Comptes Rendus Mathématique

Toshizumi Fukui

Atsufumi Honda

Masaaki Umehara

Comptes Rendus
Mathématique
Toshizumi Fukui, Atsufumi Honda and Masaaki Umehara
Convexity of space-like projections of submanifolds with co-dimension 2 in
Lorentz–Minkowski space
Volume 363 (2025), p. 109-113
Online since: 6 March 2025
https://doi.org/10.5802/crmath.704
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 109-113
https://doi.org/10.5802/crmath.704
Research article / Article de recherche
Geometry and Topology / Géométrie et Topologie
Convexity of space-like projections of
submanifolds with co-dimension 2 in
Lorentz–Minkowski space
Convexité des projections de sous-variétés de
co-dimension 2 dans l’espace de Lorentz–Minkowski
Toshizumi Fukui a, Atsufumi Honda b andMasaaki Umehara c
a Department of Mathematics, Saitama University, Saitama 338-8570, Japan
b Department of Applied Mathematics, Yokohama National University, Yokohama
240-8501, Japan
c Department of Mathematical and Computing Sciences, Institute of Science Tokyo,
Tokyo 152-8552, Japan
E-mails: tfukui@rimath.saitama-u.ac.jp, honda-atsufumi-kp@ynu.ac.jp,
umehara@comp.isct.ac.jp
Abstract. In this paper, we give a necessary and sufficient condition that any space-like projection of a sub-
manifold with co-dimension 2 in Lorentz–Minkowski space is locally strictly convex, and give its applications.
Résumé. Dans cet article, nous donnons une condition nécessaire et suffisante pour que toute projection
d’une sous-variété de co-dimension 2 dans l’espace de Lorentz–Minkowski soit localement strictement
convexe, et nous donnons ses applications.
Funding. The first and the third author were supported in part by Grant-in-Aid for Scientific Research (C)
No. 23K03106 and (B) No. 23K20794, respectively. The second author was supported in part by Grant-in-Aid
for Scientific Research (C) No. 24K06709 and (B) No. 23K20794.
Manuscript received 30 July 2024, revised 13 November 2024, accepted 18 November 2024.
1. Introduction
Let Rn+11 be Lorentz–Minkowski (n+1)-space with signature (+·· ·+−) and set Rn0 :=
{
(x,0) ; x ∈ Rn}
which can be identified with Euclidean n-space. We fix a connected (n−1)-manifold M n−1 (n ≥ 2)
without boundary. It is well-known that an immersion f : M n−1 → Rn0 is locally strictly convex at
p ∈ M n−1 if and only if the second fundamental form α fp of f at p is definite. The following fact is
well-known:
Fact 1 ([3]). If M n−1 (n ≥ 3) is compact and f is locally strictly convex at each point of M n−1, then
M n−1 is diffeomorphic to the (n −1)-sphere Sn−1 and f (M n−1) bounds a convex domain in Rn0 .
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
110 Toshizumi Fukui, Atsufumi Honda and Masaaki Umehara
Let F : N m → Rn+11 be an immersion defined on an m-manifold N m . A point p ∈ N m is called
time-like, light-like or space-like if the restriction of the canonical Lorentzian metric 〈·, ·〉 of Rn+11
to dF (Tp N m) is Lorentzian, degenerate or Riemannian, respectively. A point p ∈ N m is said to be
cocausal if it is not time-like. Moreover, the immersion F is said to be cocausal (resp. space-like)
if all points on N m are cocausal (resp. space-like). We denote by ∇ the Levi-Civita connection on
Rn+11 . Let F : M
n−1 → Rn+11 be an immersion and let N (p) (⊂ Rn+11 ) be the normal space of F at
p ∈ M n−1. We set
βFp,n(x) :=
〈∇xdF (X ),n〉 (x ∈ Tp M n−1, n ∈N (p)),
where X is a vector field of M n−1 satisfying Xp = x. The right-hand side is independent of the
choice of X . If p is not light-like, the second fundamental form αFp is well-defined as an N (p)-
valued bilinear form so that
〈
αFp (x,x),n
〉=βFp,n(x) (x ∈ Tp M n−1, n ∈N (p)).
Definition 1. We say that F is essentially cocausal at p if p is a cocausal point and each βFp,n(x)
(x ∈ Tp M n−1 \ {0}) does not vanish for any choice of space-like vector n ∈N (p). In this setting, F is
called an essentially cocausal immersion if it is essentially cocausal for each p ∈ M n−1.
One cannot replace “for any choice” with “for some choice”: in fact, if n = 2 and G(t ) =
(t + t 2,0, t − t 2) (t ∈ R), then βGp,n(x0) = 〈G ′′(0),n〉 (x0 := d/dt
∣∣
t=0) does not vanish when n =
(1,1,1) ∈N (0) but does vanish if n = (0,1,0) ∈N (0), where ′ means d/dt . We remark that, when
n = 2, F is a regular curve in R31 which is essentially cocausal at t = t0 if and only if the osculating
plane P of F at t = t0 exists (i.e. F ′(t0) and F ′′(t0) are linearly independent) and the plane P is
not time-like (in the case of G , the osculating plane P is time-like). At a space-like point, the
following simplification is possible:
Proposition 2. A cocausal immersion F : M n−1 → Rn+11 is essentially cocausal at a space-like point
p if and only if αFp (x,x) does not vanish and is not time-like for each x ∈ Tp M n−1 \ {0}.
Proof. Since v := αFp (x,x) belongs to N (p), v is time-like if and only if there exists a space-
like vector n ∈ N (p) such that 〈v,n〉 (= βFp,n(x)) vanishes. Using this, one can easily obtain the
assertion. □
Example 3. Consider a space-like immersion F : M n−1 → Rn+11 (n ≥ 4) into the light-cone Λn .
The shape operator A satisfies 〈A(x),x〉 =−S(x,x) (x ∈ T M n−1), where S is the Schouten tensor on
the conformally flat manifold M n−1 (cf. [2]). By [5, (1.2)],
〈
αFp (x,x),α
F
p (x,x)
〉= 2〈x,x〉S(x,x) holds.
By Proposition 2, F is essentially cocausal if only if S is positive semi-definite on M n−1.
If F (M n−1) is a strictly convex hypersurface lying in a space-like hyperplane Π in Rn+11 , then it
is clear that F and any small deformation of F in Rn+11 towards the outside of Π are essentially
cocausal. So, there are many examples of essentially cocausal immersions not lying in any
hyperplane in Rn+11 . For a time-like vector v ∈ Rn+11 , Π(v) :=
{
x ∈ Rn+11 ; 〈x,v〉 = 0
}
is a space-like
hyperplane in Rn+11 . We denote by πv : R
n+1
1 → Π(v) the orthogonal projection. In this paper,
such a projection is called a space-like projection. Let F : M n−1 → Rn+11 be an immersion. We set
fv :=πv ◦F .
Theorem 4. For p ∈ M n−1, F is an essentially cocausal immersion at p if and only if fv is a locally
strictly convex immersion around p for any time-like vector v in Rn+11 .
Proof. We fix a unit time-like vector v arbitrarily. Since we can write fv = F +〈F,v〉v,
d fv(X ) = dF (X )+
〈
dF (X ),v
〉
v and ∇X d fv(X ) =∇X dF (X )+
〈∇X dF (X ),v〉v (1)
Toshizumi Fukui, Atsufumi Honda and Masaaki Umehara 111
hold for any vector field X of M n−1. We can give the following construction (∗) of a unit normal
vector field n of fv for the time-like vector v:
We assume p is not time-like. Then, there exists a neighborhood U of p such that
Π(v)∩N (q) (q ∈U ) is one dimensional, sinceΠ(v) and N (q) have different causalities.
So there exists a space-like normal vector field n on U which is perpendicular to v. We
set n := n/p〈n,n〉.
(∗)
Then, n is a unit normal vector field of fv inΠ(v) (cf. the first equality of (1)). We have
(h(x,x) :=)〈∇xd fv(X ),np〉= 〈∇xdF (X ),np〉=βFp,np (x) (x ∈ Tp M n−1), (2)
where X is a vector field on U satisfying Xp = x. We remark that h can be identified with the
second fundamental form of fv.
We assume the essential cocausality of F and fix a unit time-like vector v. ThenΠ(v) is a space-
like hyperplane in Rn+11 . Since F is essentially cocausal at p, (2) with the fact that np is a space-
