ANALYSIS & PDE. mathematical sciences publishers DIMITRI SHLYAKHTENKO LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION

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1 ANALYSIS & PDE Volume 2 No DIMITRI SHLYAKHTENKO LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION mathematical sciences publishers

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3 ANALYSIS AND PDE Vol. 2, No. 2, 2009 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION DIMITRI SHLYAKHTENKO By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X,..., X n. In particular, we show that δ 0 X,..., X n ) dim M M o V, where M = W X,..., X n ) and V = { X ),..., X n )) : } is the set of values of derivations A = [X,... X n ] A A with the property that A) A. We show that for q sufficiently small depending on n) and X,..., X n a q-semicircular family, δ 0 X,..., X n ) >. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani and also studied in the free case by Biane and Voiculescu).. Introduction We present in this paper a general technique for proving lower estimates for Voiculescu s microstates free entropy dimension δ 0. The free entropy dimension δ 0 was introduced in [Voiculescu 994; 996] and is a number associated to an n-tuple of self-adoint elements X,..., X n in a tracial von Neumann algebra. This quantity has been used by various authors [Voiculescu 996; Ge 998; Ge and Shen 2002; Ştefan 2005; Jung 2007] to prove a number of very important results in von Neumann algebras. These results often take the form: If δ 0 X,..., X n ) >, then M = W X,..., X n ) cannot have certain decomposition properties for example, is non-ɣ, has no Cartan subalgebras, is not a nontrivial tensor product and so on). For this reason, it is important to know if some given von Neumann algebra has a set of generators with the property that δ 0 >. We prove that this is the case for small values of q) for the q-deformed free group factors of [Bożeko and Speicher 99]. Theorem. For a fixed N and all q < 4N 3 + 2), the q-semicircular family X,..., X N satisfies δ 0 X,..., X N ) > and δ 0 X,..., X N ) N q 2 N q 2 N) ). The theorem applies for q if N = 2. Combined with the available results on free entropy dimension, we obtain that, in this range of values of q, the algebras Ɣ q N ) = W X,..., X N ) have no Cartan subalgebras or, more generally, that Ɣ q N ), when viewed as a bimodule over any of its abelian subalgebras, contain a coarse subbimodule). Theorem also implies that these algebras are prime this was already proved in [Shlyakhtenko 2004] using the techniques of [Ozawa 2004]). The free entropy dimension δ 0 is closely related to L 2 Betti numbers [Connes and Shlyakhtenko 2005; Mineyev and Shlyakhtenko 2005] more precisely, with Murray von Neumann dimensions of MSC2000: 46L54. Keywords: free stochastic calculus, free probability, von Neumann algebras, q-semicircular elements. Research supported by NSF grant DMS

