On spectral gaps of Markov maps

0.34 MB·

by  José Manuel Conde-Alonso

ON SPECTRAL GAPS OF MARKOV MAPS
JOSE´ M.CONDE-ALONSO,JAVIERPARCET,ANDE´RICRICARD
Abstract. ItisshownthatifaMarkovmapT onanoncommutativeprobabilityspaceMhasa
spectralgaponL2(M),thenitalsohasoneonLp(M)for1<p<∞. Forfixedp,theconverse
alsoholdsifT isfactorizable. Someresultsarealsonewforclassicalprobabilityspaces.
7
1
0
2 Introduction

b
e Many definitions of spectral gaps have been considered for linear operators. They are intersting
F astheyoftenyieldnicepropertiesforfunctionalcalculusorergodictheory. Inthisnoteweconsider
contractive linear maps T on (noncommutative) L -spaces whose fixed points are 1-complemented
6 p
1 by some projection E. Then we say that T has a Lp-spectral gap when (cid:107)T(1−E)(cid:107)<1. Of course,
in this situation, 1 is at a positive distance from the rest of the spectrum of T. When T and E can

] be considered simultaneously on all L (1(cid:54)p(cid:54)∞) it is a natural question to know if L -spectral
R p p
gaps can be interpolated. This is precisely the topic what we address in this note. Our motivation
P
comes from the paper [4], where this was the key to obtain certain interpolation results that in
.
h turn yielded Calder´on-Zygmund estimates in nondoubling contexts. We focus on the particular
t
a class of Markov maps. In other words, unital, completely positive, trace preserving maps acting on
m noncommutative probability spaces. In the commutative situation, they exactly correspond to the
[ usual Markov operators. Our main result reads as follows (precise definitions below):

3 Theorem A. Given any Markov map T and 1<p<∞:
v
3 (1) If T has an L -spectral gap, then it also has an L -spectral gap.
2 p
4 (2) If T has an L -spectral gap and is factorizable, then it also has an L -spectral gap.
p 2
0
0
Anotherwayofformulatingourmainresultissayingthat,undertheadditional(andverynatural
0
. in examples) condition of being factorizable, if T has an Lp-spectral gap for some 1<p<∞ then
2
it also does for all 1 < q < ∞. When the underlying space is a classical probability space, the
0
assumption is automatic. The rest of the paper is devoted to developing the necessary machinery
7
1 and definitions and to the proof of Theorem A. We will also show an application to interpolation
: theory in a context —inspired by the aforementioned [4]— where we have two L spaces over the
v p
i same measure space quotiented by two different subalgebras.
X
InmostexamplesL -spectralgapsareeasytodetermine,thinkforinstanceofFouriermultipliers
r 2
a over the torus. It turns out that they also behave quite well with respect to algebraic operations.
For instance if T and S have an L -spectral gap, then the tensor product map T ⊗S also does.
2
Thus, Theorem A can be used to produce many examples of L -spectral gaps.
p
2010Mathematics Subject Classification: 46L51;47A30.
Key words: NoncommutativeLp spaces,Markovmaps,spectralgap.
JM Conde-Alonso was supported in part by ERC Grant32501. J Parcet was supported in part by CSIC Grant
PIE201650E030(Spain).
1
2 J.M.CONDE-ALONSO,J.PARCET,ANDE´.RICARD
To prove Theorem A one may be tempted to use an ultraproduct argument and Mazur maps.
Thisprobablycouldbedonebutwouldrequirelotsoftechnicalities,especiallyinthenoncommuta-
tivesituationasultraproductsofnoncommutativeprobabilityspacesarenotprobabiltyspacesany
longer(onehastodealwithtypeIIIalgebras). Ourapproachhastheadvantagetogivequantitative
estimates for the spectral gaps.
K. Oleszkiewicz kindly informed us that (1) is also proved in [6] in the commutative situation.
The proof given there also works in the noncommutative setting.
1. Markov maps and noncommutative L spaces
p
We work in the general setting of noncommutative integration, for which a rather complete
introduction and definitions can be found in [7]. Let (M,τ) be a noncommutative probability
space, so that M is a finite von Neumann algebra equipped with a normal faithful tracial state τ.