like vector belonging to N (p) implies that h(x,x) ̸= 0 for each x ∈ Tp M n−1 \ {0}, proving the local
strict convexity.
We then assume the local strict convexity. If p is time-like, then there exists x ∈ Tp M n−1 such
that v := dFp (x) is a time-like vector, and πv ◦F is not an immersion at p, contradiction. So we
may assume p is not time-like. We fix a space-like normal vector n0 ∈ N (p) arbitrarily. Let n⊥0
be the vector subspace of Rn+11 consisting of the vectors perpendicular to n0. Since n
⊥
0 is time-
like, there exists a time-like vector v ∈ n⊥0 . By (∗), the vector n0 can be considered as a normal
vector of fv at p. Since n0 is space-like, (2) implies the essential cocausality of F . In fact, we have
0 ̸= h(x,x) =βp,n0 (x) for each x ∈ Tp M n−1 \ {0}. □
2. Applications
Proposition 5. Suppose that Γ : R → R31 is an l -periodic (l > 0) regular curve which is cocausal.
Then for any future-pointing time-like vector v ∈ R31, the map γv : [0, l ] ∋ t 7→ πv ◦Γ(t ) ∈ Π(v) is a
closed regular curve on Π(v). Moreover, the value ιΓ of the rotation index of the map γv as a plane
curve is independent of the choice of v (called the rotation index of Γ).
Proof. Since γv is a regular closed curve in Π(v), the rotation index ιγv depends continuously on
v and we obtain the conclusion. □
Remark 6. In [4], the rotation index is defined for closed space-like regular curve in R31.
As an application of Theorem 4, we give the following:
Theorem 7. Suppose that M n−1 (n ≥ 2) is compact and F : M n−1 → Rn+11 is an essentially cocausal
immersion. If n = 2, we assume also that ιF =±1. Then:
(i) M n−1 is diffeomorphic to the (n −1)-sphere Sn−1 and F is an embedding.
(ii) πv ◦F (M n−1) bounds a convex domain inΠ(v) for any time-like vector v of Rn+11 .
Remark 8. In Ye and Ma [6], a space-like closed regular curve such that its osculating plane at
each point is space-like is said to be “strong space-like”, which implies essentially cocausality,
but the converse is not true. When n = 2 and F is strong space-like, the theorem was shown in [6,
Lemma 2.1].
Proof of Theorem 7. If n ≥ 3, then fv :=πv◦F is an embedding and fv(M n−1) is the boundary of a
convex domain by Fact 1, proving the theorem. So we may assume n = 2 and the rotation index of
F is equal to ±1. Then fv is locally strictly convex with index ±1 by Theorem 4 and Proposition 5.
In particular, F is an embedding. □
112 Toshizumi Fukui, Atsufumi Honda and Masaaki Umehara
Example 9. If we consider the map F0 : R/2πZ ∋ t 7→
(
cos t , sin t , (sin3t )/9
) ∈ R31, then it satisfies
the assumption of Theorem 7. In fact, F ′0 and F
′′
0 are linearly independent at each point of R/2πZ.
Since
〈
F ′0(t ),F
′
0(t )
〉= (17−cos6t)/18 and 〈F ′′0 (t ),F ′′0 (t )〉= cos2(3t ), F0 is essentially cocausal. For
a time-like vector v := (1,0,2)/p3, the image of πv ◦ F0
(
[0,2π]
)
is a strictly convex curve in the
planeΠ(v) as in Figure 1, right.
It is well-known that a compact conformally flat space-like hypersurface in the light-cone Λn
is diffeomorphic to the (n −1)-sphere Sn−1 (cf. [1]). By Example 3, we obtain the following:
Corollary 10. Let F : M n−1 → Rn+11 (n ≥ 4) be a space-like immersion with positive semi-definite
Schouten tensor S whose image is lying on the light-cone Λn in Rn+11 . If M
n−1 is compact, then
πv ◦F (M n−1) is a boundary of a convex domain inΠ(v) for any time-like vector v ∈ Rn+11 .
For example, a section of Λn by a space-like hyperplane satisfies the assumption. If M n−1 is
complete and the trace of S is bounded below by a positive constant c, then the Ricci tensor of
M n−1 satisfies Ric ≥ cds2, where ds2 is the first fundamental form. By Myers’s Theorem with
Corollary 10, M n−1 is diffeomorphic to Sn−1. In [1, Prop. 5.2], another condition for a given
complete hypersurface inΛn to be compact is given.
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Figure 1. The image of F0 and the image of πv ◦F0
(
[0,2π]
)
for v := (1,0,2)/p3.
Ye and Ma [6] showed the existence of a compact space-like maximal surface bounded by
a given strong space-like curve of rotation index ±1 embedded in R31. Moreover, they showed
that such a surface is a graph over a convex domain, this last conclusion follows also from the
following:
Corollary 11. Let N n (n ≥ 2) be a compact connected manifold with boundary M n−1, and let
Φ : N n → Rn+11 be a cocausal immersion. Suppose that the restriction F := Φ
∣∣
M n−1 is essentially
cocausal. When n = 2, suppose also that |ιF | is equal to one. Then M n−1 is diffeomorphic to Sn−1,
andΦ(N n) is a graph over the closed convex domain πv ◦Φ(N n) for any choice of a time-like vector
v of Rn+11 .
Proof. By Theorem 7, F (M n−1) bounds a convex domain D in Π(v). Since Φ is cocausal, ϕ :=
πv ◦Φ : N n → Π(v) is an immersion. If there exists p ∈ N n \ M n−1 such that ϕ(p) does not lie
on D, then there must exist q ∈ N n \ M n−1 giving the maximum of the continuous function
δ(x) := d(D,ϕ(x)) (x ∈ N n), where d is the canonical Euclidean distance function on Π(v). Then
q must be a critical point of the map ϕ, which contradicts that ϕ is an immersion. Thus,
ϕ
(
N n \ M n−1
)
is a subset of D, and we can conclude that ϕ(N n) coincides with the closure D
in Π(v). We then suppose that ϕ−1(y) is an infinite set for some y ∈ D. Since N n is compact,
we can find an accumulation point p of ϕ−1(y) in N n . However, it contradicts the fact that ϕ is
an immersion at p. So ϕ−1(y) is a finite set for each y ∈ D. Since N n is connected and ϕ is an
immersion, the cardinality of the set ϕ−1(y) does not depend on y . So we denote it by m. By
Theorem 7, we know that ϕ
∣∣
M n−1 is injective and M
n−1 is diffeomorphic to Sn−1. In particular,
we have m = 1, that is, ϕ is also an injection. Thus, Φ(N n) is a graph over the closed domain D,
proving the assertion. □
Toshizumi Fukui, Atsufumi Honda and Masaaki Umehara 113
The authors hope that “essential cocausality” will be widely recognized by geometers and
physicists as a new convexity for submanifolds of co-dimension 2. These statements are no
longer expected when the ambient space is the Euclidean space (F0 as in the left of Figure 1 into
the xz-plane is not convex).
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] L. J. Alías, V. L. Cánovas and M. Rigoli, “Codimension two spacelike submanifolds of the
Lorentz-Minkowski spacetime into the light cone”, Proc. R. Soc. Edinb., Sect. A, Math. 149
(2019), no. 6, pp. 1523–1553.
[2] A. C. Asperti and M. Dajczer, “Conformally flat Riemannian manifolds as hypersurfaces of
the light cone”, Can. Math. Bull. 32 (1989), no. 3, pp. 281–285.
[3] J. Hadamard, “Sur certaines propriétés des trajectoires en dynamique”, Journ. de Math. 3
(1897), pp. 331–388.
[4] S. Izumiya, M. Kikuchi and M. Takahashi, “Global properties of spacelike curves in
Minkowski 3-space”, J. Knot Theory Ramifications 15 (2006), no. 7, pp. 869–881.
[5] R. Kishida, “The volume of conformally flat manifolds as hypersurfaces in the light-cone”,
Differ. Geom. Appl. 96 (2024), article no. 102173 (15 pages).
[6] N. Ye and X. Ma, “Closed strong spacelike curves, Fenchel theorem and plateau problem in
the 3-dimensional Minkowski space”, Chin. Ann. Math., Ser. B 40 (2019), no. 2, pp. 217–226.


