∈ n=1 Y exists in the operator norm topology, and the inequality X 1 eku−1k1 1 holds. Thus it is k u − k∞ ≤ − enough to show that Γ(u)= Xu. For this, let us consider

Γ(u)a(ξ )∗ a(ξ )∗Ω= a(uξ )∗ a(uξ )∗Ω ℓ1 ··· ℓj ℓ1 ··· ℓj for any natural numbers ℓ <ℓ < <ℓ with j 1. Note that, by the CARs, if n = ℓ, we have 1 2 ··· j ≥ 6 a(ξ )∗a(ξ ) a(ξ )∗ = a(ξ )∗a(ξ )∗a(ξ )= a(ξ )∗ a(ξ )∗a(ξ ), n n · ℓ − n ℓ n ℓ · n n and if n = ℓ, we have

a(ξ )∗a(ξ ) a(ξ )∗ = a(ξ )∗a(ξ )∗a(ξ )+ a(ξ )∗ = a(ξ )∗. n n · ℓ − n ℓ n n n

Hence for all N>ℓj, we obtain

N 1 + (λ 1)a(ξ )∗a(ξ ) a(ξ )∗ a(ξ )∗Ω= λ a(ξ )∗ λ a(ξ )∗ λ a(ξ )∗Ω { n − n n }· ℓ1 ··· ℓj ℓ1 ℓ1 · ℓ2 ℓ2 ··· ℓj ℓj n=1 Y = a(uξ )∗ a(uξ )∗Ω=Γ(u) a(ξ )∗ a(ξ )∗Ω. ℓ1 ··· ℓj · ℓ1 ··· ℓj

Therefore Γ(u)= Xu follows. Step 2. the image Γ (ℓ2) is closed and the map Γ is a homeomorphism. U1 ∞ 2 F To see this, take any sequence  (un)n=1 of 1(ℓ ) and any bounded operator T on satisfying Γ(un) n→∞ U 2 k n→∞ − T 0. It is enough to show that there exists u 1(ℓ ) such that T = Γ(u) and un u 1 0. Byk Lemma→ 3.18, we have ∈ U k − k → n,m→∞ u u Γ(u ) Γ(u ) 0, k m − nk∞ ≤k m − n k∞ −→ 2 n→∞ 2 thus there exists u ∞(ℓ ) such that un u ∞ 0. Since the map Γ: (ℓ )s (F )s is n→∞∈ U k − k → U → U continuous, Γ(u ) Γ(u) (SOT). Thus T = Γ(u) and n → Γ(u∗u ) 1 = Γ(u ) Γ(u) n→∞ 0. k n − k∞ k n − k∞ → ∗ This, together with Lemma 3.19 implies that there exists n0 N such that for any n n0, we have u un 2 ∗ n→∞ 2 ∈ 2 ≥ n→∞ ∈ 1(ℓ ) and u un 1 1 0. Since un 1(ℓ ), we conclude that u 1(ℓ ) and un u 1 0. ThisU finishesk the proof.− k → ∈ U ∈ U k − k →

2 Remark 3.20. We do not know if 1(ℓ ) is SUR. We also remark that Boyer [Bo80, Bo88] studied a family of strongly continuous unitaryU representations of (ℓ2) and (ℓ2). U1 U2

12 4 Boundedness, Property (OB), (T) and (FH)

In this section, we consider structural questions about specific groups of unitaries. As is explained in 2, (T) (FH) and bounded (OB) (FH) hold in general. § ⇒ ⇒ ⇒ 4.1 Boundedness and Property (OB) ∗ 2 We consider the following question: let A be a unital C -algebra. Atkin [At91] has shown that (ℓ )u (hence also (ℓ2) ) is bounded. When is (A) bounded? By a simple argument using spectral theoryU one U s U u can show that in fact (M)u is bounded for every von Neumann algebra and for every unital AF-algebra. This is a folklore resultU (see e.g., [Neeb14, Proposition 1.3]), but because the proof can be used to prove the boundedness for more general unitary groups, we provide a proof.

Proposition 4.1 (Folklore). Let A be a unital AF algebra. Then the group (A) is bounded. U u Remark 4.2. Dowerk [Do18] has shown that the unitary group of a II1 factor or a properly infinite von Neumann algebra has a much stronger property called the Bergman’s property. Namely, whenever W1 W2 is an exhaustive increasing sequence of proper symmetric subsets of (M) (no topological ⊂ ⊂ · · · k U condition is imposed on Wn’s), there exist n, k N such that Wn = (M) holds. The property (OB) is considered to be (and initially introduced as∈ such by Rosendal [Ro09U ]) a topological version of the Bergman’s property. We need a preparation.

2 Definition 4.3. For 0

Lemma 4.4. Let M be a von Neumann algebra on a Hilbert space H. Then (M)u is bounded. More precisely, for any 0

13 were n Z 0 for which [g] = n[ι] = [ιn], there would exist a continuous mapg ˜: [0, 1] R such that exp(2∈ πi\{g˜(t))} = g(e2πit) (t [0, 1)) andg ˜(1) = n = 0. By considering g∗ instead of g if→ necessary, we may assume that n 1. Since∈ g ˜ is continuous, there6 exists by mean value theorem a t R such ≥ 0 ∈ thatg ˜(t ) = 1 . Thus g(e2πit0 ) = exp(2πig˜(t )) = 1. But this contradicts the assumption that g V . 0 2 0 − ∈ Therefore [g]=0. Let fi(1) = zi (1 i k) and let ni Z be given by [f˜i]= ni[ι], where f˜i(z)= fi(z)zi. Fix N Z n ,...,n . We show that≤ ≤f = ιN / F V n∈. Assume by contradiction that f F V n. Choose ∈ \{ 1 k} ∈ ∈ i 1,...,k and g1,...,gn V such that f(z)= fi(z)g1(z) gn(z) (z T). Let z0 = f(1), wi = gi(1), ˜∈{ } ∈ ··· n∈ ˜ ˜ n f(z)= f(z)z0, andg ˜j (z)= gj(z)wj (z T) (1 j n). Then z0 = zi j=1 wj , so that f = fi j=1 g˜j is the product of loops based at 1. Therefore∈ ≤ ≤ Q Q n [f˜]= N[ι] = [f˜i]+ [˜gj ]= ni[ι], j=1 X which contradicts N = n . Therefore G = (C(T)) is not bounded. 6 i U u Next we characterize abelian unital C∗-algebras having bounded unitary groups. Proposition 4.6. Let X be a compact Hausdorff space. Then the following conditions are equivalent. (i) (C(X)) is bounded U u (ii) X is totally disconnected. Remark 4.7. In view of Theorem 4.13 below, Proposition 4.6 follows also from Phillips’ work [Ph95, Corollary 3.3], where he shows that cel(C(X) Mn(C)) < if and only if X is totally disconnected (here, cel stands for the exponential length (see Definition⊗ 4.10∞ below). Note that if X is totally disconnected, then C(X) M (C) is a unital AF algebra and therefore its unitary group (C(X) M (C)) is connected. ⊗ n U ⊗ n Proof of Proposition 4.6. Let G = (C(X)). (ii) (i) Assume that X is totally disconnected.U Then C(X) is a unital AF-algebra. Thus G is bounded by⇒ Proposition 4.1. (i) (ii) We show the contrapositive. Assume that X is not totally disconnected. Then there exists ⇒ a connected component X0 X with X0 2. Let V = u G; u 1 < √2 N (G). Let ⊂ | | ≥ { ∈ kn − k } ∈ F = g1,...,gk G be a finite set and n N. We show that G = F V . Define D = z T; arg(z) ( π ,{π ) . Then} ⊂i log: D eit t ( ∈π , π ) is a continuous6 function. For each g{ ∈V , we have∈ − 2 2 } − ∋ 7→ ∈ − 2 2 ∈ that g X0 C(X0,D). Therefore we can define the map θ C(X0, R) by θg(x) = log(g(x)) (x X0) | ∈ iθg(x) ∈ ∈ such that g(x) = e (x X0). We call g G liftable on X0 if there exists θ C(X0, R) such that g(x) = eiθ(x) (x X ), and∈ denote by G the∈ set of all g G which are liftable on∈ X . Then V G . ∈ 0 1 ∈ 0 ⊂ 1 Note that G1 is a (possibly proper) subgroup of G. Define

iθ d(g)= max θ(x) min θ(x), g = e G1. x∈X0 − x∈X0 ∈

For g G if there are two θ ,θ C(X , R) such that g(x) = eiθ1(x) = eiθ2(x) (x X ), then X ∈ 1 1 2 ∈ 0 ∈ 0 0 ∋ x θ1(x) θ2(x) 2πZ is constant, because X0 is connected and θ1,θ2 are continuous. Thus d(g) is independent7→ − of the∈ choice of a lifting θ. Moreover, we have d(g g ) d(g )+ d(g ) for all g ,g G . 1 2 ≤ 1 2 1 2 ∈ 1 Also observe that if g V , then d(g) < π (use the canonical lift θg). This shows that d(g) < nπ for all g V n. Let C = max∈ d(g )+ nπ. Since X 2, pick x , x X with x = x . By Urysohn’s ∈ 1≤i≤k i | 0| ≥ 0 1 ∈ 0 0 6 1 Lemma, there exists θ C(X, R) with 0 θ(x) 2C (x X) such that θ(x0)=0,θ(x1)=2C. Define g G by g(x) = eiθ(x∈) (x X). Then d≤(g) 2≤C. We show∈ that g / F V n. Indeed, if g F V n, then ∈ 1 ∈ ≥ ∈ ∈ g = gih1 hn for some i 1,...,k and hj V (1 j n). Since g,hj G1 (1 j n) and G1 is a ··· ∈{ } ∈ ≤ ≤ n ∈ ≤ ≤ subgroup of G, we have gi G1 as well, so that d(g) d(gi)+ j=1 d(hj ) C, contradicting d(g) 2C. This shows that g / F V n.∈ Therefore G = F V n. Since≤ F G and n are arbitrary,≤ G is not bounded.≥ ∈ 6 ⊂ P We note the following basic facts. Lemma 4.8. Let G be a locally connected topological group. Then G is bounded if and only if the identity component G0 is bounded and G has finitely many connected components. Proof. See [Ro13, Proposition 1.14] (G has finitely many connected components, if and only if every open subgroup of G has finite index).