Given 1(cid:54)p<∞, the noncommutative L spaces associated to (M,τ) are defined as
p
L (M)=(cid:110)f ∈L (M,τ) : (cid:107)f(cid:107) =τ(cid:0)|f|p(cid:1)p1 <∞(cid:111).
p 0 p
Above, L (M,τ) denotes the set of τ-measurable operators. Strictly speaking, we should refer
0
to the trace τ in the notation for L (M), but this will not be relevant here. As usual, we can
p
think that M = L (M) is represented in B(L (M)), the bounded linear operators on L (M),
∞ 2 2
by left multiplications. It is possible to avoid L (M,τ) in the definition of the L spaces in our
0 p
situation: L (M) is just the completion of M in the L norm, because the finiteness of τ yields
p p
L (M) ⊂ L (M). In the commutative situation, M is just L (Ω) over some probability space
∞ p ∞
(Ω,µ), τ is the integration against µ and L (M)=L (Ω) for 1<p<∞.
p p
Noncommutative L spaces share many properties of classical L , but usually inequalities for
p p
operators are more difficult to deal with. To overcome some of the difficulties that arise due to
noncommutativity, the main technical tools we will be relying on are estimates on Mazur maps.
The Mazur map is the classical norm preserving map
M :L (M)→L (M) givenby M (f)=f|f|p/q−1,
p,q p q p,q
where as usual we take |f|2 =f∗f, as in the definition of the noncommutative L norm. We know
p
from [8] that the map M is min{1,p}-H¨older continuous on spheres, just as in the commutative
p,q q
case. We are interested in working with a particular set of maps.
Definition 1.1. A map T :M→M is called Markov on (M,τ) when:
i) T is unital: T(1 )=1 ,
M M
ii) T is completely positive: (T(x ))∈M (M) for all (x )∈M (M) ,
i,j n + i,j n +
iii) T is trace preserving: τ ◦T =τ.
It is then classical that T :L (M)→L (M) admits a unique contractive extension to L (M)
∞ ∞ p
for 1(cid:54)p(cid:54)∞ that we will still denote by T. More generally, one can give a definition of a Markov
map T : M → N between two semifinite von Neumann algebras. Let us see now some standard
examples of Markov maps:
(1) Given a classical probability space (Ω,µ), any unital, positive and measure preserving map
T : L (Ω) → L (Ω) is Markov. For instance, it may be given by a composition operator
∞ ∞
T(f)(x)=f(ϕ(x)), where ϕ:Ω→Ω is any measure preserving transformation.
ON SPECTRAL GAPS OF MARKOV MAPS 3
(2) Let G be a discrete group. Its associated group von Neumann algebra
(cid:110) (cid:111)(cid:48)(cid:48)
L(G)= λ(g):g ∈G ⊂B((cid:96) (G))
2
is the von Neumann algebra generated by the left regular representation λ(g). It can
be naturally viewed as a noncommutative probability space with the trace given by the
vector state associated to δ , where e is the unit in G. Any normalized positive definite
e
function c : G → C gives rise to a Fourier multiplier F (λ(g)) = c λ(g) that is a Markov
c g
map. Moreover, when G is abelian with compact Pontryagin dual G(cid:98) and normalized Haar
measure µ, then Lp(L(G))=Lp(G(cid:98),µ). Any Markov Fourier multiplier Fc as above is then
given by the convolution on G(cid:98) with a probability measure γ such that γ(cid:98)(g)=cg.
(3) If M=M , the family of n×n matrices equipped with its normalized trace, any Markov
n
map is given by T(x)=(cid:80)N a xa∗ with a ∈M such that
(cid:96)=1 (cid:96) (cid:96) (cid:96) n
N N
(cid:88) (cid:88)
1Mn = a(cid:96)a∗(cid:96) = a∗(cid:96)a(cid:96).
(cid:96)=1 (cid:96)=1
For instance, if S((x )) = (s x ) is a Schur multiplier, it is Markov if and only if
i,j i,j i,j
(s )(cid:62)0 and s =1 for all i.
i,j i,i
(4) Any ∗-representation π :M→M is a Markov map if and only if it is trace preserving.
(5) If M is finite and N ⊂M is a von Neumann subalgebra, then the trace preserving condi-
tional expectation onto N, E :M→M, is a Markov map.