Convexity of space-like projections of submanifolds with co-dimension 2 in Lorentz–Minkowski space

Cyril Houdayer

Comptes Rendus
Mathématique
Cyril Houdayer
Operator algebraic characterization of the noncommutative Poisson boundary
Volume 363 (2025), p. 199-204
Online since: 25 March 2025
https://doi.org/10.5802/crmath.715
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 199-204
https://doi.org/10.5802/crmath.715
Research article / Article de recherche
Functional analysis / Analyse fonctionnelle
Operator algebraic characterization of the
noncommutative Poisson boundary
Caractérisation en algèbres d’opérateurs de la frontière
de Poisson noncommutative
Cyril Houdayer a
a École Normale Supérieure, Département de Mathématiques et Applications,
Université Paris-Saclay, 45 rue d’Ulm, 75230 Paris Cedex 05, France
E-mail: cyril.houdayer@ens.psl.eu
Abstract. We obtain an operator algebraic characterization of the noncommutative Furstenberg–Poisson
boundary L(Γ) ⊂ L(Γ↷B) associated with an admissible probability measureµ ∈ Prob(Γ) for which the (Γ,µ)-
Furstenberg–Poisson boundary (B ,νB ) is uniquely µ-stationary. This is a noncommutative generalization of
Nevo–Sageev’s structure theorem [14]. We apply this result in combination with previous works to provide
further evidence towards Connes’ rigidity conjecture for higher rank lattices.
Résumé. Nous obtenons une caractérisation en algèbres opérateurs de la frontière de Furstenberg–Poisson
noncommutative L(Γ) ⊂ L(Γ ↷ B) associée à une mesure de probabilité admissible µ ∈ Prob(Γ) pour
laquelle la (Γ,µ)-frontière de Furstenberg–Poisson (B ,νB ) est uniquement µ-stationnaire. Il s’agit d’une
généralisation noncommutative du théorème de structure de Nevo–Sageev [14]. Nous appliquons ce résultat
en combinaison avec des travaux antérieurs pour fournir des pistes supplémentaires afin de résoudre la
conjecture de rigidité de Connes pour les réseaux de rang supérieur.
Keywords. Connes’ rigidity conjecture, higher rank lattices, noncommutative Furstenberg–Poisson bound-
aries, von Neumann algebras.
Mots-clés. Conjecture de rigidité de Connes, réseaux de rang supérieur, frontières de Furstenberg–Poisson
noncommutatives, algèbres de von Neumann.
2020 Mathematics Subject Classification. 20G25, 22D25, 46L10, 46L55.
Funding. The author is supported by ERC Advanced Grant NET 101141693.
Manuscript received 4 November 2024, accepted 19 December 2024.
1. Introduction and statement of the main results
Let Γ be a countable discrete group and µ ∈ Prob(Γ) an admissible probability measure in the
sense that
⋃
n≥1 supp(µ)n = Γ. Denote by (B ,νB ) the (Γ,µ)-Furstenberg–Poisson boundary [3,6].
Recall that (B ,νB ) is the unique (Γ,µ)-space, up to isomorphism, for which the Γ-equivariant
Poisson transform
Pµ : L
∞(B ,νB ) −→ Har∞(Γ,µ) : f 7−→
(
γ 7→
∫
B
f (γb)dνB (b)
)
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
200 Cyril Houdayer
is onto and isometric. Here, Har∞(Γ,µ) ⊂ ℓ∞(Γ) denotes the subspace of bounded (right) µ-
harmonic functions. We say that the (Γ,µ)-Furstenberg–Poisson boundary (B ,νB ) is µ-uniquely
stationary if there exists a compact metrizable model K of (B ,νB ) for which there exists a unique
µ-stationary Borel probability measure νK ∈ Prob(K ) and (K ,νK ) ∼= (B ,νB ) as (Γ,µ)-spaces. In [14,
Theorem 9.2], Nevo–Sageev showed that if the (Γ,µ)-Furstenberg–Poisson boundary (B ,νB ) is µ-
uniquely stationary, then for any amenable (Γ,µ)-space (X ,νX ), there exists a relatively measure
preserving Γ-equivariant measurable factor map (X ,νX ) → (B ,νB ).
Denote by M = L(Γ) the group von Neumann algebra and by B = L(Γ↷B) the group measure
space construction associated with the nonsingular action Γ↷ (B ,νB ). Then we have the natural
inclusion M ⊂ B. We also consider the conjugation action Γ↷ B and we denote by EB : B →
L∞(B ,νB ) the canonical Γ-equivariant faithful normal conditional expectation. Then the faithful
normal stateψB = νB ◦EB ∈B∗ is µ-stationary. Regard L2(M ,τ) ⊂ L2(B,ψB ) as a closed subspace
and simply denote by eM : L2(B,ψB ) → L2(M ,τ) the orthogonal projection. Define the normal
state ϕµ ∈ B
(
ℓ2(Γ)
)
∗ by the formula ϕµ(T ) =
∑
γ∈Γµ(γ)〈Tδγ,δγ〉 for T ∈ B
(
ℓ2(Γ)
)
. Define the
normal ucp (unital completely positive) map
Φµ : B
(
ℓ2(Γ)
)−→ B(ℓ2(Γ)) : T 7−→ ∑
γ∈Γ
µ(γ)Jλ(γ)JT Jλ(γ)∗ J .
Then we have ϕµ(T ) =
〈
Φµ(T )δe ,δe
〉
for all T ∈ B(ℓ2(Γ)). Denote by Har(Φµ) = {T ∈
B
(
ℓ2(Γ)
)∣∣Φµ(T ) = T } the subspace of Φµ-harmonic elements. Following [5,11], the noncommu-
tative Poisson transform
P̂µ : B −→ Har(Φµ) : T 7−→ eM TeM
is a normal onto isometric ucp map such that P̂µ
∣∣
M = idM . Then we refer to the inclusion M ⊂B
as the (M ,ϕµ)-noncommutative Furstenberg–Poisson boundary.
Let M ⊂ M be an inclusion of von Neumann algebras and consider the conjugation action
Γ↷M . By [2, Proposition 4.2], to any (faithful) normal µ-stationary state ψ ∈ M∗ corresponds