14 ∗ Corollary 4.9. Let A be a unital C -algebra. Then (A)u is locally connected. Consequently, (A)u is bounded if and only if (A) has finitely many connectedU components and (A) is bounded. U U u U0 u Proof. Note that if u (A) satisfies σ(u) = T, then u (A). In particular, V = u (A) ∈ U 6 ∈ U0 r { ∈ U | u 1 < r 0(A) for every 0

∗ Definition 4.10 ([Ri91]). Let A be a unital C -algebra. The exponential length cel(u) of u 0(A) is given as follows: ∈ U

n cel(u) = inf hk u = exp(ih1) exp(ihn), h1,...,hn Asa . ( k k ··· ∈ ) k=1 X

We then define the exponential length of A by cel(A) = sup cel(u) u (A) . { | ∈U0 } The exponential length can be defined for non-unital C∗-algebra, but we do not need it here. We need the following results due to Ringrose (see [Ri91, Corollary 2.5, Theorem 2.6 (i)(ii), Corollary 2.7, Proposition 2.9]) and the result due to Lin [Li93] (see also the work by Phillips [Ph94] for the purely infinite simple case). Theorem 4.11 ([Ri91]). Let A be a unital C∗-algebra and n N. ∈ (i) If h A is an element with h π, then cel(eih)= h . ∈ sa k k≤ k k (ii) If u (A) satisfies cel(u) < kπ, then there exist h ,...,h A such that h < π (1 i ∈U0 1 k ∈ sa k ik ≤ ≤ k) and u = eih1 eihk . In particular, cer(u) k. ··· ≤ Theorem 4.12 ([Li93]). Let A be a C∗-algebra of real rank zero. Then A has property weak (FU): The set eia; a A˜ has finite spectrum is norm-dense in (A˜) , where A˜ is the unitization of A. { ∈ sa } U 0 Theorem 4.13. Let A be a unital C∗-algebra. Then the following conditions are equivalent. (i) (A) is bounded. U0 (ii) cel(A) < . ∞ In particular, (A) is bounded if A is a unital C∗-algebra of real rank zero. U0 Proof. (ii) (i) Fix the smallest d N such that cel(A) < dπ. Let u 0(A). We show that for every ⇒ ∈ n(r,d) ∈ U 0

Thus n(r, d)= dn0(r) works. (i) (ii) Assume that 0(A) is bounded, and fix 0

15 Remark 4.14. Let X be a compact Hausdorff space, then the conditions (i) (ii) in Proposition 4.6 are also equivalent to the condition (ii) (C(X)) is bounded. U0 u Indeed, (i) (iii) follows from Corollary 4.9. For (iii) (i), 0(C(X)) is bounded, then by Theorem 4.13, cel(C(X)) ⇒< holds. Then by [Ph95, Corollary 3.3],⇒X isU totally disconnected. In this case, C(X) is a unital AF algebra∞ and therefore (C(X)) = (C(X)) is bounded. U U0 Example 4.15. Although 0(C(T))u is not bounded (Example 4.5 and Remark 4.14), Phillips [Ph93, U 2 Theorem 6.5 (2)] (cf. [Ph92]) showed that for A = C(T) B(ℓ ), cel(A) = 2. Therefore 0(A)u is bounded by Theorem 4.13. Note that A does not have real rank⊗ zero (a more general result canU be found in Kodaka–Osaka’s work [KO95]). This shows that the real rank zero property is not a necessary property for the identity component of the unitary group of a unital C∗-algebra to be bounded. Finally we make another remark about the relationship between the property (OB) introduced and studied by Rosendal [Ro09] and boundedness in the sense of Hejcman [He58] for groups of the form (A). In general, boundedness implies property (OB), but the converse may fail. However, if A has realU rank zero (here, A is said to have real rank zero, if invertible self-adjoint elements are dense in Asa), then they are equivalent (cf. Remark 4.15): Corollary 4.16. Let A be a unital separable C∗-algebra of real rank zero. Then the following three conditions are equivalent. (i) (A) is bounded. U u (ii) (A) has property (OB). U u (iii) (A) has finitely many connected components. U u Proof. (i) (ii) is clear. (ii) (iii)⇒ Assume (iii) does not hold, so Γ = (A)/ (A) is a countable infinite discrete group ( (A) ⇒ U U0 U 0,u is an open subgroup because (A)u is locally connected). Let q : (A) Γ be the quotient map. If Γ has property (T), then Γ is finitelyU generated. Then Γ acts by leftU multip→lication on its Cayley graph with respect to a finite generating set, which has unbounded orbit. If Γ does not have property (T), then it has some affine isometric action on a real Hilbert space with an unbounded orbit. In either case, we have an isometric action α: Γ y X on a metric space with an unbounded orbit. Then β(u)= α(q(u)) (u (A)) defines an isometric action β : (A) y X with an unbounded orbit. Thus (A) does not have property∈U (OB). U U (iii) (i) By Theorem 4.13, (A) is bounded. Then (i) follows from Lemma 4.8. ⇒ U0 4.2 Property (FH) 2 In [Pe18, Example 6.7], Pestov shows that the group 2(ℓ ) does not have property (T), and it was left 2 U unanswered (see [Pe18, 3.5]) whether 2(ℓ ) has property (FH). In this section, we show that it does not have property (FH).§ Actually we showU stronger results. Let M be a properly infinite von Neumann algebra with a normal faithful semifinite trace τ. We will show that the group p(M, τ) (1 p < ) does not have property (FH), hence they do not have property (T) either. We needU two different≤ proofs∞ depending on whether 1 p 2 or2

16 defines an isomorphism from G onto a closed subgroup of (Me). In particular, G is a Polish group of finite type. U We need the following well-known Lemma. Lemma 4.18. Let M be a finite von Neumann algebra with a normal faithful tracial state τ. Then on bounded subsets of M, the topology given by the p-norm agrees with the SOT for any 1 p< . k·kp ≤ ∞ ∞ Proof. It is well-known that 2 agrees with the SOT on bounded subsets of M. Let (xn)n=1 be a bounded sequence in M convergingk·k to 0 in SOT. Then because M is of finite type, the -operation is ∗ n→∞ ∗ SOT-continuous on bounded subsets of M. Thus, xnxn 0 (SOT). Hence (use Stone–Weierstrass p n→∞ p 2 p → p n→∞ Theorem) xn 2 0 (SOT), whence xn 2 2 = τ( xn ) = xn p 0. Converesely, assume that | |p → p k| | k | | k k → p 2 p 2 n→∞ n→∞ n→∞ x = x 2 0. Then x 2 0 (SOT), whence x = ( x 2 ) p 0 (SOT). Therefore k nkp k| n| k2 → | n| → | n| | n| → x = x n→∞ 0, so that x n→∞ 0 (SOT). k nk2 k| n|k2 → n → Proof of Theorem 4.17. We regard L2(M, τ) as a real Hilbert space by taking the real part of the inner product. Define an affine isometric action of G on L2(M, τ) by u x = ux + (u 1), u G, x L2(M, τ). · − ∈ ∈ This action is well-defined because the inequality τ( u 1 2)= τ( u 1 2−p u 1 p) u 1 2−p τ( u 1 p) < | − | | − | · | − | ≤ | − | ∞ | − | ∞ implies u 1 L2(M, τ). Note that the above argument shows that we have − ∈ 1/2 u 1 u 1 2−p u 1 p/2 21−p/2 u 1 p/2, u G. k − k2 ≤ | − | ∞ k − kp ≤ k − kp ∈ n→∞ 2 n→∞ 2 To prove the continuity of the action, let G un u G and L (M, τ) xn x L (M, τ). We see that ∋ → ∈ ∋ → ∈ u x u x = [u x + (u 1)] [ux + (u 1)] k n · n − · k2 k n n n − − − k2 u x ux + u u ≤k n n − k2 k n − k2 u x u x + u x ux + u u ≤k n n − n k2 k n − k2 k n − k2 x x + u u x + u u ≤k n − k2 k n − k2k k∞ k n − k2 x x +21−p/2 u u p/2 x +21−p/2 u u p/2 ≤k n − k2 k n − kp k k∞ k n − kp n→∞ 0, → whence the action is continuous. 2 Since G has property (FH), there is a fixed point x0 L (M, τ). Then we have (u 1)(x0 +1)= 0 for 2 ∈ ∗ − any u G. Thus the projection e of L (M, τ) onto ker (x0 + 1) reduces all elements in G. We show that ∈ ∗ ∗ ∗ e is a τ-finite projection in M. Since ker (x0 +1) = ker( x0 +1 ), e is in M. It is clear that x0e = e, thus ex = e, which means that ex x∗e = e. Then | | − 0 − 0 0 τ(e)= τ(ex x∗e)= τ(x∗ex ) τ(x∗x ) < , 0 0 0 0 ≤ 0 0 ∞ whence e is a τ-finite projection. The map G (M ) , u u →U e s 7→ |ran(e) is a topological group isomorphism onto a subgroup of (eMe). Here, we use the fact that for u G, the U ∈ identity u 1 p = (u 1)e p holds, and that on (eMe), the p-norm agrees with the SOT by Lemma 4.18. Sincek −G andk k(eMe−) arek Polish, the image ofU the map is closed. This finishes the proof. U Corollary 4.19. Let 1 p 2. Then p(M, τ) has property (FH) if and only if M is a finite von Neumann algebra. ≤ ≤ U Proof. If M is a finite von Neumann algebra, then by Lemma 4.4, (M, τ)= (M) is bounded, whence Up U s it has property (FH). Conversely, assume that p(M, τ) has property (FH). We apply Theorem 4.17 to G = (M, τ) to find a τ-finite projection e UM such that (M, τ) u u (e) (M ) is an Up ∈ Up ∋ 7→ |ran ∈ U e s isomorphism onto its image. Since p(M, τ) generates M, the projection e commutes with all elements U ⊥ in M. Thus M is a von Neumann subalgebra of a finite von Neumann algebra Me Ce . Hence M is a finite von Neumann algebra as well. ⊕