4 20 DIMITRI SHLYAKHTENKO spaces of certain derivations. For example, the nonmicrostates free entropy dimension δ which is the nonmicrostates relative of δ 0 ) is in many cases equal to L 2 Betti numbers of the underlying nonclosed) algebra [Mineyev and Shlyakhtenko 2005; Shlyakhtenko 2006]. It is known that δ 0 δ and thus it is important to find lower estimates for δ 0 in terms of dimensions of spaces of derivations. To this end we prove. Theorem 2. Let A, τ) be a finitely-generated algebra with a positive trace τ and generators X,...,X N, and let Der c A; A A) denote the space of derivations from A to A A which are L 2 closable and such that X ) A. Consider the A,A-bimodule V = { δx ),..., δx n )) : δ Der c A; A A) } A A) N. Finally, assume that M = W A, τ) can be embedded into the ultrapower of the hyperfinite II factor. Then δ 0 X,..., X n ) dim M M o V L2 A A,τ τ) N. We actually prove Theorem 2 under a less restrictive assumption: we require that δx ) and δ δx ) be analytic as functions of X,..., X N ; more precisely, there should exist noncommutative power series and ξ with sufficiently large multiradii of convergence so that δx ) = X,..., X N ) and δ δx ) = ξ X,..., X N ); see Theorem 6 below for a precise statement. This theorem is a rich source of lower estimates for δ 0. For example, if T A A, then δ : X [X, T ] = XT T X is a derivation in Der c A; A A). If W A) is diffuse, then the map L 2 A A) T [T, X ],..., [T, X N ] ) L 2 A A) N is inective and thus the dimension over M M o of its image is the same as the dimension of L 2 A A), that is,. Hence dim M M o V and so δ 0 X,..., X n ) if W A) is R ω embeddable hyperfinite monotonicity in [Jung 2003b]). If the two tuples X,..., X m and X m+,... X N are freely independent and each generates a diffuse von Neumann algebra, then for T A A the derivation δ defined by δx ) = [X, T ] for m and δx ) = 0 for m + N is also in Der c A). Then one easily gets that dim M M o V > indeed, V contains vectors of the form [T, X ],..., [T, X m ], 0,..., 0 ), T L 2 A A), and so its closure is strictly larger than the closure of the set of all vectors [T, X ],..., [T, X N ] ), T L 2 A A)). Thus δ 0 X,..., X N ) > if W A) is R ω embeddable. If X,..., X N are such that their conugate variables [Voiculescu 998] are polynomials, then the difference quotient derivations are in Der c and thus V = A A) N, and so δ 0 = N if W A) is R ω embeddable). In the case that X,..., X N are generators of the group algebra Ɣ of a discrete group Ɣ, δ X,..., X N ) = β 2) Ɣ) β2) Ɣ) +, where β 2) are the L 2 Betti numbers of Ɣ see [Lück 2002] for a definition). It is therefore natural to ask whether the same holds true for δ 0 instead of δ for some class of groups. If this is true, then knowing that 0

5 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 2 β 2) Ɣ) = 0 implies that δ 0 > and thus the group algebra has a variety of properties that we explained above see also [Peterson 2009]). It is clearly necessary for the equality δ 0 = β 2) β 2) 0 + that Ɣ can be embedded into the ultrapower of the hyperfinite II factor because otherwise δ 0 would be ). In particular, one is tempted to conecture that equality holds at least in the case when Ɣ is residually finite. Theorem 2 implies a result like the one in [Brown et al. 2008]: Theorem 3. Assume that Ɣ is embeddable into the unitary group of the ultrapower of the hyperfinite II factor. Then δ 0 Ɣ) dim LƔ) {c : Ɣ Ɣ cocycle}. In particular, if Ɣ belongs to the class of groups containing all groups with β 2) = 0 and closed under amalgamated free products over finite subgroups, passage to finite index subgroups and finite extensions, then δ 0 Ɣ) = β 2) Ɣ) β2) 0 Ɣ) +. Let us now describe the main idea of the present paper. Our main result states that if the von Neumann algebra M = W X,..., X n ) can be embedded into the ultrapower of the hyperfinite II factor, then δ 0 X,..., X n ) dim M M o V, -) where V = { X ),..., X n )) : }L2 and is some class of derivations from the algebra of noncommutative polynomials [X,..., X n ] to L 2 M) L 2 M o ), which will be made precise later. The quantity δ 0 X,..., X n ) is, very roughly, a kind of Minkowski dimension relative to R ω ) of the set of embeddings of M into R ω, the ultrapower of the hyperfinite II factor indeed, the set of such embeddings can be identified with the set of images under the embedding of the generators X,..., X n, that is, with the set of microstates for X,..., X n ). On the other hand, dim M M o V is a linear dimension relative to M M o ) of a certain vector space. If we could find an interpretation for V as a subspace of a tangent space to, then the inequality -) takes the form of the inequality linking the Minkowski dimension of a manifold with the linear dimension of its tangent space. One natural proof of such an inequality would involve proving that a linear homomorphism of the tangent space to a manifold at some point can be exponentiated to a local diffeomorphism of a neighborhood of that point. Thus an essential step in proving a lower inequality on free entropy dimension is to find an analog of such an exponential map. This leads to the idea, given a matrix Q i L 2 M) L 2 M o ) ) n of values of derivations so that Q i = X i ) for some n-tuple of derivation belonging to our class ), to try to associate to Q a one-parameter deformation α t of a given embedding α = α 0 of M into R ω. It turns out that there are two related) ways to do this. The first approach comes from the idea that we at least in principle) know how to exponentiate derivations from an algebra to itself the result should be a one-parameter automorphism group of the algebra). We thus try to extend = n to a derivation of a larger algebra = [X,..., X n, S,..., S n ], where S,..., S n are free from X,..., X n and form a free semicircular family. The key point is that the closure in L 2 ) of spanm S M + + M S n M) is isomorphic to [L 2 M) L 2 M)] n. The inverse of this isomorphism takes an n-tuple a = a b,..., a n b n ) to a S b, which we denote by