a unique Γ-equivariant (faithful) normal ucp map Θ : M → L∞(B ,νB ) such that νB ◦Θ = ψ.
Moreover, by [5, Proposition 2.1], to any (faithful) normal µ-stationary state ψ ∈M∗ corresponds
a unique (faithful) normal ucp map Θ̂ : M → B such that Θ̂∣∣M = idM , ψ = νB ◦ EB ◦ Θ̂ and
Θ= EB ◦ Θ̂.
Whenever Φ : A → B is a ucp map between unital C∗-algebras, we denote by mult(Φ) ⊂ A the
multiplicative domain of Φ. Our main result is the following noncommutative generalization of
Nevo–Sageev’s structure theorem [14, Theorem 9.2].
Theorem 1. Keep the same notation as above with M = L(Γ) and B = L(Γ↷ B). Assume that the
(Γ,µ)-Furstenberg–Poisson boundary (B ,νB ) is µ-uniquely stationary. Let M ⊂M be an inclusion
of von Neumann algebras. Assume that M is amenable and that there exists a faithful normal ucp
map Θ̂ : M →B such that Θ̂∣∣M = idM .
Then Θ̂
∣∣
mult(Θ̂) : mult(Θ̂) → B is a unital normal onto ∗-isomorphism. Therefore, we may
regard B ⊂ M as a von Neumann subalgebra and Θ̂ : M → B as a faithful normal conditional
expectation.
In particular, lettingψ= νB◦EB ◦Θ̂ ∈M∗ andψB = νB◦EB ∈B∗, it follows from [15, Theorem A]
and [12] that the inclusion M ⊂M has maximal ϕµ-entropy in the sense of [5], that is,
hϕµ
(
M ⊂M ,ψ)= hϕµ(M ⊂B,ψB )= h(µ) = limn 1n H(µ∗n).
When Γ is moreover an infinite icc group (infinite conjugacy classes), we deduce the fol-
lowing operator algebraic characterization of the (M ,ϕµ)-noncommutative Furstenberg–Poisson
boundary M ⊂B.
Cyril Houdayer 201
Corollary 2. Keep the same notation as above with M = L(Γ) and B = L(Γ↷B). Assume that Γ is
infinite icc and that the (Γ,µ)-Furstenberg–Poisson boundary (B ,νB ) is µ-uniquely stationary. Let
M ⊂M be an inclusion of von Neumann algebras satisfying the following conditions:
(i) M ′∩M =C1.
(ii) There exists a normal ucp map Θ̂ : M →B such that Θ̂∣∣M = idM .
(iii) M is amenable.
(iv) Whenever M ⊂N ⊂M is an intermediate von Neumann subalgebra for which there exists
a normal conditional expectation E: M →N , we have N =M .
Then Θ̂ : M →B is a unital normal onto ∗-isomorphism.
Let us point out that the (M ,ϕµ)-noncommutative Furstenberg–Poisson boundary M ⊂ B
satisfies Conditions (i), (ii), (iii), (iv) by [5]. Thus, Corollary 2 gives an abstract operator algebraic
characterization of the inclusion M ⊂B.
We also apply Theorem 1 to provide further evidence towards Connes’ rigidity conjecture for
higher rank lattices.
Connes’ rigidity conjecture. For every i ∈ {1,2}, let Gi be a semisimple connected real Lie group
with trivial center, no compact factor such that rkR(Gi ) ≥ 2 and let Γi <Gi be an irreducible lattice.
If L(Γ1) ∼= L(Γ2), then G1 ∼=G2 and in particular rkR(G1) = rkR(G2).
For every i ∈ {1,2}, choose a Furstenberg measure µi ∈ Prob(Γi ) so that (Gi /Pi ,νPi ) is the
(Γi ,µi )-Furstenberg–Poisson boundary [7]. Here, νPi ∈ Prob(Gi /Pi ) denotes the unique Ki -
invariant Borel probability measure, where Ki < Gi is a maximal compact subgroup and Gi =
Ki Pi . It is well-known that (Gi /Pi ,νPi ) is µi -uniquely stationary [8]. Set Mi = L(Γi ) and Bi =
L(Γi ↷Gi /Pi ).
Assume that M1 ∼= M2 and set M = M1 = M2. Since B2 is an amenable (hence injective)
von Neumann algebra and since M ⊂ B1, Arveson’s extension theorem implies that there exists
a ucp map Φ : B1 → B2 such that Φ
∣∣
M = idM . We show that if Φ : B1 → B2 is normal, then
rkR(G1) = rkR(G2).
Theorem 3. Keep the same notation as above. Assume that there exists a normal ucp map
Φ : B1 → B2 such that Φ
∣∣
M = idM . Then Φ : B1 → B2 is a unital normal onto ∗-isomorphism.
In particular, we have
rkR(G1) = rkR(G2).
2. Proofs of the main results
We record the following well-known fact on ucp maps.
Lemma 4. Let A,B,C be unital C∗-algebras andΦ : A→B,Ψ : B→C be ucp maps. Assume that
Ψ is faithful and that Ψ ◦Φ is a unital ∗-homomorphism. Then Φ is a unital ∗-homomorphism
andΦ(A) ⊂ mult(Ψ).
Proof. By Kadison’s inequality and sinceΨ◦Φ is a unital ∗-homomorphism, for every a ∈A, we
have
Ψ
(
Φ(a∗a)
)=Ψ(Φ(a∗))Ψ(Φ(a))≤Ψ(Φ(a)∗Φ(a))≤Ψ(Φ(a∗a)).
Since Ψ is faithful, it follows that Φ(a)∗Φ(a) = Φ(a∗a). Then mult(Φ) = A and so Φ is a unital
∗-homomorphism. Moreover, for every a ∈A, we have
Ψ
(
Φ(a)∗
)
Ψ
(
Φ(a)
)=Ψ(Φ(a)∗Φ(a)).
This further implies thatΦ(A) ⊂ mult(Ψ). □
202 Cyril Houdayer
Let Γ be a countable discrete group and µ ∈ Prob(Γ) an admissible probability measure. De-
note by (B ,νB ) the (Γ,µ)-Furstenberg–Poisson boundary. Set M = L(Γ) and B = L(Γ↷B). Denote
by EB : B → L∞(B ,νB ) the canonical Γ-equivariant faithful normal conditional expectation.
Let M ⊂M be an inclusion of von Neumann algebras. Let Θ̂ : M →B be a faithful normal ucp
map such that Θ̂
∣∣
M = idM . Then Θ = EB ◦ Θ̂ : M → L∞(B ,νB ) is a Γ-equivariant faithful normal
ucp map.
Lemma 5. Keep the same notation as above. We have
Θ
(
mult(Θ)
)
⋊Γ⊂ Θ̂(mult(Θ̂))⊂ L∞(B ,νB )⋊Γ=B.
In particular, if Θ
(
mult(Θ)