17 2 Corollary 4.20. Let 1 p 2, and let G be a closed subgroup of p(ℓ ) with property (FH). Then G is a compact Lie group. ≤ ≤ U

2 2 Proof. Since every finite projection e in B(ℓ ) is of finite rank, the group (B(ℓ )e) is isomorphic to the unitary group of degree dimran(e). Then by Theorem 4.17, G is isomorphicU to a closed subgroup of a finite-dimensional unitary group, so it is a compact Lie group.

4.4 Case 2 duality. See Proposition 4.23. Hopefully this and some of the other lemmas used in the first− proof might be useful in other purposes. The second proof (Theorem 4.28) is simpler and in a sense a refinement of the argument in Theorem 4.17. Below, SOT stands for the -strong operator topology. ∗ ∗ Theorem 4.21. Let 2

2 r 2−r r (u u)x = τ( (u u)x 2 (u u)x (u u)x 2 ) k n − k2 | n − | | n − | | n − | (u u)x 2−r (u u)x r ≤k n − k∞ k n − kr n→∞ 22−r x 2−r u u r x r 0. ≤ k k∞ k n − kpk k2 → This finishes the proof. Proposition 4.23. Let 2 0 and (u 1)z / L2(M, τ). − ∈ We need the following classical result due to Leach.

1 1 Theorem 4.24 ([Le56]). Let 1

M = Q Q (3) j ⊕ 0 Mj∈J where J N is a subset and each Qj (j N) is a type I∞ factor (if J = . The case J = is allowed, in which⊂ case (3) means M = Q is diffuse)∈ and Q is either 0 (in which6 ∅ case (3) means∅ J = and 0 0 { } 6 ∅ M = j∈J Qj is completely atomic) or a diffuse properly infinite semifinite von Neumann algebra. We treat the case J = and Q0 = 0 . The other cases are easier to handle. Set J0 = J 0 and we write L 6 ∅ 6 { } ∪{ }

18 x M as x = (x ) , where x Q (j J ), so that x p = x p (whether or not the both j j∈J0 j j 0 p j∈J0 j p sides∈ are finite). ∈ ∈ k k k k P Let u = T λ de(λ) be the spectral resolution of u. Then for each j J0, uj = T λ dej (λ) is the ∈ π i 2 spectral resolution of uj, where u = (uj )j∈J0 and ej ( )=1Qj e( )= e( )1Qj . Let λ∞ = λ∞+1 = e . For R · ·3 · R −n+ 1 iθn π 2 2 each n N, define λn = e , θn (0, 2 ] so that λn 1 =2 holds. Let A∞ = λ T λ 1 2 ∈ iθ ∈ | − | { ∈ | | − |≥ } and An = e θn+1 θ <θn (n N). Also let pn = e(An), pj,n = ej (An) for each n N and j J . Note{ that| T =≤1 | | } ∈ A is a partition of T. ∈ ∪ {∞} ∈ 0 { } ⊔ k∈N∪{∞} k p F p By u 1 / L (M, τ), we have uj 1Q = . For each j J, define cj = τ(qj ), where qj − ∈ j∈J0 k − j kp ∞ ∈ is a minimal projection in the type I∞ factor Qj, which is independent of the choice of qj . We also set P c = 1. For each J˜ J , we define c ˜ = inf ˜ c . 0 ⊂ 0 J j∈J j ∞ For each j J, the spectral resolution of uj takes the form uj = k=1 λj (k)ej (k), where λj (k) T (k N) ∈∞ ∞ ∈ ∈ and (ej (k))k=1 is a sequence of mutually orthogonal minimal projections in Qj with k=1 ej (k)=1. We further divide the situation into two cases. P P p Case 1. There exists J˜ J such that c ˜ > 0 and ˜ u 1 = . ⊂ J j∈J k j − Qj kp ∞ Define a σ-finite measure space (X,µ) = (J˜ N, c δ ), which satisfies the condition (i) of × P (j,k)∈X j j,k Theorem 4.24. Define f : X C by f(j, k)= λ (k) 1 2 ((j, k) X). Then → | j P− | ∈ p 2 p p f p = λj (k) 1 cj = uj 1Qj p = , k k 2 | − | k − k ∞ ˜ (j,kX)∈X Xj∈J p q 1 and f / L 2 (X,µ) holds. By Theorem 4.24, there exists g L 2 (X,µ) such that fg / L (X,µ). Define ∈ ∈ ∈ 1 z = g(j, k) 2 e (k). | | j (j,kX)∈X Then q q q 2 2 z q = g(j, k) cj = g q < . k k | | k k 2 ∞ (j,kX)∈X q 1 Thus z L (M, τ)+. Let ε > 0 and Xε = (j, k) X g(j, k) 2 ε . Then the following inequality holds. ∈ { ∈ | | | ≥ } q q q 2 2 ε cJ˜ g(j, k) cj g q < . ≤ | | ≤k k 2 ∞ (j,kX)∈Xε (j,kX)∈Xε By cJ˜ > 0, this implies that Xε is a finite set, so that

1 (z)z 2 = g(j, k) c < . k [ε,∞) k2 | | j ∞ (j,kX)∈Xε Therefore 1 (z)z L2(M, τ) holds. Moreover, [ε,∞) ∈ (u 1)z 2 = λ (k) 1 2 g(j, k) c = fg = , k − k2 | j − | | | j k k1 ∞ (j,kX)∈X whence (u 1)z / L2(M, τ) holds. − ∈ Case 2. There is no J˜ J satisfying the condition stated in Case 1. ⊂ p In this case, for any J˜ J, the implication c ˜ > 0 ˜ u 1 < holds. Let J˜ = j ⊂ J ⇒ j∈J k j − Qj kp ∞ { ∈ J c 1 and J˜ = J˜ 0 . Then obviously c 1 > 0, so that u 1 p < . By j 0 J\J˜ P j∈J\J˜ j Qj p | ≤ } p ∪{ } ≥p k − k ∞ uj 1Q = , we also have ˜ uj 1Q = . j∈J0 k − j kp ∞ j∈J0 k − j kp ∞ P P Letp ˜ = p (k N ). ThenP by λ 1 λ 1 , (λ A , k N) and λ 1 2 (λ A ), k j∈J˜0 j,k k k ∞ we have the following inequality:∈ ∪{∞} | − | ≤ | − | ∈ ∈ | − |≤ ∈ P ∞ p p p = u 1 2 2 τ(˜p )+ λ 1 τ(˜p ), ∞ k j − Qj kp ≤ ∞ | k − | k ˜ k=1 jX∈J0 X