6 22 DIMITRI SHLYAKHTENKO a#s. We now define a new derivation of with values in L 2 ) by X ) = X )#S. To be able to exponentiate, we need to make sure that it is antihermitian as an unbounded operator on L 2 ), which naturally leads to the equation S ) = ζ ), where ζ = 0,...,,..., 0) -th entry nonzero). One can check that if ζ is in the domain of for all, then is a closable operator which has an antihermitian extension, and so it can be exponentiated to a one-parameter group of automorphisms α t of L 2 ). Unfortunately, unless we know more about the derivation such as, for example, assuming that ) ), we cannot prove that α t takes W ) to W ). However, if this is the case, then we do get a one-parameter family of embeddings α t M : M M L n)) R ω. We explain this approach in more detail in the Appendix. The second approach was suggested to us by A. Guionnet, to whom we are indebted for generously allowing us to publish it. The idea involves considering the free stochastic differential equation d X t) = i Q i X t),..., X n t))#d S i 2 ξ X t),..., X n t)), X 0) = X, -2) where X ) = Q,..., Q n ) L 2 M) L 2 M o ) ) n and ξ X,..., X n ) = X ). One difficulty in even phrasing the problem is that it is not quite clear what is meant by Q i and ξ applied to their arguments in the classical case, this would mean a function applied to the random variable X t)). However, if this equation can be formulated and has a stationary solution X t) namely one for which the law does not depend on t), then the map α t : X X t 2 ) determines a one-parameter family of embeddings of the von Neumann algebra M into some other von Neumann algebra generated by all X t) : t 0). This can be carried out successfully if Q and ξ are sufficiently nice; this is this is the case, for example, when X,..., X n are q-semicircular variables, in which case Q and ξ can be taken to be analytic noncommutative power series. Let us assume now that takes = [X,..., X n ] to o and also ) this is the case, for example, if X,..., X n have polynomial conugate variables [Voiculescu 998]). Then both approaches work to actually give one a stronger statement: one gets a one-parameter family of embeddings α t : M R ω so that α t X ) X + t i Q i #S i ) 2 = Ot 2 ). Let us assume for the moment that Q i = δ i, so that our estimate reads α t X ) X + t S ) 2 = Ot 2 ). -3) An estimate of this kind was used as a crucial step by Otto and Villani in their work on the classical transportation cost inequality [Otto and Villani 2000, 4 Lemma 2]; a free version for n = ) is the key ingredient in the proof of free transportation cost inequality and free Wasserstein distance given in [Biane and Voiculescu 200]. Indeed, since the law of α t X ) is the same as X, one obtains after working out the error bounds an estimate on the noncommutative Wasserstein distance between the laws µ X,...,X n and µ X +t S,...,X n +t S n : d W µ X,...,X n, µ X +t S,...,X n +t S n ) 2 X,..., X n ) /2 t + Ot 2 ). We now point out that this estimate is of direct relevance to a lower estimate on δ 0. Indeed, suppose that some n-tuple of k k matrices x,..., x n has as its law approximately the law of X,..., X n that is, x,..., x n ) ƔX,..., X n ; k, l, ε) in the notation of [Voiculescu 994]). Then -3) implies that by approximating α t X ) with polynomials in X,..., X n, S,..., S n, one can find another n-tuple