) = L∞(B ,νB ), then Θ̂(mult(Θ̂)) = B. In that case, we may regard B ⊂
M as a von Neumann subalgebra and Θ̂ : M →B as a faithful normal conditional expectation.
Proof. By definition, Θ
∣∣
mult(Θ) : mult(Θ) → L∞(B ,νB ) is a Γ-equivariant unital normal ∗-
isomorphism and Θ̂
∣∣
mult(Θ̂) : mult(Θ̂) → B is a unital normal ∗-isomorphism such that
M ⊂ Θ̂(mult(Θ̂)) ⊂ B. Since EB ◦ Θ̂ = Θ and since EB is faithful, Lemma 4 implies that
Θ̂
∣∣
mult(Θ) : mult(Θ) → B is a unital normal ∗-isomorphism and Θ̂
(
mult(Θ)
) ⊂ mult(EB ). Then
we have mult(Θ) ⊂ mult(Θ̂) and so M ∨mult(Θ) ⊂ mult(Θ̂). Since mult(EB ) = L∞(B ,νB ) ⊂ B, it
follows that Θ̂
(
mult(Θ)
)⊂ L∞(B ,νB ) ⊂B and thatΘ∣∣mult(Θ) = Θ̂∣∣mult(Θ). Then we have
Θ
(
mult(Θ)
)
⋊Γ⊂ Θ̂(mult(Θ̂))⊂ L∞(B ,νB )⋊Γ=B.
Moreover, assume that Θ
(
mult(Θ)
) = L∞(B ,νB ). Then we necessarily have Θ̂(mult(Θ̂)) = B.
Define the unital normal ∗-isomorphism ι = (Θ̂∣∣mult(Θ̂))−1 : B → mult(Θ̂). Then ι ◦Θ : M →
mult(Θ̂) is a faithful normal conditional expectation. □
Proof of Theorem 1. Keep the same notation as above with M = L(Γ) and B = L(Γ↷B). Assume
moreover that the (Γ,µ)-Furstenberg–Poisson boundary (B ,νB ) is µ-uniquely stationary. We still
denote by B the compact metrizable model and we assume that B = supp(νB ). Then the identity
map idC(B) : C(B) → L∞(B ,νB ) is the unique Γ-equivariant ucp map (see [13, Corollary VI.2.10]
and [9, Theorem 3.4]). Denote by λ : Γ→ U (M) : γ 7→ λ(γ) the left regular representation. We
naturally identify L2(M) = ℓ2(Γ). Denote by M ′ the commutant of M in B(L2(M)). Since M ⊂M ,
we may regard L2(M ) as a Hilbert left M-module. Then [1, Proposition 8.2.3] implies that there
exists a projection e ∈ M ′ ⊗B(ℓ2) and a unitary mapping V : L2(M ) → e(L2(M)⊗ℓ2) such that
V (aξ) = (a ⊗1)V (ξ) for every a ∈ M and every ξ ∈ L2(M ). Then the mapping
π : B
(
L2(M)
)−→ B(L2(M )) : T 7−→V ∗e(T ⊗1)eV
is a normal ucp map such that π(aT b) = aπ(T )b for all T ∈ B(L2(M)) and all a,b ∈ M .
Since M ⊂ B(L2(M )) is amenable, there exists a (possibly non normal) conditional ex-
pectation E: B
(
L2(M )
) → M . Choose a point b ∈ B and define the Γ-equivariant unital ∗-
homomorphism θ : C(B) → ℓ∞(Γ) : F 7→ (γ 7→ F (γb)). Regard ℓ∞(Γ) ⊂ B(ℓ2(Γ)) = B(L2(M)) and
define the ucp map ι= E◦π◦θ : C(B) →M . Then for every F ∈ C(B) and every γ ∈ Γ, we have
ι(F ◦γ−1) = E
(
π
(
θ(F )◦γ−1))= E(π(λ(γ)θ(F )λ(γ)∗))=λ(γ)ι(F )λ(γ)∗.
Therefore, the ucp map ι : C(B) → M is Γ-equivariant with respect to the conjugation action
Γ↷M .
By composition, Θ ◦ ι : C(B) → L∞(B ,νB ) is a Γ-equivariant ucp map. Then we have Θ ◦ ι =
idC(B). Since Θ : M → L∞(B ,νB ) is faithful, Lemma 4 implies that ι : C(B) → M is a unital ∗-
homomorphism and ι
(
C(B)
) ⊂ mult(Θ). Set ψ = νB ◦Θ ∈ M∗ and note that ψ is a µ-stationary
faithful normal state. Since ψ ◦ ι = νB , we may uniquely extend ι : L∞(B ,νB ) → M to a faithful
normal unital ∗-homomorphism such that Θ ◦ ι = idL∞(B ,νB ). Then we have ι
(
L∞(B ,νB )
) ⊂
mult(Θ), L∞(B ,νB ) =Θ
(
ι
(
L∞(B ,νB )
))
and so Θ
(
mult(Θ)
)= L∞(B ,νB ). The moreover part follows
from Lemma 5. □
Cyril Houdayer 203
Proof of Corollary 2. Set M = L(Γ) and B = L(Γ↷ B). Assume that Γ is infinite icc and that
the (Γ,µ)-Furstenberg–Poisson boundary (B ,νB ) is µ-uniquely stationary. Denote by λ : Γ →
U (M) : γ 7→ λ(γ) the left regular representation. Let Θ̂ : M →B be a normal ucp map such that
Θ̂
∣∣
M = idM as given by Condition (iii). Denote by p ∈ M the support projection of Θ̂ : M → B.
Then for γ ∈ Γ, we have Θ̂(λ(γ)pλ(γ)∗) = λ(γ)Θ̂(p)λ(γ)∗ = 1 and so p ≤ λ(γ)pλ(γ)∗. Since this
holds for every γ ∈ Γ, we infer that p ∈ L(Γ)′∩M . Condition (i) further implies that p = 1. Thus,
Θ̂ : M →B is a faithful normal ucp map. By Condition (ii) and Theorem 1, we infer that B ⊂M
can be regarded as a von Neumann subalgebra and Θ̂ : M →B as a faithful normal conditional
expectation. Then Condition (iv) finally implies that B =M and so Θ̂ : M →B is a unital normal
onto ∗-isomorphism. □
Proof of Theorem 3. Consider the inclusion M ⊂ B1. Since M is a type II1 factor, [5, Proposi-
tion 2.7] implies that M ′∩B1 =C1. By assumption, B1 is amenable andΦ : B1 →B2 is a normal
ucp map such that Φ
∣∣
M = idM . By [5, Theorem 4.1], for every intermediate von Neumann sub-
algebra M ⊂ N ⊂ B1 for which there exists a normal conditional expectation E: B1 → N , we
have N =B1. We may apply Corollary 2 to the inclusion M ⊂B1 to conclude that Φ : B1 →B2
is a unital normal onto ∗-isomorphism. By [10, Corollary F] and [4, Theorem B], we infer that
rkR(G1) = rkR(G2). □
Acknowledgments
I am grateful to Amos Nevo for thought provoking discussions during the IHP trimester Group
actions and rigidity: around the Zimmer program and for pointing out the reference [14].
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
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Operator algebraic characterization of the noncommutative Poisson boundary