N
Given any Markov map T : M → M, the set of its fixed points is a von Neumann sub-algebra
N ⊂M. We know from [3] that this algebra is exactly the multiplicative domain of T. The same
holds for the L extension of T: the points fixed by T on L (M) coincide with L (N). Moreover,
p p p
the conditional expectation E onto N commutes with T. This can be seen by applying the von
N
Neumann ergodic theorem to T on L (M). We now make precise the notion of L -spectral gap
2 p
that we shall be using. To that end, we need to introduce the following notation:
(cid:110) (cid:111)
L0(M)= x∈L (M) : E (x)=0 .
p p N
L0(M) is complemented in L (M) by Id−E . The notion of L -spectral gap is then given by
p p N p
certain norm estimates:
Definition 1.2. We say that a Markov map T :M→M with fixed points algebra N has a spectral
gap on L (M) if
p
cp :=(cid:13)(cid:13)T :L0p(M)→L0p(M)(cid:13)(cid:13)<1,
that is, if there is a constant c<1 such that for any x∈L (M) with E x=0, we have (cid:107)T(x)(cid:107) (cid:54)
p N p
c(cid:107)x(cid:107) .
p
We can now justify the fact that having an L -spectral gap with constant c < 1 implies that 1
p
has to be an isolated point of the spectrum. Indeed, suppose not and that for (cid:15) arbitrarily small
(any (cid:15)<(1−c) suffices) λ is an element of the spectrum of T such that |1−λ|<(cid:15) with associated
eigenvector x. Then we may consider the vector z = x−E x, which belongs to L0(M). Since T
N p
commutes with E ,
N
T(z)=T(x)−T(E (x))=T(x)−E (T(x))=λ(x−E x)=λz,
N N N
4 J.M.CONDE-ALONSO,J.PARCET,ANDE´.RICARD
soz isalsoaneigenvectorassociatedtothesameeigenvalueand(cid:107)Tz(cid:107) =|λ|(cid:107)z(cid:107) >c(cid:107)z(cid:107) ,violating
p p p
the L -spectral gap condition.
p
Duetocomplementation,oneistemptedtotrytorelatespectralgapsusingcomplexinterpolation
and prove Theorem A in that way. But since (cid:107)Id−E :M→M(cid:107)=2 in general, this only gives
N
thatc (cid:54)c2/p21−2/p forp>2,whichisusuallynotenough. Thisiswhyweneedtoemployanother
p 2
approach based on the use of Mazur maps mentioned above. Also, for the backwards direction of
Theorem A, we will need to consider a particular type of Markov maps:
Definition 1.3. A Markov map T on (M,τ) is factorizable if there exist a bigger finite von Neu-
mann algebra (M˜,τ˜)⊃(M,τ) and a ∗-representation π :M→M˜ such that
∀x∈M, τ(x)=τ˜(x)=τ˜(π(x)) and T(x)=E π(x).
M
Theorem A then states that if T is factorizable in the sense of definition 1.3 then having a
spectral gap in L (M) is equivalent to having a spectral gap in L (M). This notion appeared in
p 2
[1] and turned out to be quite useful to deal with analytical problems. It follows from [5] that
there are Markov maps that are not factorizable. However, most natural examples are, see [10]. If
M=L (Ω) then all Markov maps are factorizable. This corresponds to the basic construction of
∞
Markov chains. On the other hand, a Fourier multiplier F on a discrete group G is factorizable if
c
c : G → R is positive definite, and a Schur multiplier S is factorizable if it is a Markov map and
(m )∈M (R). Finally, a product of factorizable maps is still factorizable.
i,j n
2. L -spectral gap implies L -spectral gap
2 p
This section is devoted to the proof of the forward direction of Theorem A, which is contained
in the next result. We keep all notations from the previous paragraphs.
Theorem2.1. AssumethataMarkovmapT on(M,τ)hasaspectralgaponL (M)withconstant
2
c < 1. Then T also admits a spectral gap in L (M) for any 1 < p < ∞ with constant c =
2 p p
c(p,c )<1. Moreover, the following estimates hold for some universal C >0:
2

p−1 if p<2,
1−c 
lim p (cid:62)C 1(cid:16)log2(cid:17)p
c2→11−c2 p 2p if p>2.
The proof of Theorem 2.1 requires a couple of short auxiliary lemmas.
Lemma 2.2. Let 1<p<2, x∈L+(M) and T as above. Then
p
(cid:107)T(x)(cid:107) (cid:54)(cid:107)T(xp)(cid:107)2/p−1(cid:107)T(xp/2)(cid:107)2−2/p.
p 1 2
Proof. This is a standard application of complex interpolation. If θ = 2 − 2, then L (M) is
p p
the interpolated space (L (M),L (M)) . Consider the function F(z) = T(xp−zp/2), which is
1 2 θ
holomorphic on the strip S ={0<Rez <1} and continuous on S. By interpolation
(cid:107)T(x)(cid:107) =(cid:107)F(θ)(cid:107) (cid:54)sup(cid:107)F(it)(cid:107)1−θsup(cid:107)F(1+it)(cid:107)θ.
p p 1 2
t∈R t∈R
Recall the following factorization: for any y ∈L (M) there exists a contraction γ ∈M such that
p
T(y)=T(|y∗|)1/2γT(|y|)1/2.