7 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 23 x,..., x n with almost the same law as X,..., X n, and so that x x +ts ) Ct 2 here s,..., s n are some matrices whose law is approximately that of S,..., S n, and which are approximately free from x,..., x n ). But this means that if one moves along a line starting at x,..., x n in the direction of s,..., s n, then the distance to the set ƔX,..., X n ; k, l, ε) grows quadratically. Thus this line is tangent to the set ƔX,..., X n ; k, l, ε). From this one can derive estimates relating the packing numbers of ƔX,..., X n ; k, l, ε) and ƔX + t S,..., X n + t S n ; k, l, ε) which can be converted into a lower estimate on δ 0. In conclusion, it is worth pointing out that the main obstacle that we face in trying to extend the estimate -) to larger classes of derivations is the question of existence of stationary solutions of -2) for more general classes of functions Q and ξ and not, surprisingly enough, the usual difficulties in dealing with sets of microstates). 2. Existence of stationary solutions 2.. Free SDEs with analytic coefficients. The main result of this section states that a free stochastic differential equation of the form d X t = #d S t 2 ξ tdt where X t is an N-tuple of random variables has a stationary solution, as long as the coefficients and ξ are analytic that is, they are noncommutative power series with sufficient radii of convergence) Estimates on certain operators appearing in free Ito calculus. Let f be a noncommutative power series in N variables. We denote by c f n) the maximal modulus of a coefficient of a monomial of degree n in f. Thus if f = f i...i n X i X in, then c f n) = max i...i n f i...i n. We also write φ f z) = c f n)z n. Then φ f z) is a formal power series in z. If ρ is the radius of convergence of φ f, we ll say that R = ρ/n is the multiradius of convergence of f. We also write f ρ = n 0 c f n)n n ρ n [0, + ]. Note that f ρ = sup z Nρ φ f z) since all of the coefficients in the power series φ f z) are real and positive). We denote by R) the collection of all power series f for which the multiradius of convergence is at least R. In other words, we require f ρ < for all ρ < R. Note that R is a complete topological vector space when endowed with the topology such that T i T if and only if T i T ρ 0 for all ρ < R. Let be a noncommutative power series in N variables having the form fi,...,i k ;,..., l Y i Y ik Y Y l. We call a formal noncommutative power series with values in Y,..., Y N 2. We write c m, n) the maximal modulus of a coefficient of a monomial of the form Y i Y im Y Y n in. We let

8 24 DIMITRI SHLYAKHTENKO φ z, w) = n,m c ψm,n)z m w n. We put ρ = sup z, w Nρ φ z, w) = φ Nρ, Nρ) = n 0 k+l=n ) c k, l) N n ρ n [0, + ]. We denote by R) the collection of all noncommutative power series for which ρ < for all ρ < R. It will be convenient to use the following notation. Let φz,..., z n ), ψz,..., z n ) be two formal power series in commuting variables). We say that φ ψ if all coefficients in φ, ψ are real and positive, and for each k,..., k n, the coefficient of z k zk n n in φ is less than or equal to the corresponding coefficient in ψ. If is a unital Banach algebra, Y,..., Y N and Y < ρ for all, then gy,..., Y n ) g ρ whenever g is in any one of the spaces R), or R) here the norm gy,..., Y n ) denotes the norm on or on the proective tensor product 2, as appropriate). We now collect some facts about power series: Let f, g R). Then φ f g φ f φ g. In particular, f g R) and f g ρ f ρ g ρ. Let f = f i...i n X i X in R) and denote by i f the formal power series i f = δ ik =iδ il = f i...i n X ik+ X il X il+ X in X i X ik. i...i n k<l Since a monomial X i X ik X X r could arise in the expression for i f in at most r + ways, c f a, b) b + )c f a + b + 2). Denote by ˆφ f the power series ˆφ f z, w) = n,m n + )c f n + m + 2)z m w n. Then φ f ˆφ f. Since ˆφ f z, z) φ f z), we conclude that and in particular i f R). i f ρ sup φ f z) z Nρ Let = i...i n ;... m X i X in X X m R), and let Consider = i...i n ;... m X i... X in X... X m. # in = t...t a,s,...,s b i...i n ;... m X i X in X t X ta X s X sb X X m. In the simple case that = A B and = P Q, where A, B, P, Q are monomials, we have # in = P A B Q, that is, # in is the inside multiplication on R)). Then c #in n, m) c k, r)c l, s), k+l=n r+s=m