Tatsuki Kuwagaki

Vivek Shende

Adjoints, wrapping, and morphisms at infinity

Aleksei B. Aleksandrov

Vladimir V. Peller

Comptes Rendus
Mathématique
Aleksei B. Aleksandrov and Vladimir V. Peller
Relatively bounded and relatively trace class perturbations
Volume 363 (2025), p. 377-382
Online since: 20 May 2025
https://doi.org/10.5802/crmath.722
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 377-382
https://doi.org/10.5802/crmath.722
Research article / Article de recherche
Functional analysis, Harmonic analysis / Analyse fonctionnelle, Analyse
harmonique
Relatively bounded and relatively trace class
perturbations
Perturbations relativement bornées et relativement de
classe trace
Aleksei B. Aleksandrov a,b and Vladimir V. Peller ,a,b
a St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St. Petersburg, Russia
b St. Petersburg Department, Steklov Institute of Mathematics, Russian Academy of Sciences,
Fontanka 27, 191023 St. Petersburg, Russia
E-mails: aall54eexx@gmail.com, peller@math.msu.edu
Abstract. In this note we study the behaviour of functions of self-adjoint operators under relatively bounded
and relatively trace class perturbations. We introduce and study the class of relatively operator Lipschitz
functions. We obtain a trace formula in the case of relatively trace class perturbations and show that this
class of functions is the maximal class of functions for which the trace formula holds. Our method also gives
us a new approach to the inequality
∫ |ξ(t )|(1+ |t |)−1 dt < ∞ for the spectral shift function ξ in the case of
relatively trace class perturbations.
Résumé. Dans cette note nous étudions le comportement des fonctions d’opérateurs auto-adjoints sous des
perturbations relativement dans la classe des opérateurs à trace et relativement bornées. Nous introduisons
et étudions la classe de fonctions relativement lipschitziennes opératorielles. Nous obtenons une formule de
trace dans le cas de perturbations relativement à trace et montrons que cette classe de fonctions est maximale
pour la validité de la formule de trace. Notre méthode nous donne une nouvelle approche de l’inégalité∫ |ξ(t )|(1+|t |)−1 dt <∞ pour la fonction ξ de décalage spectral.
Keywords. Relatively bounded perturbation, relatively trace class perturbation, relatively operator Lipschitz
class, trace formula, self-adjoint operators, double operator integrals.
Mots-clés. Perturbations relativement bornées, perturbations relativement à trace, fonctions relativement
lipschitziennes opératorielles, formules des traces, opérateurs auto-adjoints, intégrales d’opérateurs dou-
bles.
2020 Mathematics Subject Classification. 47A55.
Funding. The research on Sections 2–4 is supported by the Russian Science Foundation (grant number 23-
11-00153). The research on Sections 5–6 is supported by a grant of the Government of the Russian Federation
for the state support of scientific research, carried out under the supervision of leading scientists, agreement
075-15-2024-631.
Manuscript received 19 December 2024, revised 23 January 2025, accepted 24 January 2025.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
378 Aleksei B. Aleksandrov and Vladimir V. Peller
1. Introduction
The purpose of this note is to study the behaviour of functions of self-adjoint operators under
relatively bounded and relatively trace class perturbations. It turns out that the natural class
of functions to be considered in this connection is the class of relatively operator Lipschitz
functions, which we are going to define in the introduction. We are going to obtain various
characterizations of this class. An important role will be played by double operator integrals.
Let A be a not necessarily bounded self-adjoint operator in Hilbert space. A self-adjoint
operator K (also, not necessarily bounded) is called a relatively bounded perturbation of A if
∥K v∥ ≤ const(∥v∥+∥Av∥), v ∈ Dom(A), (1)
where Dom(A) stands for the domain of the operator A. It is easy to see that property (1) is
equivalent to the fact that the operator
C
def= K (A+ iI )−1 (2)
is bounded. We say that a self-adjoint operator K is a relatively trace class perturbation of A if the
operator C defined by (2) belongs to the trace class S1.
We consider parametric families At = A + tK , t ∈ R, where A and K are self-adjoint operators
and K is a relatively bounded perturbation of A and we obtain a formula for the derivative of the
function t 7→ f (At )− f (A) in the strong operator topology in terms of double operator integrals.
This will be used to obtain a trace formula and compute the trace of f (A +K )− f (A) in the
case when K is a relatively trace class perturbation of A and f is a relatively operator Lipschitz
function. Our results allow us to give a new approach to the inequality
∫ |ξ(t )|(1+ |t |)−1 dt <∞,
where ξ is the spectral shift function that corresponds to the pair {A, A+K }.
Definition 1. A continuous function f on the real line R is called relatively operator Lipschitz if
there is a positive number k such that∥∥ f (B)− f (A)∥∥≤ k∥∥(B − A)(A+ iI )−1∥∥ (3)
whenever A and B are self-adjoint operators such that B − A is a relatively bounded perturbation
of A.
We denote by ROL the class of relatively operator Lipschitz functions on R.
Lemma 2. Let f be a continuous function on R. It is relatively operator Lipschitz if and only if
f (B)− f (A) ∈ S1
whenever A and B are self-adjoint operators such that B − A is a relatively trace class perturbation
of A.
It is easy to verify that f is a relatively operator Lipschitz function if it is a resolvent Lipschitz
function, i.e., ∥∥ f (B)− f (A)∥∥≤ const∥∥(B + iI )−1 − (A+ iI )−1∥∥,
for arbitrary self-adjoint operators A and B .
On the other hand, it is obvious that if f is a relatively operator Lipschitz function, then it is an
operator Lipschitz function, i.e., ∥∥ f (B)− f (A)∥∥≤ const∥A−B∥ (4)
for arbitrary self-adjoint operators A and B , no matter bounded or unbounded. The class of
operator Lipschitz functions on R is denoted by OL; the norm ∥ f ∥OL of a function f in OL is, by
definition, the smallest possible constant on the right-hand side of (4).
Aleksei B. Aleksandrov and Vladimir V. Peller 379
We refer the reader to the survey [1] for detailed information about operator Lipschitz func-
tions. It is well known (see [1]) that operator Lipschitz functions must be differentiable every-
where on the real line R, and so relatively operator Lipschitz functions must also be differentiable
on R.
2. Characterizations of the class of relatively operator Lipschitz functions
In this section we give several necessary and sufficient conditions for a function on R to be
relatively operator Lipschitz. We also obtain similar characterizations of the class of resolvent
Lipschitz functions. In Section 4 we obtain another characterization of the class of relatively
operator Lipschitz functions in terms of Schur multipliers. Yet another characterization of the
class ROL (in terms of commutator estimates) will be given in Section 5.
Definition 3. We say that a function f on R is a multiplier of the space OL of operator Lipschitz
functions if
g ∈ OL =⇒ f g ∈ OL.
We denote by MOL the class of all multipliers of OL.
Theorem 4. Let f be a continuous function on R. The following are equivalent:
(a) f ∈ ROL;
(b) the function x 7→ (x + i) f (x) is an operator Lipschitz function;
(c) f ∈MOL.
The following theorem provides a similar characterization of the class of resolvent Lipschitz
functions.
Theorem 5. Let f be a continuous function on R. The following are equivalent:
(a) f is a resolvent Lipschitz function;
(b) the function x 7→ (x + i)2( f (x)− c) is an operator Lipschitz function for some c ∈C;
(c) the function x 7→ (x + i)( f (x)− c) belongs to MOL for some c ∈C.
If c ∈C satisfies (b) or (c), then c must be equal to lim|x|→∞ f (x).
3. Double operator integrals
As in the case of operator Lipschitz functions (see [1]), an important role to study the class of rel-
atively operator Lipschitz functions is played by double operator integrals. Double operator inte-
grals appeared first in [7]; later Birman and Solomyak developed in [3–5] a rigorous mathematical
theory of double operator integrals.
Double operator integrals are expressions of the formÏ
Φ(x, y)dE1(x)Q dE2(y). (5)
Here E1 and E2 are spectral measures on Hilbert space,Φ is a bounded measurable function and
Q is a bounded linear operator on Hilbert space.
The starting point of Birman and Solomyak in [3] was to define double operator integrals of the
form (5) in the case when Q is a Hilbert–Schmidt operator, i.e., Q ∈ S2. In this case
Î
ΦdE1Q dE2
also belongs to S2 and ∥∥∥∥Ï ΦdE1Q dE2∥∥∥∥
S2
≤ ∥Φ∥L∞∥Q∥S2 .
380 Aleksei B. Aleksandrov and Vladimir V. Peller
IfΦ has the property
Q ∈ S1 =⇒
Ï
ΦdE1Q dE2 ∈ S1,
we say that Φ is a Schur multiplier with respect to the spectral measures E1 and E2. In this case
we can define by duality the double operator integral (5) for an arbitrary bounded linear operator
Q, see [1,3] for more detail.
We use the notation M(R2) for the class of functions on R2 that are Schur multipliers with
respect to arbitrary Borel spectral measures.
4. A characterization of the class of relatively operator Lipschitz functions in terms of
Schur multipliers
Recall that for a differentiable function f on R, the divided difference D f is defined on R2 by
(D f )(x, y)
def=