ON SPECTRAL GAPS OF MARKOV MAPS 5
Therefore,ify isnormalthenbyH¨older’sinequalityweknowthat(cid:107)T(y)(cid:107) (cid:54)(cid:107)T(|y|)(cid:107) . Wededuce
q q
that for any t∈R, (cid:107)F(it)(cid:107) (cid:54)(cid:107)T(xp)(cid:107) and (cid:107)F(1+it)(cid:107) (cid:54)(cid:107)T(xp/2)(cid:107) . (cid:3)
1 1 2 2
Lemma 2.3. Let α(cid:62)1, p(cid:62)1 and T as above. Then for all x∈L+ (M):
pα
(cid:107)T(xα)(cid:107) (cid:62)(cid:107)T(x)α(cid:107) .
p p
Proof. The fact that p(cid:62)1 ensures that all elements are well defined. By operator convexity of the
map t(cid:55)→tα, the result is obvious if α ∈[1,2] because 0 (cid:54)T(x)α (cid:54)T(xα). Therefore, to conclude
it suffices to note that if the lemma holds for α, it also holds for 2α. Indeed,
(cid:107)T(x)2α(cid:107) =(cid:107)T(x)2(cid:107)α (cid:54)(cid:107)T(x2)(cid:107)α =(cid:107)T(x2)α(cid:107) (cid:54)(cid:107)T(x2α)(cid:107) .
p αp αp p p
(cid:3)
Proof of Theorem 2.1. Let x ∈ L0(M) with (cid:107)x(cid:107) = 1, and assume (cid:107)T(x)(cid:107) = γ. We will give
p p p
an upper bound for γ. To do so, we first notice that we can assume x = x∗. This can be justified
by the use of the so-called 2×2 trick. Indeed, consider
(cid:18) (cid:19)
1 0 x
x˜= ∈L (M ⊗M).
21/p x∗ 0 p 2
Then one has (cid:107)x˜(cid:107)p =1 and (cid:107)IdM2 ⊗T(x˜)(cid:107)p =γ, x˜∗ =x˜, IdM2 ⊗EN(x˜)=0 and IdM2 ⊗T is still a
Markov map on M ⊗M.
2
Using that x is self-adjoint, we write x = x −x the decomposition of x into its positive and
+ −
negative parts. Without loss of generality we can assume (cid:107)x (cid:107)p (cid:62) 1. Define γ by (cid:107)T(x )(cid:107) =
+ p 2 ± ± p
γ (cid:107)x (cid:107) . We next use the fact that if a,b(cid:62)0 then (cid:107)a−b(cid:107)p (cid:54)(cid:107)a(cid:107)p+(cid:107)b(cid:107)p. Applying it to T(x )
± ± p p p p +
and T(x ) yields
−
(cid:107)T(x )(cid:107)p+(cid:107)T(x )(cid:107)p (cid:62)(cid:107)T(x)(cid:107)p =γp.
+ p − p p
Therefore we have γp(cid:107)x (cid:107)p + γp(cid:107)x (cid:107)p (cid:62) γp. At this point we need to distinguish two cases
+ + p − − p
according to the value of p.
Case p<2. Lemma 2.2 applied to x gives (cid:107)T(xp/2)(cid:107)2−2/p(cid:107)T(xp)(cid:107)2/p−1 (cid:62)γ (cid:107)x (cid:107) . Since T is
+ + 2 + 1 + + p
a contraction on L (M), we get
1
p
(cid:107)T(xp/2)(cid:107) (cid:62)γ2p−2(cid:107)x (cid:107)p/2.
+ 2 + + p
On the other hand, by orthogonality we have
(cid:107)x (cid:107)p =(cid:107)xp/2(cid:107)2 =(cid:107)xp/2−E (xp/2)(cid:107)2+(cid:107)E (xp/2)(cid:107)2.