19 we need to consider two cases. Case 2-1. τ(˜pk)= for some k N . ∞ ∈ ∪ {∞}˜ ∞ Assume first that τ(pj,k) < for all j J0. In this case, we may find a sequence (en)n=1 of mutually ∞ ∈ ∞ orthogonal projections in M, such that for each n N, en is of the form en = ˜ fj,n, where (J˜n) ∈ j∈Jn n=1 is a sequence of (possibly non-disjoint) subsets of J˜ , and for each j J˜ , f is a subprojection of 0 n Pj,n p Q with 1 τ(e ) 2 (the upper bound τ(e ) 2 can be made∈ possible thanks to the condition j,k ∈ j ≤ n ≤ n ≤ that c 1 (j J˜) and the diffuseness of Q ). Define j ≤ ∈ 0 ∞ − 1 z = n 2 en. n=1 X ∞ q q − 2 q Then z q = n=1 n τ(en) < by q > 2 and τ(en) 2 (n N). Therefore z L (M, τ)+. For each ε> 0,k k ∞ ≤ ∈ ∈ P 2 −1 1[ε,∞)(z)z 2 = n τ(en) < k k −2 ∞ nX≤ε 2 because the right hand side is a finite sum. Thus 1[ε,∞)(z)z L (M, τ) holds. Moreover, thanks to λ 1 λ 1 (λ A ) and τ(e ) 1 (n N), we see that∈ | − | ≥ | k+1 − | ∈ k n ≥ ∈ 2 ∞ 2 − 1 (u 1)z = (u 1 ) n 2 f k − k2 j − Qj · j,n j∈J0 n=1 ˜ X X jX∈Jn 2 ∞

= n−1 (u 1 )f 2 k j − Qj j,nk2 n=1 ˜ X jX∈Jn ∞ = n−1 λ 1 2 dτ(e (λ)f ) | − | j j,n n=1 ˜ Ak X jX∈Jn Z ∞ λ 1 2 n−1τ(e )= . ≥ | k+1 − | n ∞ n=1 X Therefore (u 1)z / L2(M, τ). − ∈ Next, assume that τ(pj,k) = for some j J˜0. Then by using the assumption that Qj is either a ∞ ∈ ∞ type I∞ factor or diffuse, we can find a sequence (en)n=1 of mutually orthogonal projections in Qj with ∞ ∞ 1 ∞ q − 2 q − 2 τ(en) = cj (n N), such that n=1 en = pj,k. Set z = n=1 n en. Then z q = n=1 n cj < , q∈ k k2 ∞ whence z L (M, τ)+. By a similar argument as above, we have 1[ε,∞)(z)z L (M, τ) and ∈ P P ∈ P ∞ (u 1)z 2 λ 1 2 n−1c = . k − k2 ≥ | k+1 − | j ∞ n=1 X Thus (u 1)z / L2(M, τ) holds. − ∈ Case 2-2 τ(˜pk) < for all k N . ∞ ∞ p ∈ ∪ {∞} ∞ p −1 In this case, k=1 λk 1 τ(˜pk)= , hence k=1 λk+1 1 τ(˜pk)= holds by λk+1 1 =2 λk | − | −k∞+ 3 | − | ∞ | − | | − 1 (k N). In view of λ 1 =2 2 (k N), we have | ∈ P | k − | ∈ P λ 1 pτ(˜p ) < . | k+1 − | k ∞ k:τX(˜pk)<1

This implies that the set I = k N 1 τ(˜pk)(< ) is infinite. Let k1 < k2 < be the enumeration { ∈ | ≤ ∞ } ∞ ··· of I. Consider the σ-finite measure space (X,µ) = (N, j=1 aj δj ), where aj = τ(˜pkj ) (j N), which 2 ∈ satisfies the condition (i) of Theorem 4.24. Define f : X C by f(j) = λkj +1 1 (j N). Then by p P→ q | − | ∈ 1 assumption, f / L 2 (X,µ). Then by Theorem 4.24, there exists g L 2 (X,µ) such that fg / L (X,µ). Define ∈ ∈ ∈ ∞ 1 z = g(j) 2 p˜ . | | kj j=1 X

20 q q q 1 2 N 2 Since z q = X g dµ< , z L (M, τ)+ holds. For each ε> 0, consider Jε = j g(j) ε . Then byk kz Lq(M,| | τ), we∞ have (use∈ a 1 (j N)) { ∈ | | | ≥ } ∈ R j ≥ ∈ ∞ q q q q ε g(j) 2 a g(j) 2 a = z < . ≤ | | j ≤ | | j k kq ∞ j=1 jX∈Jε jX∈Jε X 2 Thus Jε is finite and consequently 1 (z)z = g(j) aj < . Moreover, by λ 1 λk+1 k [ε,∞) k2 j∈Jε | | ∞ | − | ≥ | − 1 (λ Ak, k N), we obtain | ∈ ∈ ∞P (u 1)z 2 f(j) g(j) a = k − k2 ≥ | | j ∞ j=1 X by fg / L1(X,µ). Therefore (u 1)z / L2(M, τ) holds. ∈ − ∈ Lemma 4.25. Let 2 0. If sup N (u 1)z < , then (u 1)z L (M, τ) holds. n∈ k n − k2 ∞ − ∈ ∞ Proof. By assumption, the sequence ((un 1)z)n=1 is a bounded sequence in the separable Hilbert space 2 − ∞ 2 L (M, τ), so there exists a subsequence ((unk 1)z)k=1 converging weakly to some y L (M, τ). Let ∞ − ∈ z = λ de(λ) be the spectral resolution of z. For each m N, define em = 1[ 1 ,∞)(z) M. By 0 ∈ m ∈ 2 2 k→∞ assumption,R emz = zem L (M, τ) holds. Let x L (M, τ). Then for each k,m N, unk u (SOT) implies that ∈ ∈ ∈ → (u 1)e z, x k→∞ (u 1)e z, x . |h nk − m i| → |h − m i| On the other hand, by using the trace property, we also have

(u 1)e z, x = (u 1)ze , x = (u 1)z,xe |h nk − m i| |h nk − m i| |h nk − mi| k→∞ y,xe → |h mi| y xe y x . ≤k k2k mk2 ≤k k2k k2 This shows that (u 1)e z, x y x , m N. |h − m i|≤k k2k k2 ∈ 2 Since x L (M, τ) is arbitrary, this implies that (u 1)emz 2 = (u 1)zem 2 y 2. for every m N. By the normality∈ of the trace and (u 1)z e (uk −1)z k(u k1)z −2 (m k )≤k (SOT),k we obtain ∈ | − | m| − | ր | − | → ∞ (u 1)ze 2 = τ(e (u 1)z 2e )= τ( (u 1)z e (u 1)z ) k − mk2 m| − | m | − | m| − | τ( (u 1)z 2) (m ). ր | − | → ∞ Thus (u 1)z y holds. This shows that (u 1)z L2(M, τ). k − k2 ≤k k2 − ∈ 1 1 1 q Proof of Theorem 4.21. Take q > 2 such that p + q = 2 . By Proposition 4.23, there exists z L (M, τ)+ 2 2 ∈ such that 1[ε,∞)(z)z L (M, τ) for every ε> 0 and (u 1)z / L (M, τ) holds. Define an affine isometric∈ action of G on L2(M, τ) by− ∈

u x = ux + (u 1)z, u G, x L2(M, τ) (4) · − ∈ ∈ Thanks to the noncommutative H¨older’s inequality, the action is well-defined (by (u 1)z L2(M, τ)). ∞ − ∈ ∞ We show that the action is continuous. Let (vn)n=1 be a sequence in G converging to v G and (xn)n=1 be a sequence in L2(M, τ) converging to x L2(M, τ). Then by Lemma 4.22, we have ∈ ∈ v x v x = v x + (v 1)z vx (v 1)z k n · n − · k2 k n n n − − − − k2 v (x x) + (v v)x + (v v)z ≤k n n − k2 k n − k2 k n − k2 x x + (v v)x + v v z ≤k n − k2 k n − k2 k n − kpk kq n→∞ 0. → This shows that the action is continuous. If G had property (FH), there would exist a G-fixed point ∗SOT x L2(M, τ). By G (M, τ), there exists u (M) (M, τ) and a sequence (u )∞ in G 0 ∈ 6⊂ Up ∈ U \Up n n=1

21 n→∞ such that un u ( SOT). Then for each n N, we have un x0 = x0, whence (un 1)x0 = (un 1)z. In particular, → ∗ ∈ · − − − sup (un 1)z 2 = sup (un 1)x0 2 2 x0 2 < . n∈N k − k n∈N k − k ≤ k k ∞ However, this is impossible by Lemma 4.25 and (u 1)z / L2(M, τ). Therefore G does not have property (FH). For each n N, define p = ∞ e and−u =∈p p⊥ (n N), where (e )∞ is a sequence of ∈ n k=n+1 k n n − n ∈ k k=1 mutually orthogonal projections in M such that τ(e )=1(k N). Then u (M, τ), and u n→∞ 1 P k n p n (SOT). Therefore, by 1 / (M, τ), (M, τ) does not have∈ property (FH).∈U → − − ∈Up Up Corollary 4.26. Let 2 0, let −tku−1k2 (πt,Ht, ξt) be the cyclic representation of (M, τ)2 associated with the character ϕt(u)= e 2 (u ∞ U ∈ p(M, τ)). Define π = π 1 . Then one can easily show that the trivial representation π0 is weakly U n=1 n contained in π, but because each π is a II factor representation [EI16, Theorem 1.6], π does not contain L t 1 π . Therefore (M, τ) does not have property (T). 0 Up Next, we give the second proof of the absence of property (FH) for p(M, τ). This may be considered to be an analogue of Theorem 4.17. U Theorem 4.28. Let 2