f (x)− f (y)
x − y , if x ̸= y,
f ′(x), if x = y.
Put
(D f )x(x, y)
def= (x + i)(D f )(x, y), (x, y) ∈R2.
The following result describes the class ROL in terms of Schur multipliers.
Theorem 6. Let f be a continuous function on R. Then f is relatively operator Lipschitz if and
only if it is differentiable on R and (D f )x ∈M(R2).
Note that the part “if” was established in [4, Theorem 4.6].
Now we are in a position to obtain a useful formula for f (B)− f (A) in the case when B − A is a
relatively bounded perturbation of A and f is a relatively operator Lipschitz function on R.
Theorem 7. Let f be a relatively operator Lipschitz function on R. If A and B are self-adjoint
operators such that B − A is a relatively bounded perturbation of A, then
f (B)− f (A) =
Ï
R×R
(D f )(x, y)(y + i)dEB (x)(B − A)(A+ iI )−1 dE A(y).
5. A characterization of the class of relatively operator Lipschitz functions in terms of
commutators
In this section we are going to obtain a characterization of the class of relatively operator
Lipschitz functions in terms of commutator and quasi-commutator estimates. The following
result means, in particular, that a function is relatively operator Lipschitz if and only if it is
relatively commutator Lipschitz, i.e. satisfies statement (b) of the following theorem.
Theorem 8. Let f : R→C be a differentiable function. The following statements are equivalent:
(a) f is a relatively operator Lipschitz function;
(b) ∥∥ f (A)R −R f (A)∥∥≤ const∥∥(AR −R A)(A+ iI )−1∥∥
for any bounded operator R and any self-adjoint operator A;
(c) ∥∥ f (B)R −R f (A)∥∥≤ const∥∥(BR −R A)(A+ iI )−1∥∥
for any bounded operator R and any self-adjoint operators A and B.
Aleksei B. Aleksandrov and Vladimir V. Peller 381
6. Differentiation in a parameter
To obtain a trace formula for f (B)− f (A) in the case when K def= B − A is a self-adjoint relatively
trace class perturbation of A, we are going to consider a parametric family At
def= A + tK , t ∈ R,
and differentiate the function
t 7→ f (At )− f (A)
in the parameter t in the strong operator topology.
Theorem 9. Let f be a relatively operator Lipschitz function. Suppose that K is a relatively
bounded perturbation of a self-adjoint operator A. Then the function t 7→ f (At )− f (A) is differen-
tiable on R in the strong operator topology and
d
ds
(
f (As )− f (At )
)∣∣∣∣
s=t
=
Ï
R×R
f (x)− f (y)
x − y (y + i)dE At (x)K (At + iI )
−1 dE At (y) (6)
for any t ∈R.
7. The trace formula
For a pair of self-adjoint operators {A,B} with trace class difference the very important Lifshits–
Krein trace formula [8,9] gives the following expression for the trace of the difference f (A)− f (B):
trace
(
f (A)− f (B))= ∫
R
f ′(t )ξ(t )dt
for nice functions f , where ξ is the so-called spectral shift function for the pair {A,B}. It is real-
valued and belongs to L1(R). It was established in [10] that the maximal class of functions f for
which the Lifshits–Krein trace formula holds for arbitrary pairs of self-adjoint operators with trace
class difference coincides with the class of operator Lipschitz functions OL.
To obtain a trace formula in the case of relatively operator Lipschitz functions and relatively
trace class perturbations, we are going to use an idea of Birman and Solomyak [5]. Namely, we
are going to use formula (6) to compute the derivative of the function t 7→ f (At )− f (A) in the
strong operator topology. Then we compute the trace of the derivative and use a version of the
Newton–Leibnitz formula to compute the trace of f (B)− f (A). We also describe the maximal
class of functions for which the trace formula holds in the case of arbitrary relatively trace class
perturbations.
Theorem 10. Let A be a self-adjoint operator and let K be a relatively trace class perturbation of A.
Then there exists a unique measurable real-valued function ξ on R satisfying∫
R
|ξ(t )|
1+|t | dt <∞ (7)
and such that for an arbitrary relatively operator Lipschitz function f on R the following trace
formula holds
trace
(
f (A+K )− f (A))= ∫
R
f ′(t )ξ(t )dt . (8)
The function ξ satisfying (8) and (7) is called the spectral shift function for the pair {A, A+K }.
The fact that the spectral shift function in the case of a relatively trace class perturbation must
satisfy (7) was established earlier in [6].
Theorem 11. Let f be a differentiable function onR such that trace formula (8) holds for arbitrary
self-adjoint operators A and K such that K is a relatively trace class perturbation of A. Then
f ∈ ROL.
The details of this note can be found in [2].
382 Aleksei B. Aleksandrov and Vladimir V. Peller
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] A. B. Aleksandrov and V. V. Peller, “Operator Lipschitz functions”, Russ. Math. Surv. 71
(2016), no. 4, pp. 605–702.
[2] A. B. Aleksandrov and V. V. Peller, “Functions of self-adjoint operators under relatively
bounded and relatively trace class perturbations. Relatively operator Lipschitz functions”,
2025. Online at https://arxiv.org/abs/2411.01901.
[3] M. S. Birman and M. Z. Solomyak, “Double Stieltjes operator integrals”, Probl. Mat. Fiz. 1
(1966), pp. 33–67.
[4] M. S. Birman and M. Z. Solomyak, “Double Stieltjes operator integrals. III”, Probl. Mat. Fiz.
6 (1973), pp. 27–53.
[5] M. S. Birman and M. Z. Solomyak, “Remarks on the spectral shift function”, J. Sov. Math. 3
(1975), pp. 408–419.
[6] A. Chattopadhyay and A. Skripka, “Trace formulas for relative Schatten class perturba-
tions”, J. Funct. Anal. 274 (2018), pp. 3377–3410.
[7] Y. L. Daletskĭı and S. G. Kreı̆n, “Integration and differentiation of functions of Hermitian
operators and application to the theory of perturbations (Russian)”, Voronež. Gos. Univ.
Trudy Sem. Funkcional. Anal. 1 (1956), pp. 81–105.
[8] M. G. Kreı̆n, “On the trace formula in perturbation theory (Russian)”, Mat. Sb. 33 (1953),
pp. 597–626.
[9] I. M. Lifshits, “On a problem in perturbation theory connected with quantum statistics
(Russian)”, Usp. Mat. Nauk 7 (1952), pp. 171–180.
[10] V. V. Peller, “The Lifshits–Krein trace formula and operator Lipschitz functions”, Proc. Am.
Math. Soc. 144 (2016), pp. 5207–5215.