+ p + 2 + N + 2 N + 2
Next, we write T(xp/2)=(cid:0)T(xp/2)−E T(xp/2)(cid:1)+E T(xp/2). Then, by orthogonality again and
+ + N + N +
the L -spectral gap assumption, we get
2
2p
γ2p−2(cid:107)x (cid:107)p (cid:54)c2(cid:107)xp/2−E (xp/2)(cid:107)2+(cid:107)E (xp/2)(cid:107)2,
+ + p 2 + N + 2 N + 2
which yields
(cid:0)γ2p2−p2 −c2(cid:1)(cid:107)xp/2−E (xp/2)(cid:107)2 (cid:54)(cid:0)1−γ2p2−p2)(cid:107)E (xp/2)(cid:107)2 (cid:54)(cid:0)1−γ2p2−p2)(cid:107)x (cid:107)p.
+ 2 + N + 2 + N + 2 + + p
We can go back now to L (M) by raising to the power 2/p, see Lemma 2.2 in [8]. This means that
p
we get
(cid:107)x −E (xp/2)2/p(cid:107) (cid:54)3(cid:107)xp/2−E (xp/2)(cid:107) (cid:107)x (cid:107)1−p/2,
+ N + p + N + 2 + p
6 J.M.CONDE-ALONSO,J.PARCET,ANDE´.RICARD
and since E (xp/2)2/p ∈L (N) we arrive at
N + p
(cid:107)x −E (x )(cid:107) (cid:54)2(cid:107)x −E (xp/2)2/p(cid:107) (cid:54)6(cid:107)xp/2−E (xp/2)(cid:107) (cid:107)x (cid:107)1−p/2.
+ N + p + N + p + N + 2 + p
We conclude that either γ (cid:54)c(2p−2)/p, in which case we are done, or
+ 2
(cid:118)
(cid:117) 2p
(cid:117) 1−γ2p−2
(cid:107)x −E (x )(cid:107) (cid:54)6(cid:117) + (cid:107)x (cid:107) .
+ N + p (cid:116) 2p + p
γ2p−2 −c2
+ 2
(cid:112)
Obviously, the same estimate holds for x and γ . Denote ϕ(t) = (1−t)/(t−c2). Since we
− − 2
know that E (x)=E (x )−E (x )=0, we also have 1(cid:54)(cid:107)x −E (x )(cid:107) +(cid:107)x −E (x )(cid:107) .
N N + N − + N + p − N − p
Ifγ (cid:54)c(2p−2)/p,thensinceγp (cid:54)(1+γp)/2,wegetanupperestimateandwearedone. Therefore,
+ 2 +
we can assume that γ >c(2p−2)/p.
+ 2
Wenowsplitagainintotwocases. First,if(cid:107)x (cid:107) (cid:54)1/4,then6(cid:54)(cid:107)x −E (x )(cid:107) andtherefore
− p + N + p
2p
ϕ(γ2p−2)(cid:62)1/12. That means
+
γ (cid:54)(cid:16)122+c22(cid:17)2p2−p2 andhence γp (cid:54) 1+γ+p (cid:54) 1+(cid:16)112222++c122(cid:17)2p2−p2.
+ 122+1 2 2
Finally, if (cid:107)x (cid:107) > 1/4, then γp (cid:54) 1−(1−γp)/(4p) and as above we can assume γ > c(2p−2)/p.
− p − − 2
2p 2p
But then 1/6(cid:54)ϕ(γ2p−2)+ϕ(γ2p−2) and one of these two terms has to be bigger than 1/2. Hence
+ −
γ ∧γ (cid:54)(cid:16)122+c22(cid:17)2p2−p2, whichalsogivesthebound γp (cid:54)1− 1−(cid:16)112222++c122(cid:17)2p2−2.
+ − 122+1 4p
This is the worst possible bound. Regarding the quantitative behavior when c → 1, it is easy to
2
check that there is some C >0 independent of p∈]1,2] so that
1−c
lim p (cid:62)C(p−1).
c2→11−c2
Case p>2. We use Lemma 2.3 this time to get
(cid:107)T(xp/2)(cid:107) (cid:62)γp2(cid:107)x (cid:107)p/2.
+ 2 + + p
As in the situation when p<2, from the above display we derive
(cid:0)γp −c2(cid:1)(cid:107)xp/2−E (xp/2)(cid:107)2 (cid:54)(cid:0)1−γp)(cid:107)x (cid:107)p.
+ 2 + N + 2 + + p
In this case, the way to go back to L (M) by raising to power 2/p is via Ando’s inequality (see
p
Lemma 2.2 in [2]):
(cid:107)x −E (x )(cid:107) (cid:54)2(cid:107)x −E (xp/2)2/p(cid:107) (cid:54)2(cid:107)xp/2−E (xp/2)(cid:107)2/p.
