Subgaussian 1-cocycles on discrete groups
We prove the Poincaré inequalities with constant for -cocycles on countable discrete groups under Bakry–Emery’s -criterion. These inequalities determine an analogue of subgaussian behavior for 1-cocycles. Our theorem improves some of our previous results in this direction, and in particular implies Efraim and Lust-Piquard’s Poincaré type inequalities for the Walsh system. The key new ingredient in our proof is a decoupling argument. As complementary results, we also show that the spectral gap inequality implies the Poincaré inequalities with constant under some conditions in the noncommutative setting. New examples which satisfy the -criterion are provided as well.
Key words and phrases:Decoupling, (noncommutative) Burkholder inequality, (noncommutative) Poincaré inequality, (noncommutative) transportation inequality, noncommutative spaces, -criterion, group von Neumann algebras, 1-cocycles, spectral gap
2010 Mathematics Subject Classification:46L53, 60E15, 22D25
Subgaussian behavior of random variables and stochastic processes is an important topic in probability theory. It is closely related to the concentration of measure phenomenon; see e.g. [Ver]. Functional inequalities – including log-Sobolev inequality, Poincaré inequality, transportation-entropy inequalities – have played a critical role in the development of this theory in the last two decades; see [Ta96, BG, OV, BGL] and the references therein for the extensive literature. More recently, this theory has been applied to study random matrices; see e.g. [Ver, Gui]. In this paper, we want to connect this well-known theory in classical probability to 1-cocycles on groups, which is important in both group theory (Kazhdan’s Property (T), the Haagerup property, etc.) and operator algebras; see e.g. [BO]. We are interested in determining a class of 1-cocycles which satisfy an analogue of the subgaussian growth condition via Poincaré type inequalities. Recall that a random variable is subgaussian if and only if for all . Here and in the following we use , etc. to denote absolute constants which may vary from line to line. To generalize this notion, we consider the following Poincaré inequalities for a probability space ,
for all and differentiable . Observe that (1.1) resembles subgaussian growth of random variables. In particular, choosing when we recover the classical definition except for .
As a classical example, the Gaussian measure on satisfies (1.1) due to Pisier [Pis86]; see [JZ12] for another proof. More classical examples satisfying (1.1) can be found in [AW13] and the references therein. In fact, one way to generalize (1.1) is via the semigroup theory of operators. The analogue of gradient in this context is Meyer’s “carré du champs”. We can even go further and consider an analogue of (1.1) in a noncommutative probability space. Recall from [VDN] that is a probability space if is a von Neumann algebra and is a normal state. We also assume that is finite and is tracial and faithful. Throughout we always work with a standard semigroup acting on with generator . Here a standard semigroup is pointwise -weak (weak) continuous such that every is normal unital completely positive and symmetric on . We define the gradient form associated to (Meyer’s “carré du champs”) as
for in a suitable involutive subalgebra of the domain of the generator, which is supposed to exist. In the following, we may simply write for if the generator under consideration is clear. Let be the fixed point algebra of . It was shown in [JX07] that is a von Neumann subalgebra of . Thus there exists a unique conditional expectation . Recall that the noncommutative space is defined as the closure of in the norm given by for and for , where is the operator norm. We usually write for short. It is well known that is a Banach space for ; see [PX03] for more details.
A standard semigroup acting on is said to be subgaussian if the following Poincaré inequalities
hold for and in a suitable involutive subalgebra of the domain of the generator.
For simplicity, in the following we may say the above inequality holds for all , since it is automatically true if the right-hand side is infinity. Since the gradient form coincides with the modulus of the gradient if is the Laplacian of a Euclidean space, (1.2) is indeed a generalization of (1.1). It is known that for classical diffusion semigroups, log-Sobolev inequality implies (1.2); see [AS94] and also [AW13]. Efraim and Lust-Piquard proved that (1.2) holds for Walsh systems and CAR algebras in [ELP]. In fact, we started to study the subgaussian behavior (1.2) of semigroups acting on a general noncommutative probability space in [JZ12]. It was shown in [Z13] that the group measure space satisfies (1.2), where the action and the gaussian measure are associated to an orthogonal representation of on a real Hilbert space, and the semigroup acting on is a natural extension of the Ornstein-Uhlenbeck semigroup on . A remarkable consequence of (1.2) is that one can get concentration inequalities, exponential integrability and transportation cost inequalities; see [ELP, Z13]. Our goal here is to prove (1.2) for group von Neumann algebras under some conditions on the 1-cocycles of groups and to elaborate on the relationship between the spectral gap of and Poincaré inequalities for semigroups acting on a probability space .