G (M ), u u →U e 7→ |ran (e) defines an isomorphism from G onto a closed subgroup of (eMe). In particular, G is a Polish group of finite type. U In the rest of this section, we fix (M, τ) and G satisfying the hypothesis of Theorem 4.28. We 2 2 represent M on L (M, τ) by the standard representation given by τ. Consider a map ΨG : L (M, τ) 2 → u∈G L (M, τ) given by Ψ (x) = ((u 1)x) , x L2(M, τ). Q G − u∈G ∈ Also fix 2

22 By Lemma 4.30, for each z X, one can define a continuous affine isometric action αz of G on ⊥ ∈ ker(ΨG) (regarded as a real Hilbert space by taking the real part of the inner product) by

α (u)x = ux + (u 1)z, u G, x ker(Ψ )⊥. z − ∈ ∈ G ⊥ Because G is assumed to have property (FH), there exists an αz-fixed point in ker(ΨG) . Let x, y ker(Ψ )⊥ be two α -fixed points. Then for every u G, (u 1)x = (u 1)y, whence x y belongs∈ G z ∈ − − − to ker(ΨG), which implies that x = y. Thus, there exists a unique αz-fixed point, which we denote by f(z) ker(Ψ )⊥. ∈ G Lemma 4.31. The map f : X z f(z) ker(Ψ )⊥ is an Lq-L2 continuous linear map. ∋ 7→ ∈ G ⊥ Proof. By the uniqueness of the fixed point, it follows that f is a linear map. Because X (resp. ker(ΨG) ) is a closed subspace of the Banach space Lq(M, τ) (resp. L2(M, τ)), thanks to the closed graph theorem, ∞ we only have to prove that f is closed. Suppose (zn)n=1 is a sequence in X converging to z X and that n→∞ ∈ f(z ) y 0 for some y ker(Ψ )⊥. Then for each u G, we have k n − k2 → ∈ G ∈ (u 1)(y f(z )) 2 y f(z ) n→∞ 0, k − − n k2 ≤ k − n k2 → while (u 1)(z z) u 1 z z n→∞ 0. Thus, it follows that in the L2-topology, we have k − n − k2 ≤k − kpk n − kq →

(u 1)y = lim (u 1)f(zn) = lim (u 1)zn = (u 1)z. − n→∞ − n→∞ − − − −

This shows that αz(u)y = y. Then by the uniqueness of the αz-fixed point, y = f(z) holds. This finishes the proof. Lemma 4.32. e M is a τ-finite projection. ∈ Proof. Assume by contradiction that τ(e) = . As in the proof of Proposition 4.23, we write M = Q Q as the direct sum of type I ∞factors and a diffuse part, and for each j J, define j∈J j ⊕ 0 ∞ ∈ cj = τ(qj ), where qj is a minimal projection and c0 = 1. Write e = (ej )j∈J . We treat the case J = and L 0 6 ∅ Q0 = 0 . By assumption, τ(ej )= holds. 6 { } j∈J0 ∞ Case 1. There exists j JPsuch that τ(e )= . ∈ 0 j ∞ Then by using either the diffuseness of Q0 or the fact that Qj (j J) is a type I∞ factor, there exists a ∞ ∈ sequence (ej,n)n=1 of mutually orthogonal projections in Qj such that τ(ej,n) = cj for every n N and ∞ ∈ ej = n=1 ej,n. Define n − 1 P z := k 2 e , n N. n j,k ∈ Xk=1 2 q Then for each n N, zn M L (M, τ) L (M, τ) holds. Moreover, because e is the projection onto ⊥ ∈ ∈ ∩ ⊂ 2 ⊥ ker(ΨG) and ej,n en e, we see that (we regard ej,n L (M, τ)) zn M ker(ΨG) , hence zn X ≤ ≤ ⊥ ∈ ∈ ∩ ∈ holds. Thus, because zn ker(ΨG) , the uniqueness of the αzn -fixed point implies that f(zn) = zn. On the other hand, by q>∈ 2, for each n N, we have − ∈ n ∞ q − q − q z = k 2 c k 2 c (< ), k nkq j ≤ j ∞ Xk=1 kX=1 whence sup N z < . On the other hand, n∈ k nkq ∞ n f(z ) 2 = z 2 = k−1c n→∞ . k n k2 k nk2 j → ∞ Xk=1 This contradicts the continuity of f by Lemma 4.31. Therefore, τ(e) < holds. ∞

Case 2. τ(ej ) < for all j J0. Discarding all j ∞J for which∈ τ(e ) = 0, we may assume that τ(e ) = 0 for every j J . In this case, ∈ 0 j j 6 ∈ 0 for each j J, there exists d N such that e = dj e holds, where (e )dj is a family of mutually ∈ j ∈ j p=1 j,p j,p p=1 orthogonal minimal projections in Qj . P Case 2-1. τ(ej ) < . j: cj <1 ∞ P 23 In this case, τ(ej )= holds. Let j1 < j2 < . . . be the enumeration of the set J˜ = j J cj j: cj ≥1 ∞ { ∈ | ≥ 1 . Define m(1,p)= p (1 p d ) and m(k,p)= m(k 1, d )+ p (1 p d ) for k 2, and set } P ≤ ≤ j1 − jk−1 ≤ ≤ jk ≥

n dj k 1 1 − q − 2 zn = m(k,p) cjk ej,p. p=1 Xk=1 X Then z X (n N) as in Case 1, and n ∈ ∈

n djk q − q −1 zn = m(k,p) 2 c cj k kq jk k p=1 Xk=1 X m(n,djn ) ∞ − 2 − 2 = k q k q < , ≤ ∞ kX=1 kX=1 whence sup N z < . On the other hand, n∈ k nkq ∞

n dj k 2 2 −1 − q zn = m(k,p) c cj k k2 jk k p=1 Xk=1 X m(n,djn ) n 1− 2 n→∞ = k−1c q k−1 jk ≥ → ∞ kX=1 Xk=1 by q> 2 and c 1 (k N). Thus, as in Case 1, we get a contradiction. jk ≥ ∈ Case 2-2. τ(ej )= . Let j1 < j2 < . . . be the enumeration of the set J˜ = j J cj < 1 . We j: cj <1 ∞ { ∈ | } may find a sequence (e )∞ of mutually orthogonal projections in M such that for each n N, e is of P n n=1 n f , where J J˜ and f e (it is possible that J J = for some∈n = m) and the form j∈Jn j,n n j,n j n m ⊂ ≤ n − 1 ∩ 6 ∅ 6 that 1 τ(e ) 2 for every n N. Then define z = k 2 e X. By1 τ(e ) 2, we have ≤ P n ≤ ∈ n k=1 k ∈ ≤ n ≤ n P ∞ q − q − q z = k 2 τ(e ) 2 k 2 < , k nkq k ≤ ∞ Xk=1 Xk=1 and n n z 2 = k−1τ(e ) k−1 n→∞ k nk2 k ≥ → ∞ kX=1 kX=1 then as in Case 1, we get a contradiction. Proof of Theorem 4.28. By Lemma 4.29 and Lemma 4.32, e is a τ-finite projection in M such that the map G u u ran(e) (eMe) defines a topological group isomorphism of G onto a subgroup of (eMe)(cf.∋ Lemma7→ | 4.18).∈ Since U G and (eMe) are Polish, the image of this isomorphism is closed. U U 2 Corollary 4.33. Let 2

4.5 Property (T) We have seen that there are many examples of unital separable C∗-algebras whose unitary group with the norm topology are bounded as Polish groups. Consequently, such groups have property (FH). It is interesting to know whether they also have property (T). Pestov [Pe18, Example 6.6] showed that 2 the Fredholm unitary group ∞(ℓ ) does not have property (T), while it has property (OB) by Atkin’s result [At89, At91]. The notionU of property (T) for C∗-algebras has been introduced by Bekka [Be06] and recently Bekka and Ng [BN19] have shown that if G is a locally compact group, then it has property (T) if and only if the full group C∗-algebra C∗(G) has property (T), and if moreover G is an [IN]-group, then this condition is also equivalent to the strong property (T) (in the sense of Ng [Ng14]) for the reduced ∗ ∗ group C -algebra Cr (G). Still, the relationship between the property (T) for the unitary group and the C∗-algebra itself is not well understood. We show here that for many A, its unitary group fails to have