9 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 25 and hence the coefficient of z n w m in φ #in z, w) is dominated by the coefficient of z n w m in φ z, w)φ z, w). Consequently, φ #in φ φ and # in ρ ρ ρ. In particular, # in R). Similar estimates and conclusion of course hold for the outside multiplication # out, defined by # out = s,...,s b ;t...t a i...i n ;... m X t X ta X i X in X X m X s X sb. In that case we get φ #out z, w) φ w, z)φ z, w) and # out ρ ρ ρ. Let τ be a linear functional on the algebra of noncommutative polynomials in n variables, so that τx i X in ) R0 n for all n. Given = i...i n ;... m X i X in X X m R), assume that R 0 < R and consider the formal sum τ) ) = ) i...i n ;... m τx X m ) X i X in. m,,..., m n,i,...,i n More precisely, we consider the formal power series in which the coefficient of X i X in is given by the sum m,,..., m i...i n ;... m τx X m ). But since τx X m ) R m 0, this sum is bounded by the coefficient of zn in the power series expansion of φz, N R 0 ) as a function of z), and is convergent. Thus φ τ) ) z) φ z, N R 0 ) and we readily see that τ) ) is well-defined, belongs to R), and moreover whenever ρ > R 0. τ) ) ρ ρ, Let f = f i...i n X i X in R) and consider the -th cyclic partial derivative [Voiculescu 999; 2002b] f = n δ il = X il+... X in X i X il. i...i n l= Then we see that φ f φ f ) and f R). We now combine these estimates: Lemma 4. Let τ as above be a linear functional on the space of noncommutative polynomials in N variables satisfying τx i X in ) R0 n. Let R > R 0 and assume that ξ R), =,..., N, = i ) M N N R). For f R) let ) τ) f ) = τ) k # in ki # out i f )) 2 ξ f. i k

10 26 DIMITRI SHLYAKHTENKO Then τ) f ) R) and moreover for any R 0 < ρ < R, τ) f ) ρ k ρ ki ρ sup φ f z) + 2 ξ ρ sup φ f, i k z Nρ z Nρ φ τ) f )z) φ k z, N R 0 )φ ki N R 0, z) ˆφ f z, N R 0 ) + 2 φ ξ z)φ f z), i k where ˆφ f z, w) = n,m n + )c f n + m + 2)z m w n. For φ a power series in z, w,..., w k with multiradius of convergence bigger than ρ and all coefficients of monomials nonnegative, let φ w,...,w k z) = φz, w,..., w k ). Set Qφz, w,... w k+ ) = φ w,...,w k z, w k+ ) and Dφz, w,..., w k ) = z 2 φz, w,..., w k ). We note that ˆφz, z) φ z), and that Q, D and are monotone for the ordering. It follows that if κ, λ are some power series with radius of convergence bigger than ρ and positive coefficients, then for any a, b,..., a k, b k 0 and any R < ρ, [ Q a κ z)d b λ z)µ Q a 2 κ 2 z)d b 2 λ 2 z) D b k λ k ] z=w = =w b k =R [ D a κ z)d b λ z)d a 2 κ 2 z)d b 2 λ 2 z) D b k λ k ] z=w = =w b k =R. Now define ˆ φz) = φ ψ k z, N R 0 )φ ki N R 0, z)φ z) + 2 φ ξ z)φ z). i k Then we have obtained the inequality which we record as: φ n f N R 0 ) ˆ n φ f N R 0 ), Lemma 5. Let ˆ φz) = i k φ ψ k z, N R 0 )φ ki N R 0, z)φ z) + 2 so that for any monomial P, τp) < R0 n, n = deg P. Then τ n f ) ˆ n φ f N R 0 ). φ ξ z)φz) and let τ be a trace Analyticity of X ). Let us now assume that =,..., N ) R). Let X,..., X N ) be an N-tuple of self-adoint operators in a tracial von Neumann algebra M, τ) and assume that X < R for all. Let : L 2 M) L 2 M) L 2 M) be the derivation densely defined on polynomials in X,..., X N by X ) = X,..., X N ). We assume that belongs to the domain of and that there exists some ζ R) so that ) = ζx,..., X N ). Lemma 6. With the assumptions above, there exist ξ R), =,..., N, so that ξ X,..., X N ) = X ). Proof. It follows from [Voiculescu 998; Shlyakhtenko 998] that under these assumptions, is closable. Moreover, for any a, b polynomials in X,..., X N, a b belongs to the domain of and a b) = aζ b + τ)[ a)]b + aτ )[ b)],