Let us be more precise. Let be a countable discrete group. Recall that a (generic) conditionally negative length (or cn-length for short) function on determines a 1-cocycle on with coefficients in an orthogonal representation of , and vice versa. Let be the left regular representation given by for , where ’s form a unit vector basis of . The group von Neumann algebra is the closure of linear span of in the weak operator topology. It is well known that admits a canonical normal faithful tracial state given by for , where is the identity element of . Consider the semigroup acting on defined by for . Then is a standard semigroup on . Thus extends to a strongly continuous semigroup of contractions on and the generator is given by . We say that a 1-cocycle on with coefficients in the orthogonal representation is subgaussian if the semigroup given by is subgaussian in the sense of Definition 1.1, i.e.,
holds for all and .
For readers who are not familiar with von Neumann algebras, (1.3) can be formulated in a more algebraic way, i.e.,
for , , where is the group algebra of (thus is a finite linear combination),
and is the Gromov form given by
Here in the linear combination implies that . We remark that and are not equal in general because of noncommutativity. It is clear that in this formulation (1.3) is really a condition on the 1-cocycle (or the cn-length function) and involves no probability theory or semigroups of operators. However, the only way we know to prove such inequalities is to use probability in an efficient way. To state our main results, we need to introduce the well-known -criterion due to Bakry–Emery. Recall that
whenever and are in a suitable involutive subalgebra of the domain of the generator.
Let be a countable discrete group with cn-length function and its group von Neumann algebra. Suppose satisfies for some . Then for ,
Note that Theorem 1.2 is a result for individual elements. As a property of the group von Neumann algebra, we hope to show that (1.4) holds for all . Recall Bakry–Emery’s -criterion [BE85]: There exists such that for all for which both and are well-defined. As observed in [JZ12], in our context this condition is equivalent to the algebraic condition that is a positive semidefinite form. The domain of is typically smaller than that of . Therefore, it is possible that is not well defined for some element while (1.4) still holds for this element. However, by [JZ12]*Corollary 4.8, we know that (1.4) always holds for all provided for all , because is a weakly dense subalgebra of . Let us record this as the following result.
Our motivation to study this problem comes from both noncommutative harmonic analysis and probability theory. In noncommutative harmonic analysis, Poincaré inequalities are closely related to noncommutative Riesz transform and smooth Fourier multiplier theory developed in [JM10, JMP]. In probability theory, precise moment estimation of random variables could be the starting point of various results, including concentration and transportation inequalities.
Let us mention some interesting applications. As indicated in [JZ12], applying Theorem 1.2 to the group , we recover the Poincaré type inequalities for the Walsh system due to Efraim and Lust-Piquard [ELP]. By embedding the matrix algebra into the discrete Heisenberg group von Neumann algebra, we find subgaussian behavior for matrix algebras. Another immediate consequence of our main results is the following transportation type inequalities shown in [JZ12, Z13]. Let us recall some notation. Let be a semigroup acting on a noncommutative probability space with generator . Given -measurable operators and , we define the following analogues of classical Wasserstein distances
where and ; see [Z13] for a detailed discussion about these distances and their relationship to Rieffel’s quantum metric spaces. For a -measurable positive operator , we define the entropy
Suppose the -criterion holds for the cn-length function on a discrete group . Then
for all -measurable positive operators affiliated to with .
We remark that the constant of order in our Poincaré inequalities is crucial to deduce these entropy bounds as observed in [JZ12, Z13]. A constant of the order , as obtained in Section 4, is not sufficient for such entropy bounds.
Let us now point out the connection of our results to some previous ones. As is well known, the major application of -criterion is to derive Gross’ log-Sobolev inequality (LSI) under some mild condition; see [BE85] and also the lecture notes [GZ] for more details in this direction. However, as observed in [JZ12], this implication is not true in general non-diffusion setting where the sample paths are discontinuous; see e.g. [BGL14, RY] for the definition of classical diffusion semigroups and processes. In particular, the diffusion property is characterized by the Leibnitz rule on the form
for smooth functions and in the domain of the generator. On the other hand, we can deduce concentration inequalities directly from -criterion without (1.5); see (1.7) below. Of course we still need a certain regularity condition on the semigroup :
(more precisely, for all by extension).