24 property (T). This is an immediate consequence of the notion of the property Gamma of Murray and von Neumann. Recall that a type II1 factor M with normal faithful tracial state τ has property Gamma, if for each x ,...,x M and ε> 0, there exists u (M) such that ux x u <ε and τ(u) = 0. 1 n ∈ ∈U k i − i k2 Theorem 4.34. Let A be a unital separable C∗-algebra admitting a -representation π such that M = ′′ ∗ π(A) is a type II1 factor with separable predual having property Gamma. Then (A)u does not have property (T). In particular, this is the case if A is a unital separable simple infinte-dimensionalU nuclear C∗-algebra with a tracial state. Proof. Let τ be the unique faithful normal tracial state on M = π(A)′′ and we represent M on L2(M, τ) by the GNS representation associated with τ. We writea ˆ (a M) when we view a as an element in L2(M, τ). + ∈ 2 2 Recall that any ϕ N∗ is represented as ϕ = ξϕ, ξϕ for a unique element ξϕ L (M, τ)+. L (M, τ) is equipped with the∈ canonical M-M bimodule structureh · i given by x ξ y = xJy∗Jξ∈(x, y M, ξ L2(M, τ) and Jxˆ = x∗ (x M). Let H = L2(M, τ) C1.ˆ Define a unitary· representation· ρ: (∈A) ∈ (H) by ∈ ⊖ U u →U s ρ(u)ξ = π(u)ξπ(u)∗, ξ H, u (A). c ∈ ∈U

Clearly ρ is strongly continuous. Let Q be a nonempty compact subset of (A)u and ε > 0. By compactness, there exist u ,...u Q such that Q m B(u ; ε ), where UB(u; r) = v (A) 1 m ∈ ⊂ j=1 j 3 { ∈ U | u v < r . Let u Q. Because M has property Gamma, there exists v (M) such that vπ(uj ) k − k } ∈ S ∈U k − π(uj )v 2 < ε (1 j m) and τ(v) = 0. Then ξ =v ˆ H is a unit vector satisfying ρ(uj)ξ ξ 2 < ε (1 kj m). Let≤ u≤ Q and choose 1 j m such that∈ u u < ε . Then k − k ≤ ≤ ∈ ≤ ≤ k − jk 3 ρ(u)ξ ξ ρ(u)ξ ρ(u )ξ + ρ(u )ξ ξ k − k≤k − j k k j − k π(u u )ξπ(u)∗ + π(u )ξπ(u u )∗ + ε ≤k − j k k j − j k 3 < ε.

This shows that ρ almost has invariant vectors. Assume that η H is a ρ-invariant vector. Let η = v η be the polar decomposition of η regarded as a closed and densely∈ defined operator on L2(M, τ) affiliated| | 2 with M. Then v M and η L (M, τ)+. By the ρ-invariance, we have π(u)η = ηπ(u) for every u (A). Since π(∈A)′ = M ′,| this| ∈ implies that for every u (M), uηu∗ = η holds. Therefore ∈U ∈U (uvu∗)(u η u∗)= v η , u (M). | | | | ∈U By the uniqueness of the polar decomposition, it follows that v M ′ = JMJ and u η u∗ = η (u (M)). Write v = Jw∗J for w (M). By the latter identity, the functional∈ η , η is tracial| | on| M| . Therefore∈U by the uniqueness of the∈U representing vector in the positive cone, thereh · | exists| | |i a nonnegative real λ such that η = λ1.ˆ Then η = Jw∗Jλ1ˆ = λwˆ. But for each u (M), the condition uηu∗ = η implies | | ∈ U λuwu\∗ = λwˆ. Thus w (M)= C1. This shows that η C1. But η is orthogonal to 1ˆ by hypothesis, ∈ Z ∈ whence η = 0. This shows that ρ does not admit a nonzero invariant vector. This shows that (A)u does not have property (T). The last assertion is then immediate, because if τ is an extremal pointU in the tracial state space of A, then τ is faithful (by the simplicity of A) and its associated GNS representation πτ generates a hyperfinite type II1 factor, which has property Gamma. Remark 4.35. The same argument shows that (M) as a discrete group does not have property (T) U if M is a type II1 factor with separable predual and with property Gamma. This generalizes Pestov’s result [Pe18, Example 5.5] for (R), where R is the hyperfinite type II factor. It is also immediate by U 1 using Connes–Jones’ notion [CJ85] of property (T) for von Neumann algebras, that for a type II1 factor M with separable predual, if (M) as a discrete group has property (T), then M has property (T). The converse implication is howeverU unclear. Remark 4.36. As we mentioned in the beginning of this section, Pestov [Pe18, Example 6.6] showed 2 that the Fredholm unitary group ∞(ℓ ) does not have property (T). His proof is based on his result [Pe18, Theorem 6.3] that a topologicalU group which is both NUR and amenable must be maximally almost 2 periodic, and the fact that ∞(ℓ ) is amenable (in fact, it is extremely amenable by Gromov–Milman’s result [GM83]). We remarkU that the same type of argument as the one in Theorem 4.34 provides another 2 elementary proof of the absence of property (T) for ∞(ℓ ). This is likely to be known or implicit in the literature, but we include the proof for the reader’sU convenience. Consider the strongly continuous unitary representation π : (ℓ2) (S (ℓ2)) given by U∞ →U 2 π(u)x = uxu∗, u (ℓ2), x S (ℓ2). ∈U∞ ∈ 2

25 ∞ 2 Then π almost has invariant vectors (e.g. if (ξn)n=1 is an orthonormal basis for ℓ , then the rank one n→∞ 2 projection pn onto Cξn (n N) satisfies π(u)pn pn 2 0 (u ∞(ℓ )), but it is straightforward to see that there is no nonzero∈ π-invariantk vector. − k → ∈U

Let M ∞ be the UHF algebra of type 2∞ (the CAR algebra). Then (M ∞ ) is bounded by Propo- 2 U 2 u sition 4.1, hence it has property (FH). We know that (M2∞ )u does not have property (T) by Theorem 4.34. The absence of the property (T) can also be seenU from the existence of certain type III factor representations. The reason for presenting an alternative proof is our hope that this method could be used to show the absence of property (T) for C∗-algebras without tracial states. We will not pursue this approach further in this paper.

Proposition 4.37. The group (M ∞ ) does not have property (T). U 2 u For the proof, we recall the construction of the Powers factors Rλ (0 <λ< 1). For 0 <λ< 1, let 1 0 h = 1 and ψ = Tr(h ), which is a normal faithful state on M (C). Let ϕ = ∞ ψ λ 1+λ 0 λ λ λ · 2 λ n=1 λ   ∞ be the infinite tensor product state on M2 and let (Hλ, πϕλ , ξϕλ ) be the associated GNS representationN ∞ ∞ ′′ of M2 . Then Rλ = πϕλ (M2 ) is called the Powers factor (of type IIIλ). Let Jψλ (resp. ∆ψλ ) be the modular conjugation operator (resp. the modular operator) associated with ψλ defined on its GNS representation. Similarly we let Jϕλ and ∆ϕλ be the corresponding modular objects associated with ϕλ. Then ∞ ∞

Jϕλ = Jψλ , ∆ϕλ = ∆ψλ . n=1 n=1 O O ∞ We use the notation Λψλ : M2(C) Hψλ and Λϕλ : M2 Hϕλ for the canonical dense embedding. For each n N, let λ =1 1 . Set→ → ∈ n − n+1

ϕn = ϕλn , ψn = ψλn , Hn = Hϕn , πn = πϕn . Then for each a,b,c,d C, the following formulas hold: ∈ 1 1 − 2 2 a b a λ b ∆ψ Λψλ =Λψλ 1 , λ c d λ 2 c d     − 1 a b a λ 2 c Jψλ Λψλ =Λψλ 1 . c d λ 2 b d    

∞ 1 Proof of Proposition 4.37. Let A = M2 A0 = alg M2(C). Let λn = 1 n+1 (n N) and consider ⊃ − ′′ ∈ the GNS representation πn : A B(Hn) of the Powers state ϕλn . Then Mn = πn(A) is the Powers factor of type III . Let ξ H be→ the GNS vector ofNϕ . Let ∆ = ∆ , J = J , H = ∞ H , λn n ∈ n λn n ϕλn n ϕλn n=1 n π = ∞ π : A B(H) and ξ˜ = (0,..., 0, ξ , 0,... ) H. Each H is endowed with the standard n=1 n → n n ∈ n L Mn Mn bimodule structure given by −L a ξ b = aJ b∗J ξ, a,b M , ξ H . · · n n ∈ n n ∈ n Define a unitary representation ρ: (A) (H) by U →U ρ(u)(η )∞ = (π (u)η π (u)∗)∞ , u (A), (η )∞ H. (5) n n=1 n n n n=1 ∈U n n=1 ∈ We show that ρ almost has invariant vectors. To this end, we first show that

lim πn(a)ξn ξnπn(a) =0, a A0. (6) n→∞ k − k ∈ Since A is spanned by elementary tensors, it suffices to show (6) for elements of the form a = a 0 1 ⊗ ak 1 A0, where a1,...,ak M2(C) and k N. By construction, we may identify Hn as the ··· ⊗ ··· ∈ ∈ ∞ ∈ ⊗k ∞ infinite tensor product Hilbert space (M2(C), ξψλ ), and ξn = ξ ξψλ . We have m=1 n ψλn ⊗ j=k+1 n N k ∞ N π (a)ξ = Λ (a ) ξ , n n ψλn j ⊗ ψλn j=1 O j=Ok+1 k ∞ 1 2 ξnπn(a)= ∆ πn(aj )ξψλ ξψλ . ψλn n ⊗ n j=1 O j=Ok+1