11 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 27 where ζ = ζx,..., X N ) = ). Consider now formal power series in N variables having the form = i,...,i k ;,..., l ;t,...,t r Y i Y ik Y Y l Y t Y tr. We write φ z, w, v) for the power series whose coefficient of z m w n v k is equal to the maximum max { i,...,i m ;,..., n ;t,...,t k : i,..., i m,,..., n, t,..., t n {,..., N} }. We denote by R) the collection of all such power series for which φ has a multiradius of convergence at least N R. Let s) : R) R) be given by s) fi,...,i k ;,..., l Y i Y ik Y Y l = fi,...,i k ;,..., l δ i p =sy i Y i p Y i p+ Y ik Y Y l. Then clearly φ s) )z, z, w) zφ z, w) so that s) indeed lies in R) if R). Similarly, if we define for R), R) # ) in = t...t a,s,...,s b i...i n ;... m ;k...k p Y i Y in Y t Y ta Y s Y sb Y Y m Y k Y k p, p corre- then φ # ) z, v, w) φ z, v)φ z, v, w) and in particular # ) in R). Note that # ) in in sponds to multiplying around the first tensor sign in ). Finally, if τ is any linear functional so that τp) < R deg P 0 for any monomial P and we put M 2 ) = i,...,i n ;,..., m ;k,...k p Y i Y in τy Y m Y k Y k p ), then φ M2 )z) φ z, N R 0, N R 0 ) and in particular M 2 ) R) once R) and R 0 < R. In the foregoing, we ll use the trace τ of M as our functional. So if we put T = M 2 s # ) in s) ), s then T maps R) into R). Note that in the case that = A B, where A, B are monomials, T = τ) A)) B. One can similarly define T 2 : R) R); it will have the property that T 2 = Aτ ) B)). Lastly, let ζ R) and let m : R) R) be given by m ) = i,...,i n ;,..., m ζ p,...,p r Y i Y in Y p,...p r Y Y m. Once again, φ m ) z) φ z, z)φ ζ z). Let Q ) = T )+T 2 )+m ). We claim that ξ = Q ))X,..., X N ) = X,..., X N )). Note that if n is a partial sum of say obtained as the sum of all monomials in having degree at most n), then Q n )X,..., X N ) = n X,..., X N )). Moreover, as n, we have that n X,..., X N ) X,..., X N ) in L 2 and also Q n )X,..., X N ) Q )X,..., X N ) in L 2

12 28 DIMITRI SHLYAKHTENKO this can be seen by observing first that the coefficients of Q n ) converge to the coefficients of Q ) and then approximating Q ) and Q n ) by their partial sums). Since is closed, the claimed equality follows Existence of solutions. Recall that a process X t),..., X t) N M, τ) is called stationary if its law does not depend on t; that is, for any polynomial f in N noncommuting variables, τ f X t),..., X t) N )) is constant. Lemma 7. Let X 0),..., X 0) N be an N-tuple of noncommutative random variables, R 0 > max X 0) and R > R 0. Let ξ R), = i ) M N N R)) so that i Z,..., Z N ) = i Z,..., Z N ) for any self-adoint Z,..., Z N. Consider the free stochastic differential equation d X i t) = X t),..., X N t))#d S t ),..., d S t N) ) 2 ξ i X t),..., X N t) ) dt 2-) with the initial condition X 0) = X 0), =,..., n. Here d S t ),..., d S t N) is free Brownian motion, and for Q kl = ai kl bi kl M ˆ M, and Q = Q kl ) M N N M ˆ M), we write Q#W,..., W N ) = ai k W k bi k,..., ) ai Nk W bi Nk. ki Let A = W X 0),..., X 0) N ) and let : L 2 A) L 2 A A) be derivations densely defined on polynomials in X 0),..., X 0) N and determined by Assume that for all, i X 0) domain i and that X 0) i ) = i X 0),..., X 0) N ). ξ X 0),..., X 0) N ) = i i ix 0) ). Then there exists a t 0 > 0 and a stationary solution X t), 0 t < t 0. This stationary solution satisfies X t) W X,..., X N, {S s) : 0 s t} N =). We note that in view of Lemma 6, we may instead assume that domain and ) = ζ X 0),..., X 0) N ) for some ζ,..., ζ N R), since this assumption guarantees the existence of ξ R) satisfying the hypothesis of Lemma 7. Proof. We note that, because and ξ are analytic, they are Lipschitz in their arguments. Thus it follows from the standard Picard argument see [Biane and Speicher 998]) that a solution with given initial conditions) exists, at least for all values of t lying in some small interval [0, t 0 ), t 0 > 0. Now choose t 0 so that X t) R 0 < R for all 0 t < t 0 this is possible, since the solution to the SDE is locally norm-bounded).