+ N + p + N + p + N + 2
So either γ (cid:54)c2/p or
+ 2
(cid:18) 1−γp (cid:19)1/p
(cid:107)x −E (x )(cid:107) (cid:54)2 + (cid:107)x (cid:107) =:2ψ(γp)(cid:107)x (cid:107) .
+ N + p γp −c2 + p + + p
+ 2
ON SPECTRAL GAPS OF MARKOV MAPS 7
We discuss as before: if γ (cid:54) c2/p then γ (cid:54) [(1+c2)/2]1/p. Otherwise, by assumption we have
+ 2 2
(cid:107)x (cid:107) (cid:62) 1/2 so that (cid:107)x (cid:107) (cid:54) 1/21/p. But by Corollary 2.5 in [9], for a (cid:62) 0 and p > 2, we have
+ p − p
(cid:107)a−E a(cid:107) (cid:54)(cid:107)a(cid:107) . This implies that
N p p
1
1− (cid:54)1−(cid:107)x (cid:107) (cid:54)1−(cid:107)x −E (cid:107) (cid:54)(cid:107)x −E x (cid:107) ,
1 − p − N p + N + p
2p
so one gets δ := 1(1− 1 )(cid:54)ψ(γp). This leads to
p 2 21/p +
(cid:32)1+δp1+c22(cid:33)1/p
γ (cid:54) p 2 ,
1+δp
p
which is enough for our purpose. Finally, one easily checks that for some universal C,
1−c C(cid:16)log2(cid:17)p
lim p (cid:62) .
c2→11−c2 p 2p
(cid:3)
Remark 2.4. As pointed out in [4], the result above is false for p = 1,∞, even when T is a
conditional expectation.
Remark 2.5. In Theorem 2.1 the requirement that (M,τ) is a probability space can be relaxed.
If τ is only semifinite, the conclusion of the theorem holds if one adds the additional assumption
that the set of fixed points satisfies
(cid:110) (cid:111)
x:(cid:107)T(x)(cid:107) =(cid:107)x(cid:107) =L (N)
2 2 2
for some von Neumann algebra N that is semifinite for τ. Notice that this new requirement is
necessary in the general case. Indeed, consider T :B((cid:96) )→B((cid:96) ) given by x(cid:55)→sxs∗, where s is a
s 2 2
unilateral shift. Then, T is Markov and the set of fixed points is not a von Neumann algebra.
s
Remark 2.6. Forcommutativeprobabilityspaces,Theorem2.1hasanelegantproofin[6],Propo-
sition 4.1. All the arguments there carry over to von Neumann algebras. This provides a different
estimate for c , namely c (cid:54)(1−22−p∗(1−c ))1/p∗, where p∗ =max{p,p(cid:48)}. This behavior of c is
p p 2 p
better than ours for p>2 and c close to 1.
2
3. L -spectral gap implies L -spectral gap
p 2
Wekeepthesamesettingasintheprevioussection. Thistimeweonlyneedoneauxiliarylemma.
Lemma 3.1. Let T :M→M be a Markov map. Then for all y ∈L (M)
2
(cid:107)T(M (y))−M (y)(cid:107) (cid:54)C(cid:107)T(y)−y(cid:107)θ(cid:107)y(cid:107)1−θ,
2,p 2,p p 2 2
for some universal constant C >0 and θ = 1min{p,2}.
4 2 p
Proof. This is a variant of Corollary 2.4 (and Remark 2.5) in [2] where this is done for p>2. Let
1<p<2fortherestoftheproof;thisistheonlycasethatweneedtoconsider. Wewantanupper
bound for T(M (y))−M (y). As in section 2, by the 2×2-trick, we can reduce to proving the
2,p 2,p
result for y =y∗ ∈L (M). Again decompose y =y −y , so that
2 + −
(cid:104) (cid:105) (cid:104) (cid:105)
T(M (y))−M (y)=T(y2/p−y2/p)−(y2/p−y2/p)= T(y2/p)−y2/p − T(y2/p)−y2/p .
2,p 2,p + − + − + + − −
8 J.M.CONDE-ALONSO,J.PARCET,ANDE´.RICARD
We shall prove the desired estimate for y and y separately instead of working with y. To that
+ −
end, write
(cid:107)T(y2/p)−y2/p(cid:107) (cid:54)(cid:107)T(y2/p)−T(y )2/p(cid:107) +(cid:107)T(y )2/p−y2/p(cid:107) =:I+II.