This condition is introduced in [JRS] to characterize the semigroups which admit a Markov dilation with certain nice properties in analogy to classical diffusion processes. For example the Poisson semigroup on the circle satisfies (1.6), but it is not a classical diffusion semigroup. For a standard semigroup with (1.6), it was proved in [JZ12] that the -criterion implies the following Poincaré type inequalities
for all self-adjoint . The obstruction of inequalities like (1.2) in the noncommutative setting was a lack of the good Burkholder inequality with appropriate norms or constants. Indeed, with the help of the optimal Burkholder–Davis–Gundy (BDG) inequality, it was proved that the classical diffusion semigroups satisfy (1.2) under the -criterion; see [JZ12, Theorem 4.9]. This may be regarded as a shortcut of the following implication in the classical diffusion setting
Here the first implication was due to Bakry–Emery [BE85] and the second was due to Aida–Stroock [AS94].
The optimal classical BDG inequality due to Barlow and Yor [BY82] asserts that
for any continuous mean 0 martingale , where is the quadratic variation of . One way to obtain such an inequality in the noncommutative setting is through the Burkholder inequality [JX03]
where is the martingale difference associated to the martingale and is the conditional expectation. One would expect the best order of is , which is indeed the case in the commutative theory [Pin]. The difficulty in the noncommutative generality can be seen from the fact that if one requires , then the optimal order of in (1.9) is known to be [Ran, JX05], compared to in the commutative theory [Hi90]. This shows that general noncommutative martingales exhibit quite different behaviors from the classical martingales so that may not be true. Although it is still unclear to us whether can be reduced to in the general noncommutative setting, we do resolve an important case of this problem in this paper, which is good enough to establish Theorem 1.2. In this way we improve the main results of [JZ12] for the case of semigroups acting on group von Neumann algebras generated by 1-cocycles. Our proof follows the same strategy as that in [JZ12]. The difficulty mentioned above is overcome by a decoupling argument, which is the key new ingredient (Lemma 3.1) in our proof. We refer the interested reader to the monograph [dG] for various aspects of decoupling and applications.
Let us conclude the introduction by mentioning the relationship among log-Sobolev inequality, spectral gap inequality and Poincaré inequalities. It is well known that the log-Sobolev inequality implies the existence of spectral gap, or equivalently, Poincaré inequality. Conversely, the spectral gap inequality together with a defective log-Sobolev inequality yields the log-Sobolev inequality; see e.g. [GZ] for these facts. On the other hand, the Poincaré inequalities obviously imply the spectral gap inequality. It would be interesting to determine when the converse implication is possible. It is known that in the classical diffusion setting the spectral gap would imply Poincaré inequalities, but with constant ; see e.g. [Mi09, Proposition 2.5]. We show similar results in the noncommutative setting under some conditions. In Section 4, we formulate certain results in this direction and prove, e.g., the following:
Let be an ergodic standard semigroup acting on a diffuse probability space which satisfies (4.2). Suppose the generator of has a spectral gap: For ,
Then for all even integer and all ,
We overcome the lack of (1.5) in the noncommutative setting by using the derivations of noncommutative Dirichlet forms developed by Cipriani and Sauvageot [JRS] and the regularity theorem due to Olkiewicz–Zegarlinski [OZ99].
The paper is organized as follows. We recall some preliminary facts in Section 2. Then we prove the Poincaré inequalities with constant in Section 3. The relationship between the spectral gap inequality and Poincaré inequalities is discussed in Section 4. Some examples and illustrations are given in Section 5.
2.1. Crossed products
We briefly recall the crossed product construction. Our reference is [Tak, JMP]. Let be a discrete group with left regular representation . Given a noncommutative probability space , we may assume for some Hilbert space . Suppose a trace preserving action of on is given, i.e., we have a group homomorphism (the -automorphism groups of ) with for all . Identify with . Consider the representation of on given by , where is the matrix unit of . In other words, for . Then the crossed product of by , denoted by , is defined as the weak operator closure of and in . We usually drop the subscript if there is no ambiguity. Clearly, is a von Neumann subalgebra of . In the special case , the complex number algebra, reduces to the group von Neumann algebra . Therefore, is a von Neumann subalgebra of and there exists a unique conditional expectation . If , we simply write or even for . A generic element of can be written as
There is a canonical trace on given by
where we denote by the canonical trace on . The arithmetic in is given by and . In what follows, we may simply write instead of .