26 Thus

k k 1 2 πn(a)ξn ξnπn(a) = πψλ (aj )ξψλ ∆ πn(aj )ξψλ . k − k n n − ψλn n j=1 j=1 O O

n ⊗n Because the map H (η1,...,ηn) η 1 ηn H is continuous, in order to prove (6) for ∋ 7→ ⊗···⊗ ∈ 1 2 n→∞ a = a1 ak, it suffices to show that πψλ (b)ξψλ ∆ πψλ (b)ξψλ 0 for all b M2(C). ⊗···⊗ k n n − ψλn n n k → ∈ p q n→∞ Let b = M (C). Then by λ 1, we have r s ∈ 2 n →  

1 1 1 − 1 n→∞ π (b)ξ ∆ 2 π (b)ξ 2 = (1 λ 2 )2 r 2 + λ (1 λ 2 )2 q 2 0. ψλn ψλn ψλ ψλn ψλn n n n k − n k 1+ λn { − | | − | | } → This shows (6). Let K (A) be a nonempty norm-compact subset and ε > 0. Because K is totally bounded, there exist u ,...,u⊂ U K such that K m B(u ; ε ), where B(u; r)= v (A); v u < 1 m ∈ ⊂ i=1 i 3 { ∈U k − k r (u (A)). Since A0 is norm-dense in A, (A0) is norm-dense in (A) (well-known). For each } ∈ U U S ε Um 2 i =1,...m, there exists vi (A0) such that ui vi < 3 . Then K i=1 B(vi; 3 ε). By (6), for each ∈U k − k ⊂ ∗ ε i =1,...,m, there exists ni N such that πn(vi)ξn ξnπn(vi) = πn(vi)ξnπn(vi) ξn < 3 holds for all n n . Let n = max ∈ n . Then fork each i =1−,...,m, k k S − k ≥ i 1≤i≤m i

ρ(v )ξ˜ ξ˜ = π (v )ξ π (v )∗ ξ < ε . k i n − nk k n i n n i − nk 3 If u K, then u v < 2 ε for some 1 i m. Thus ∈ k − ik 3 ≤ ≤ ρ(u)ξ˜ ξ˜ (ρ(u) ρ(v ))ξ˜ + ρ(v )ξ˜ ξ˜ k n − nk≤k − i nk k i n − nk π (u v )ξ˜ π (u)∗ + π (v )ξ˜ (π (u∗ v∗) + ε ≤k n − i n n k k n i n n − i k 3 2 u v + ε < ε. ≤ k − ik 3 Since u K is arbitrary, ρ almost has invariant vectors. ∈ ∞ Next, we show that ρ does not have a nonzero invariant vector. Assume by contradition that η = (ηn)n=1 ∗ ∈ H is a nonzero ρ-invariant vector. Then there exists n N such that ηn = 0, and πn(u)ηnπn(u) = ηn for all u (A). We may assume that η = 1. Let τ =∈ η , η (M 6 )+. Then for each u, v (A), ∈U k nk n h · n ni ∈ n ∗ ∈U τ (π (u)π (v)) = π (u)π (v)η , η = π (u)η π (v), η n n n h n n n ni h n n n ni = π (u)η , η π (v)∗ = π (u)η , π (v)∗η h n n n n i h n n n ni = π (v)π (u)η , η = τ (π (v)π (u)). h n n n ni n n n Since span( (A)) = A, we have τ (ab) = τ (ba) for all a,b π (A), whence for all a,b M by the U n n ∈ n ∈ n normality of τn. This shows that τn is a tracial state. This contradicts the fact that Mn is of type IIIλn . Therefore ρ does not have a nonzero invariant vector. Hence (A) does not have property (T). U Appendix

For the reader’s convenience, we include proofs of the three well-known results we need in 3.1. § Lemma 4.38. Let X be an infinite compact Hausdorff space. Then then the Banach space C(X) does not have nontrivial cotype. Proof. We consider two cases separately. Case 1. X has a connected component X consisting of more than two points, say x, x′ X (x = x′), 0 ∈ 0 6 then since X0 is a normal space, by Urysohn’s Lemma, there exists a continuous function f : X0 [0, 1] ′ → satisfying f(x)=0, f(x ) = 1. Because X0 is connected, this shows that f(X0) = [0, 1]. By Tietze’s extension Theorem, f can be extended to a continuous function on X, which we still denote by f. Then the dual map f ∗ : C[0, 1] C(X) is an isometric embedding. Since C[0, 1] does not have nontrivial cotype, C(X) does not either.→

27 Case 2. X is totally disconnected. Since X is compact, there exist at most finitely many isolated points in X. Thus X has a connected component X0 which is totally disconnected and perfect. Since X0 is compact Hausdorff, X0 is therefore zero-dimensional, meaning that it has a neighborhood basis consisting of clopen subsets. Then because X0 is infinite, for each n N, there exists disjoint clopen sets ∞ ∈ U1,...,Un X0. Define xi = 1Ui C(X) (1 i n). Let (εn)n=1 be a Rademacher sequence and let q [2, ).⊂ Then ∈ ≤ ≤ ∈ ∞ 1 1 n q n q q 1 xi = n q , E εixi =1. k k ! i=1 i=1 X X Since n N is arbitrary, this shows that C(X) cannot have cotype q, which shows the lemma. ∈ Lemma 4.39. Let A be an infinite-dimensional unital abelian C∗-algebra. Then there exists a separable infinite-dimensional unital C∗-subalgebra B of A. Proof. Let X be the spectrum of A and identify A = C(X). X is then an infinite compact Hausdorff space. If X is not totally disconnected, then as in the proof of Lemma 4.38, C(X) contains an isometric copy of C[0, 1]. If X is totally disconnected, then because X is compact, there are at most finitely many isolated points, so that there exists an infinite closed totally disconnected perfect subset X1 of X. Then ∞ X1 is zero-dimensional, and there exists a countable disjoint family Un n=1 of nonempty clopen subsets ∞ −n { } ∗ of X1. Then the self-adjoint element x = n=1 2 1Un C(X) has infinite spectrum. Thus B = C (x, 1) is a separable infinite-dimensional C∗-subalgebra of A. ∈ P Proposition 4.40. Let A be an infinite-dimensional unital C∗-algebra and B be a maximal abelian - subalgebra of A. Then B is infinite-dimensional. ∗ Proof. This can be found e.g., in [KRI, Exercise 4.6.12]. We show the contrapositive. Assume that B is finite-dimensional. Let I = 1, 2,...,n , X be the spectrum of B, which is a finite set, say x1,...,xn . Let p =1 B. Then p{ ,...,p }is an orthogonal family of projections in B and n p{ = 1. Since} i {xi} ∈ { 1 n} i=1 i B is maximal abelian, we have piApi = Cpi for every 1 i n. In particular, there exists a state ϕi on A such that p xp = ϕ (x)p for all x A, i I. Define≤ a relation≤ on I by P i i i i ∈ ∈ ∼ i j p Ap = 0 . ∼ ⇔ j i 6 { } Then is an equivalence relation. Only the transitivity is not entirely obvious. If i j and j k, then p xp =∼ 0 and p yp = 0 for some x, y A. Note that for a A, ∼ ∼ j i 6 k j 6 ∈ ∈ (p ap )∗(p ap )= p a∗p ap = ϕ (a∗p a)p =0, (p ap )(p ap )∗ = ϕ (ap a∗)p . j i j i i j i i j i 6 j i j i j i j ∗ ∗ Therefore pj api = 0 if and only if ϕi(a pj a) = 0, if and only if ϕj (apia ) = 0. Then z = pkypjxpi pkApi satisfies 6 6 6 ∈

∗ ∗ ∗ ∗ ∗ z z = pix pj y pkypj xpi = pix ϕj (y pky)pjxpi = ϕ (x∗p x)ϕ (y∗p y)p =0 i j j k i 6 by p xp =0 = p yp . Thus i k. Next we show j i 6 6 k j ∼

Claim. If pj Api = 0 , then dim(pj Api) = 1. Fix y A such that6 {p}yp = 0. Then for every x A, we have ∈ j i 6 ∈ ∗ ∗ (pj xpiy pj )ypi = pjx(piy pj ypi) ϕ (xp y∗)p yp = ϕ (y∗p y)p xp , ⇔ j i j i i j j i whence (by ϕ (y∗p y) = 0) i j 6 p xp = ϕ (y∗p y)−1ϕ (xp y∗)p yp Cp yp . j i i j j i j i ∈ j i

Let I = I1 Id be the decomposition of I into equivalence classes. For each 1 k d ⊔···⊔ ∼ ∗ 1 ≤ ≤ and i, j Ik, fix an element xi,j A with pj xi,j pi = 0. Then pj xi,j pi = ϕi(xi,j pj xi,j ) 2 pi, so that p x p =∈ u p x p gives the polar∈ decomposition of6 p x p , where| | j i,j i ij | i i,j i| j i,j i ∗ − 1 u = ϕ (x p x ) 2 p xp A. i,j i i,j j i,j j i ∈

28 Then pjApi = Cui,j if i j and pj Api = 0 otherwise. It is also straightforward to check that ∼ { } p . Therefore for every x A we ui,j; i, j Is is a system of matrix units in qsAqs with qs = i∈Is i have{ ∈ } ∈ d P x = pj xpi = ai,j (x)ui,j s=1 i,jX∈I X i,jX∈Is for some a (x) C, and A = d M (C) is finite-dimensional. i,j ∈ ∼ s=1 |Is| L Acknowledgments

HA is supported by JSPS KAKENHI 16K17608 and Grant for Basic Science Research Projects from The Sumitomo Foundation. YM is supported by KAKENHI 26800055 and 26350231.