+ + p + + p + + p
WeshallestimateIandIIseparately. Byoperatorconvexityoft(cid:55)→tγ forγ ∈[1,2]anditsoperator
concavity for γ ∈]0,1], we get
0(cid:54)T(y2/p)−T(y )2/p (cid:54)T(y2)1/p−T(y )2/p.
+ + + +
Then, by Ando’s inequality we get
Ip (cid:54) (cid:107)T(y2)−T(y )2(cid:107)
+ + 1
= τ(T(y2)−T(y )2)=τ(y2 −T(y )2)
+ + + +
(cid:54) 2(cid:107)y −T(y )(cid:107) (cid:107)y (cid:107) .
+ + 2 + 2
On the other hand, by Lemma 2.2 in [8], II(cid:54)3(cid:107)T(y )−y (cid:107) (cid:107)y (cid:107)2/p−1, and so
+ + 2 + 2
II(cid:54)C(cid:107)T(y )−y (cid:107)1/p(cid:107)y (cid:107)1/p
+ + 2 + 2
forsomeuniversalC. Ofcourse,thesameestimatesapplytoy . Butweknowthat(cid:107)T(y )−y (cid:107)2 (cid:54)
− ± ± 2
2(cid:107)T(y)−y(cid:107) (cid:107)y(cid:107) by the proof of Corollary 2.4 in [2]. Finally, we collect everything and we get
2 2
(cid:107)T(M (y))−M (y)(cid:107) (cid:54) I+II
2,p 2,p p
(cid:16) (cid:17)
(cid:54) C (cid:107)T(y)−y(cid:107)1/p(cid:107)y(cid:107)1/p+(cid:107)T(y)−y(cid:107) (cid:107)y(cid:107)2/p−1
2 2 2 2
(cid:54) C(cid:107)T(y)−y(cid:107)1/2p(cid:107)y(cid:107)3/2p,
2 2
for some universal C >0, which is enough. (cid:3)
Remark 3.2. The exponent θ is probably not optimal.
Theorem 3.3. Assume T is a factorizable Markov map. Let 1<p(cid:54)=2<∞ and assume there is
some constant c <1 such that for all x∈L0(M)
p p
(cid:107)T(x)(cid:107) (cid:54)c (cid:107)x(cid:107) .
p p p
Then there exists a constant c =c(p,c )<1 such that for all x∈L0(M)
2 p 2
(cid:107)T(x)(cid:107) (cid:54)c (cid:107)x(cid:107) .
2 2 2
Proof. We remind the reader that the factorizability assumption means that there is another finite
von Neumann algebra (M˜,τ˜) containing (M,τ) with a trace preserving conditional expectation
E and a trace preserving ∗-representation π : M → M˜ so that T(x) = Eπ(x) for x ∈ M. Let
x∈L (M) with E (x)=0 and (cid:107)x(cid:107) =1. We want to find a lower bound for δ =(cid:107)x(cid:107) −(cid:107)T(x)(cid:107) .
2 N 2 2 2
First notice that by orthogonality δ2 =(cid:107)Eπ(x)−π(x)(cid:107)2 =δ2. This also yields that for any y ∈N,
2
π(y)=y and E(y)=y.
The idea is to use the hypothesis via the properties of the Mazur map. By Lemma 3.1,
(cid:107)EM (π(x))−M (π(x))(cid:107) (cid:54) δθ. Set λ := E (M (x)). Recalling that Eπ(λ) = π(λ) = λ
2,p 2,p p N 2,p
ON SPECTRAL GAPS OF MARKOV MAPS 9
because M (π(x))=π(M (x)), the hypothesis yields
2,p 2,p
(cid:107)E(π(M (x))−λ)(cid:107) = (cid:107)T(M (x))−λ(cid:107)
2,p p 2,p p
(cid:54) c (cid:107)M (x)−λ(cid:107) = c (cid:107)π(M (x)−λ)(cid:107)
p 2,p p p 2,p p
(cid:54) c [(cid:107)T(M (x))−λ(cid:107) +(cid:107)(Id−E)(π(M (x))−λ)(cid:107) ].
p 2,p p 2,p p
Therefore, we get
(1−c )(cid:107)M (x)−λ(cid:107) (cid:54)δθ.
p 2,p p
Takingnowγ =min{1,p},theMazurmapM isγ-H¨olderwithconstantCpbythemaintheorem
2 p,2
in [8] (for some universal C). Hence
(cid:18) δθ (cid:19)γ
1=(cid:107)x(cid:107) =(cid:107)(1−E )(x−M (λ))(cid:107) (cid:54)(cid:107)x−M (λ)(cid:107) (cid:54)Cp .