2.2. 1-cocycles on groups
Let be a countable discrete group with a conditionally negative length (cn-length) function . Recall that is a length function if and , and is conditionally negative if . Then determines an affine representation which is given by an orthogonal representation over a real Hilbert space together with a map satisfying the cocycle law, i.e., ; see e.g. [BO]. To be more concrete, let be the algebraic group algebra of . Put for and define
Then is the closure of the quotient of by the kernel of , i.e., where is the kernel of . Define for and . In this way, we obtain a 1-cocycle . Conversely, suppose that is a 1-cocycle with coefficients in an orthogonal representation of . Put for . Then is a cn-length function on . By a Gram–Schmidt procedure, we may choose an orthonormal basis of so that depends on only finitely many nonzero coordinates for all . This observation will save us from some technical problems. We write even if .
2.3. Gaussian measure space construction
Note that the Hilbert space is separable. By the well known Gaussian space construction (see e.g. [RY, Str]), there exists a probability space and a linear map
such that is Gaussian centered and
We simply write and denote by the -subalgebra of generated by , for all and . By Kolmogorov’s continuity criterion (see, e.g., [RY, Theorem I.2.1]), thus constructed is a -valued Brownian motion, where is viewed as an abstract Wiener space associated to if . Indeed, by construction the -th component of is a 1-dimensional Brownian motion with mean 0 and variance , where is the -th component of , and all the components of are independent. More explicitly, we can simply take . Then , where is the -th coordinate map at time . It is readily seen that is a random variable in with variance . Suppose is an orthogonal representation of on . By [Str, Theorem 8.3.14], determines a Gaussian measure preserving action on . By abuse of notation, we still denote by . The -action on induces an action on , such that . It follows that
for , where . Clearly, extends naturally to isometric actions on for . In the following we will consider the von Neumann algebra and simply omit the subscript in the notation. To conclude this section, we remark that although (and thus ) may be infinitely dimensional, is always a finite dimensional Brownian motion for all because only depends on finitely many nonzero coordinates.
2.4. Hardy spaces associated to martingales
We refer to [JM10, JP] for this subsection. Let be a filtration with conditional expectations . Recall that a sequence is a martingale if and . Let be the associated martingale differences. We need the conditional Hardy spaces associated to martingales given as follows. For , define
We are going to use the continuous filtration in the following. Recall that a martingale is said to have almost uniform (or a.u. for short) continuous path if for every , every there exists a projection with such that the function given by is norm continuous. Let be a partition of the interval and its cardinality. Put
and . Let be an ultrafilter refining the natural order given by inclusion on the set of all partitions of . Let . For , we define
Here the limit is taken in the weak* topology and it is shown in [JKPX] that the convergence is also true in norm for all . We define the continuous version of norms for ,
and for . Then for all
A martingale is said to be of vanishing variation if for all and all . If has a.u. continuous path, then it is of vanishing variation. In the following, we will apply these results to matrix-valued martingales driven by Brownian motions. Hence they automatically have almost uniform continuous paths.
3. Poincaré inequalities for group von Neumann algebras
Consider the semigroup acting on given by , where is a conditionally negative length function on ; see Section 2.2. satisfies (1.6). For a proof of this fact, see [JZ12]. According to [JRS], admits a Markov dilation with almost uniformly continuous path. We refer the reader to [JRS, JZ12] for the precise definition. In fact, we can write down the dilation explicitly in our setting. Following the notation of Section 2.1 and 2.3, we define
The Markov property can be checked directly because
for and , and the same equation holds for arbitrary by linearity and density. Here is the conditional expectation. It follows that
is a martingale with almost uniformly continuous path for . We will need the reversed martingale. To this end, let us fix a large constant , and define
for . It is easy to check that is a martingale.
For with finitely many nonzero coordinates, we write , where are independent Brownian motions with variance , and can be given by in the notation of Section 2.3. By Ito’s formula,
It follows that
Note that . By Ito’s formula, we have
It follows that
Let be a finite sum. Then
We consider the discretized stochastic integral (assuming ), or martingale transform
where . It is well known that this martingale converges to the stochastic integral in for . Indeed, the stochastic integral can be defined as the limit of a certain martingale transform; see e.g. [KS91]. Similar argument can be applied to the case of . We need a precise Burkholder inequality for this (noncommutative) martingale in order to derive the subgaussian property. As explained in Introduction (see also [JZ12]), however, the upper bounds in known inequalities can only result in the inequality (1.7). Our approach here relies on the decoupling technique thanks to the special structure in the martingale transform.
Let us consider the discrete time martingale given by
where is a continuous function, for any , is a martingale with independent martingale differences . In what follows we will simply write instead of and this always means a finite sum.
Lemma 3.1 (Decoupling).
Suppose is measurable with respect to for and and an independent copy of . Then for ,
To shorten the notation, we simply write for . Consider independent random selectors