References

[AK06] F. Albiac and N. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics 233, Springer (2006). [AM12-1] H. Ando and Y. Matsuzawa, Lie Group-Lie Algebra Correspondences of Unitary Groups in Finte von Neumann Algebras, Hokkaido J. Math. 41 (2012), no. 1, 31–99. [AM12-2] H. Ando and Y. Matsuzawa, On Polish groups of finite type, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 389-408. [AM15] H. Ando and Y. Matsuzawa, The Weylvon Neumann theorem and Borel complexity of unitary equivalence modulo compacts of self-adjoint operators, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 6, 1115-1144. [AMTT17] H. Ando, Y. Matsuzawa, A. Thom and A. T¨ornquist, Unitarizability, Maurey–Nikishin fac- torization and Polish groups of finite type, to appear in J. Reine Angew. Math.. DOI:10.1515/crelle- 2017-0047 [Ar18] A. Arai, Analysis on Fock spaces and mathematical theory of quantum fields, World Scientific (2018). [At89] C. J. Atkin, The Finsler geometry of certain covering groups of operator groups Hokkaido Math. J. 18 (1989), no. 1, 45-77. [At91] C. J. Atkin, Boundedness in uniform spaces, topological groups, and homogeneous spaces, Acta Math. Hungar. 57 (1991), no. 3-4, 213–232. [Ba] W. Banaszczyk, Additive subgroups of topological vector spaces, Lecture Notes in Mathematics, vol. 1466 (1991), Springer-Verlag. [Be03] B. Bekka, Kazhdan’s Property (T) for the unitary group of a separable Hilbert space, [Be06] B. Bekka, Property (T) for C∗-algebras, Bull. London Math. Soc. 38 (2006), no. 5, 857867. [BN19] B. Bekka and C-K. Ng, Property (T) for locally compact groups and C-algebras, Proc. Amer. Math. Soc. 147 (2019), no. 5, 21592169. [BHV08] B. Bekka, P. de la Harpe, and A. Valette. Kazhdan’s Property (T). Cambridge University Press, Cambridge, 2008. [BL00] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI (2000).

[Bo80] R. P. Boyer, Representation theory of the Hilbert-Lie group U(H)2, Duke Math. J. 47 (1980), no. 2, 325–344.

[Bo88] R. P. Boyer, Representation theory of U1(H) in the symmetric tensors. J. Funct. Anal. 78 (1988), no. 1, 13–23.

29 [CJ85] A. Connes and V. Jones, Property T for von Neumann algebras, Bull. London Math. Soc. 17 (1985), no. 1, 57–62. [CT18] A. Carderi and A. Thom, An exotic group as limit of finite special linear groups, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 1, 257-273 [De77] P. Delorme, 1-cohomologie des repr´esentations unitaires des groupes de Lie semi-simples et r´esolubles. Produits tensoriels continus de repr´esentations, Bull. Soc. Math. France 105 (1977), no. 3, 281-336. [Do18] P. Dowerk, The Bergman property for unitary groups of von Neumann algebras, arXiv:1804.09515. [EI16] T. Enomoto and M. Izumi, Indecomposable characters of infinite dimensional groups associated with operator algebras, J. Math. Soc. Japan 68 (2016), no. 3, 1231–1270. [Ga05] S. Gao, Unitary group actions and Hilbertian Polish metric spaces. in Logic and its applica- tions, Contemp. Math. 380, Amer. Math. Soc. Providence, RI, (2005), 53–72. [Ga09] J. Galindo, On unitary representability of topological groups, J. Math. Z. (2009), no. 1, 211–220. [GM83] M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), no. 4, 843-854. [Gu72] A. Guichardet, Sur la cohomologie des groupes topologiques. II, Bull. Sci. Math. (2) 96 (1972), 305-332. [dlH73] P. de la Harpe, Moyennabilit´ede quelques groupes topologiques de dimension infinie, C. R. Acad. Sci. Paris S´er. A-B 277 (1973), A1037-A1040. [dlH78] P. de la Harpe, Moyennabilite du groupe unitaire et propriete P de Schwartz des algebres de von Neumann, Lecture Notes in Math., 725, Springer, (1979), 220–227. [He58] J. Hejcman, Boundedness in uniform spaces and topological groups, Czechoslovak Math. J. 9 (84) (1959), 544–563. [HC75] W. Here and J. P. R. Christensen, On the existence of pathological submeasures and the con- struction of exotic topological groups, Math. Ann. 213 (1975), 203-210. [KRI] R. Kadison, J. Ringrose, Fundamentals of the theory of operator algebras. Vol. I. Elementary theory, American Mathematical Society, Providence, RI (1997). [Ka67] D. A. Kaˇzdan, On the connection of the dual space of a group with the structure of its closed subgroups (Russian), Funkcional. Anal. i Priloˇzen 1 (1967), 71–74. [KO95] K. Kodaka and H. Osaka, Real rank of tensor products of C∗-algebras. Proc. Amer. Math. Soc. 123 (1995), no. 7, 2213–2215. [Le56] E. Leach, On a converse of the H¨older inequality, Proc. Amer. Math. Soc. 7 (1956), 607-608. [Li93] H. Lin, Exponential rank of C∗-algebras with real rank zero and the Brown-Pedersen conjectures, J. Funct. Anal. 114 (1993), no. 1, 1–11. [Me08] M. Megrelishvili, Reflexively representable but not Hilbert representable compact flows and semi- topological semigroups, Colloq. Math. 110 (2008), no. 2, 383–407. [Neeb14] K. H. Neeb, Unitary representations of unitary groups, Dev. Math., 37, Springer (2014), 197– 243. [Ng14] C-K. Ng, Property (T) for general C∗-algebras, Math. Proc. Cambridge Philos. Soc. 156 (2014), no. 2, 229239 [Om97] H. Omori, Infinite-Dimensional Lie Groups, Translations of Math. Monographs 158 Amer. Math. Soc. (1997). [Pe18] V. Pestov, Amenability versus property (T) for non locally compact topological groups, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7417–7436.

30 [Ph91] N. C. Phillips, Simple C-algebras with the property weak (FU), Math. Scand. 69 (1991), no. 1, 127-151. [Ph92] N. C. Phillips, The rectifiable metric on the space of projections in a C∗-algebra, Internat. J. Math. 3 (1992), no. 5, 679–698. [Ph93] N. C. Phillips, A survey of exponential rank. Contemp. Math., 167, Amer. Math. Soc., Providence, RI, (1994), 352–399. [Ph94] N. C. Phillips, Approximation by unitaries with finite spectrum in purely infinite C∗-algebras, J. Funct. Anal. 120 (1994), no. 1, 98-106. [Ph95] N. C. Phillips, Exponential length and traces, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 1, 13–29. [Ph00] N. C. Phillips, A classification theorem for nuclear purely infinite simple C∗-algebras, Doc. Math. 5 (2000), 49–114.

[Ph02] N. C. Phillips, Real rank and exponential length of tensor products with O∞, J. Operator Theory 47 (2002), no. 1, 117–130. [Ri91] J. R. Ringrose, Exponential length and exponential rank in C∗-algebras, Current Topics in Oper- ator Algebras (H. Araki etc., ed.), World Scientific (1991), 395–413. [Ri92] J. R. Ringrose, Exponential length and exponential rank in C∗-algebras, Proc. Edinburgh Math. Soc. A., (1992), no. 1–2, 55–71. [Ri71] J. R. Ringrose, Compact non-self-adjoint operators, Van Nostrand Reinhold Co (1971). [Ro09] C. Rosendal, A topological version of the Bergman property, Forum Math. 21 (2009), no. 2, 299–332. [Ro13] C. Rosendal, Global and local boundedness of Polish groups, Indiana Univ. Math. J. 62 (2013), no. 5, 1621–1678. [Ro18] C. Rosendal, Coarse geometry of topological groups, available at author’s webpage. [RS81] M. Reed and B. Simon, Methods of modern mathematical physics, vol I: Functional Analysis, Academic Press (1981). [Sc38] I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), no. 3, 522–536. [Tak] M. Takesaki, Theory of Operator Algebras. I (2002), II (2003) and III (2003). Springer-Verlag. [Tao] T. Tao, Hilbert’s fifth problem and related topics, Graduate Studies in Mathematics, 153 (2014), American Mathematical Society.

Hiroshi Ando Department of Mathematics and Informatics, Chiba University, 1-33 Yayoi-cho, Inage, Chiba, 263- 8522, Japan [email protected]

Yasumichi Matsuzawa Department of Mathematics, Faculty of Education, Shinshu University, 6-Ro, Nishi-nagano, Nagano, 380- 8544, Japan [email protected]

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