2 N p,2 2 p,2 2 1−c
p
This means that for some universal C
(cid:18)C (cid:19)2/θ
δ (cid:62) (1−c ) =1−c .
p p 2
(cid:3)
4. An illustration
Motivated by questions from interpolation theory in the paper [4], we give an illustration of
Theorem A. Let (M,τ) be a noncommutative probability space and assume that A and B are
sub-algebras so that A∩B = N. Consider the Markov map T = E E . Our assumption yields
A B
that its fixed points algebra is exactly N. On L0(M), one can define a norm by
p
(cid:107)x(cid:107) =(cid:107)(1−E )x(cid:107) +(cid:107)(1−E )x(cid:107) .
Σ,p A p B p
Ofcourseonehas(cid:107)x(cid:107) (cid:54)4(cid:107)x(cid:107) . Oneimportantquestionin[4]wastoknowiftheyareequivalent.
Σ,p p
This is false in general:
Proposition 4.1. Assume 1 < p < ∞, then (cid:107).(cid:107) , (cid:107).(cid:107) are equivalent on L0(M) if and only if
Σ,p p p
E E has an L -spectral gap.
A B p
Proof. The direction in which we assume that T has a spectral gap was done in [4]. However, we
include the argument here. Indeed, let x∈L0(M), then
p
(cid:16) (cid:17)
(cid:107)E E x(cid:107) (cid:54)c (cid:107)x(cid:107) (cid:54)c (cid:107)x−E x(cid:107) +(cid:107)E (x−E x)(cid:107) +(cid:107)E E x(cid:107) .
A B p p p p A p A B p A B p
We deduce (cid:107)E E x(cid:107) (cid:54) cp (cid:107)x(cid:107) . Since
A B p 1−cp Σ,p
(cid:107)x(cid:107) (cid:54)(cid:107)x−E x(cid:107) +(cid:107)E (x−E x)(cid:107) +(cid:107)E E x(cid:107) ,
p A p A B p A B p
we get (cid:107)x(cid:107) (cid:54) 1 (cid:107)x(cid:107) .
p 1−cp Σ,p
Assume now T has no spectral gap. Then there exists a sequence of norm one elements (x ) in
n
L0(M) such that (cid:107)E E x (cid:107) → 1. Necessarily (cid:107)E x (cid:107) → 1, and thus by the uniform convexity
p A B n p B n p
of L (M), we must have (cid:107)E x −x (cid:107) → 0. Similarly we know that (cid:107)E E x −E x (cid:107) → 0.
p B n n p A B n B n p
10 J.M.CONDE-ALONSO,J.PARCET,ANDE´.RICARD
This implies that (cid:107)x(cid:107) → 0 due to the fact that (cid:107)E x −x (cid:107) → 0, which in turn follows from
Σ,p A n n p
the above and the decomposition
E x −x =E (x −E x )+(E E x −E x )+(E x −x ).
A n n A n B n A B n B n B n n
(cid:3)
Corollary 4.2. The following are equivalent
(1) For some 1<p<∞, (cid:107).(cid:107) , (cid:107).(cid:107) are equivalent on L0(M).
Σ,p p p
(2) For all 1<p<∞, (cid:107).(cid:107) , (cid:107).(cid:107) are equivalent on L0(M).
Σ,p p p
(3) For some 1<p<∞, E E has a L -spectral gap.
A B p
(4) For all 1<p<∞, E E has a L -spectral gap.
A B p
If this holds then the norms ((cid:107).(cid:107) ) on L0 (M) form an interpolation chain.
Σ,p 1<p<∞ ∞
Proof. Note that E E is factorizable, so this is an easy combination of Proposition 4.1 and Theo-
A B
rems 2.1 and 3.3. (cid:3)
Remark 4.3. It follows from the symmetry that if E E has an L -spectral gap, E E also does.
A B p B A
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(2008)
Departament de Matema`tiques, Facultat de Cie`ncies,
Universitat Auto`noma de Barcelona, 08193 Barcelona, Spain
E-mail address: [email protected]
InstitutodeCienciasMatema´ticasCSIC-UAM-UC3M-UCM,ConsejoSuperiordeInvestigacionesCient´ıficas
C/ Nicola´s Cabrera 13-15. 28049, Madrid. Spain
E-mail address: [email protected]
Laboratoire de Mathe´matiques Nicolas Oresme, Universite´ de Caen Normandie,14032 Caen Cedex,
France
E-mail address: [email protected]