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Comptes Rendus
Mathématique
George Peterzil and Guy Sapire
Irrationality of the length spectrum
Volume 363 (2025), p. 825-828
Online since: 4 July 2025
https://doi.org/10.5802/crmath.756
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 825-828
https://doi.org/10.5802/crmath.756
Research article / Article de recherche
Algebra, Dynamical systems / Algèbre, Systèmes dynamiques
Irrationality of the length spectrum
Irrationalité du spectre des longueurs
George Peterzil a and Guy Sapire a
a Einstein Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
E-mails: george.peterzil@mail.huji.ac.il, guy.sapire@mail.huji.ac.il
Abstract. It is a classical result of Dal’Bo that the length spectrum of a non-elementary Fuchsian group is non-
arithmetic, namely, it generates a dense additive subgroup of R. In this note we provide an elementary proof
of an extension of this theorem: a non-elementary Fuchsian group contains two elements whose lengths are
linearly independent overQ, reproving a result of Prasad and Rapinchuk [9].
Résumé. C’est un résultat classique de Dal’Bo que le spectre des longueurs d’un groupe fuchsien non
élémentaire est non arithmétique, c’est-à-dire qu’il génère un sous-groupe additif dense de R. Dans cette
note, nous fournissons une preuve élémentaire d’une extension de ce théorème : un groupe fuchsien non
élémentaire contient deux éléments dont les longueurs sont linéairement indépendantes surQ, reproduisant
un résultat de Prasad et Rapinchuk [9] dans un cas particulier.
Keywords. Length spectrum, Fuchsian group, trace field.
Mots-clés. Spectre de longueur, groupe fuchsien, corps de traces.
2020 Mathematics Subject Classification. 20H10, 37D40.
Funding. The second author was supported by the Israel Science Foundation (grant no. 2301/20).
Manuscript received 25 March 2025, revised 22 April 2025, accepted 6 May 2025.
1. Introduction
Let Γ be a non-elementary Fuchsian group, that is, a discrete subgroup of PSL2(R) which does not
contain a cyclic subgroup of finite index.
Definition 1. We say that γ ∈ Γ \ {e} is hyperbolic whenever it is diagonalizable over R. Given a
hyperbolic γ ∈ PSL2(R), we define the translation length of γ, denoted ℓ(γ), to be 2log(|λ|), where λ
is the eigenvalue of γ with |λ| ≥ 1. We define L(Γ) to be the set of translation lengths of hyperbolic
elements in Γ.
The group PSL2(R) is the group of orientation-preserving isometries of the hyperbolic plane.
Under the classical correspondence between (conjugacy classes of) hyperbolic γ ∈ Γ and closed
geodesics on the orbifoldH/Γ, the set L(Γ) can also be characterized as the set of lengths of closed
geodesics onH/Γ, where the latter is equipped with the hyperbolic metric.
The following is a well-known theorem of Dal’Bo.
Theorem 2 (Non-arithmeticity of the length spectrum [3]). Let Γ≤ PSL2(R) be a non-elementary
Fuchsian group. Then L(Γ) generates a dense additive subgroup of R.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
826 George Peterzil and Guy Sapire
Non-arithmeticity has been widely applied in homogeneous dynamics, for example in proving
the topological mixing of the geodesic flow and the density of horocycles originating from a
horocyclic limit point of Γ.
This result has been generalized to related notions of spectra in other settings as well, see
e.g. [1,4,5]. We make use of some basic results on radicals in finitely-generated field extensions
ofQ to give an elementary proof of the following result.
Theorem 3. Let Γ be a non-elementary Fuchsian group. Then L(Γ) ⊆ R contains two elements
which are linearly independent overQ.
This theorem is a special case of a result by Prasad and Rapinchuk (see [9, Remark 1]), which
was recently generalized in [2]. In fact, it is shown in [2, Proposition 1.8] that L(Γ) contains an
infinite set of lengths which is linearly independent overQ. Both theorems rely on deep results in
number theory.
It is worth noting that the results in Section 3 can be rephrased and proved in the arithmetic
language of heights, in particular using the Northcott property of Moriwaki’s height (see [7,
Section 4], and in general [10]).
2. Proof of the theorem
We first recall a fundamental notion in the theory of discrete groups.
Definition 4. Let Γ be a matrix group. The trace field of Γ is defined to be
kΓ=Q({tr(γ) : γ ∈ Γ})
We start with the following.
Lemma 5. The trace field of a finitely-generated matrix group Γ is finitely generated over Q.
Proof. If Γ = 〈γ1, . . . ,γn〉 then the trace field is contained in the extension of Q generated by the
entries of γ1, . . . ,γn , which is finitely-generated. □
Our final ingredient will be the following proposition, which follows directly from Lemma 5
and Proposition 10, which we will prove later.
Proposition 6. Let Γ ≤ PSL2(R) be a finitely-generated Fuchsian group, kΓ its trace field, r ∈ kΓ
positive with r ̸= 1, K /kΓ a finitely generated extension of kΓ. Then there are only finitely many
rational 0 < q < 1 such that r q is at most quadratic over K .
Proof of Theorem 3. Since every non-elementary Γ contains a non-elementary finitely gener-
ated subgroup, we will assume that Γ is finitely generated. Assume towards a contradiction that
L(Γ) ⊆ ℓ0Q for some ℓ0 ∈ L(Γ), and for every γ ∈ Γ, let qγ ∈ Q be such that ℓ(γ) = qγℓ0. Let
K = kΓ(e ℓ02 ) be the field generated by e ℓ02 over kΓ. We claim that for any γ ∈ Γ we have that e
{qγ}ℓ0
2
is at most quadratic over K , where {qγ} ∈ [0,1) is the fractional part of qγ. First recall that for any
γ ∈ Γ, by [6, Lemma 12.1.2] we have ∣∣tr(γ)∣∣= e ℓ(γ)2 +e− ℓ(γ)2
This means that x = e ℓ(γ)2 satisfies x + x−1 ∈ kΓ, so there is some λ ∈ kΓ such that x satisfies
x2 − xλ+1 = 0, hence e ℓ(γ)2 = e
qγℓ0
2 is at most quadratic over kΓ and thus also over K . This gives
us that e
{qγ}ℓ0
2 = e
qγℓ0
2 (e
ℓ0
2 )−⌊qγ⌋ is at most quadratic over K , where ⌊qγ⌋ denotes the integer part
of qγ. By Proposition 6 with r = e
ℓ0
2 , there is a bound n ∈ N on the denominators of such qγ,
hence L(Γ) ⊆ 1n!ℓ0Z and in particular
〈
L(Γ)
〉⊆R is a discrete subgroup, but this is a contradiction
to Theorem 2. □
George Peterzil and Guy Sapire 827
Remark 7. It is worth noting that a similar proof works for a general non-elementary group
Γ ≤ SO(n,1)+. To see this, write k ′Γ for the field generated by the entries of a (finite) generating
set of Γ. Then for every hyperbolic γ ∈ Γ with translation length ℓ, we have that eℓ is a root of the
characteristic polynomial of γ, which is of degree n +1 over k ′Γ. The rest of the proof follows the
same lines, using the general non-arithmeticity result proved in [5].
3. Radicals in finitely generated extensions ofQ
Proposition 8. Let K ⊆ R be a finitely generated extension of Q, r ∈ K positive with r ̸= 1. Then
there are only finitely many n ∈N such that r 1n ∈ K .
Proof. Write L = Q(r ) and denote by L̃ the relative algebraic closure of L in K . Note that L̃ is a
finite extension of L since it is algebraic over L and a subfield of K , which is finitely generated.
If r is algebraic, then L̃ is a number field (i.e. a finite extension of Q). Write OL̃ for the ring of
integers of L̃, that is, the integral closure ofZ in L̃, and suppose first r ∈O×
L̃
. Since r is integral and
OL̃ is integrally closed in L̃, any element of the form r
1
n is in O×
L̃
. By Dirichlet’s unit theorem [8,
Theorem 7.4] this group is free up to roots of unity. Because r > 0 and is different from 1 it is
not a root of unity, so it can only have finitely many roots in OL̃ . If r ∉ O×L̃ , recall that by the
unique factorization of fractional ideals [8, Corollary 3.9], there exist prime ideals p1, . . . ,pt ⊆ OK
and e1, . . . ,et ∈Z\ {0} such that
(r ) = pe11 · · ·pett
so for every k ∈N, if a k-th root of r is in K we get
(r 1/k )k = (r ) = pe11 · · ·pett
and so from uniqueness necessarily k
∣∣ei for every i ≤ t , meaning the only roots of r that could
be in K are d-th roots for d divisors of gcd(e1, . . . ,et ).
Otherwise r is transcendental. Because r is a free generator of L, the degree of r
1
n over L is n
and so for every n > [L̃ : L] necessarily r 1n ∉ K . □
The following basic lemma will be of use.
Lemma 9. Let K ⊆ R be a field, t ,n ∈ N and α ∈ R× such that αn ∈ K and [K (α) : K ] = t , then
αt ∈ K .
Proof. The minimal polynomial p(x) of α over K is of degree t . Since xn −αn = ∏ni=1(x − ζknα)
(for ζn a fixed primitive root of unity) has coefficients in K and has a root α, we must have
p(x)
∣∣xn −αn , hence there are 1 ≤ k1, . . . ,kt ≤ n such that p(x) =∏ti=1(x −ζkin α), and in particular
the constant coefficient of p(x) is (−1)tζk1+···+ktn αt ∈ K . Since K ⊆ R and αt ∈ R, (−1)tζk1+···+ktn
is a real root of unity hence equal to ±1, so the constant coefficient of p(x) is ±αt ∈ K and so
αt ∈ K . □
Using this, we prove the following.
Proposition 10. Let K ⊆ R be a finitely generated extension of Q, r ∈ K positive, r ̸= 1 and t ∈N.
Then there are only finitely many n ∈N such that [K (r 1n ) : K ]= t . Moreover, there are only finitely
many rational 0 < q < 1 such that r q is of degree at most t over K .
Proof. Suppose n ∈ N satisfies [K (r 1n ) : K ] = t . By Lemma 9 for α = r 1n we get r tn ∈ K and
by Proposition 8 there are only finitely such n. Let n0 be minimal such that for all n ≥ n0 we
have that r
1
n is of degree strictly greater than t over K . Assume towards a contradiction that
there is some rational 0 < q = mn < 1 such that r q ∈ K , with gcd(m,n) = 1. By Bézout’s identity,
there are integers x, y ∈ Z such that mx + ny = 1, so in particular r 1n = (r q )x r y ∈ K (r q ), so[
K (r q ) : K
]≥ [K (r 1n ) : K ]> t , arriving at a contradiction. □
828 George Peterzil and Guy Sapire
Acknowledgments
This paper is part of the PhD thesis of G. P. conducted at the Hebrew University of Jerusalem,
under the supervision of Or Landesberg. G. P. would like to thank Or for suggesting this ques-
tion and for reviewing multiple versions of this paper. G. S. would like to thank Borys Kadets for
enriching conversations, and Ari Shnidman for reviewing the paper. The authors would like to
thank Françoise Dal’Bo for several insightful comments on a previous version of the manuscript,
and to Rafael Potrie for pointing out the references [2,9].
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
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