13 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 29 Next, we note that if we adopt the notations of Lemma 4 and define for f R) τ) f ) = τ) k # in ki # out i f ))) 2 ξ f, i k then we have that τ t ) f R) here τ t refers to the trace on X t),..., X n t) obtained by restricting the trace from the von Neumann algebra containing the process X t for small values of t, that is, τ t P) = τpx t),..., X n t)))). Ito calculus shows that for any f R), d dt τ f X t),..., X N t))) = τ s τs) f )X s),..., X N s))). t=s In particular, replacing f with τ t ) f and iterating gives us the equality d n dt n τ f X t),..., X N t))) = τ s τs) ) n f )X s),..., X N s))). t=s Since ξ X 0),..., X n 0)) = i i ix 0)), τ τ 0) f X 0),..., X N 0))) ) = 0 for any f R). Applying this to f replaced with τ0) f and iterating allows us to conclude that d n dt n τ f X t),..., X N t))) = 0, n. t=0 Let as before ˆ φz) = φ ik z, N R 0 )φ k N R 0, z)φ z) + 2 φ ξ z)φ z) = αz) 2 φ z) + βz)φ z), i k where βz) is complex-valued function and αz) is a complex vector-valued analytic function, both defined on {z : z < N R}. Moreover, α and β are real for z. Consider the partial differential equation t φx, t) = ˆ φx, t), φx, 0) = φ f x), x. The solution φx, t) will be real-analytic in time at least for small values of t), because the equation is elliptic. By Lemma 5, we conclude that t n τ f X t),..., X N t))) t=s ˆ n φ)n R 0, s) = t n φn R 0, t). t=s Hence, since all derivatives of τ f X t)),..., f X N t))) vanish at zero, τ f X s),..., X N s))) τ f X 0),..., X N 0))) s s = t n τ f X t),..., f X N t)))) dr) n 0 0 t=r s s C t n φn R 0, t)) φn R 0, s) P n N R 0, s), t=r 0 0

14 30 DIMITRI SHLYAKHTENKO where P n is the n-th Taylor polynomial of φ at zero. Since φ is real-analytic in s, the right-hand side of the equation goes to zero as n for s in some interval including zero. Thus the function s τ f X s),..., X N s))) is constant and so the solution is stationary. We note that once the Equation 2-) has a stationary solution on a small interval [0, t 0 ), then it of course has a stationary solution for all time since the same lemma applied to X t0 /2 guarantees existence of the solution for up to 3t 0 /2 and so on). However, we will not need this here. 3. Otto Villani type estimates The main result of this section is an estimate on the noncommutative Biane Voiculescu Wasserstein distance between the law of an N-tuple of variables X = X,..., X N and the law of the N-tuple X + t Q#S, where S =S,..., S N ) is a free semicircular family, Q M N N L 2 W X,..., X N ) 2 ) ) is a matrix, and for Q i = k Ak) i B k) i, we denote by Q#S the N-tuple Y,..., Y N ) with Y i = A k) i S B k) i. k The sum defining Y i is L 2 convergent; in fact, the L 2 norm of Y i is the same as the L 2 norm of the element A k) i B k) i. k The estimate on Wasserstein distance Proposition 8) is obtained under the assumptions that a certain derivation, defined by X i ) = Q i,..., Q i N ) L 2 W X,..., X N ) 2) N is closable and satisfies certain further analyticity conditions see below for more precise statements). Under such assumptions, the estimate states that d W X, X + t Q#S) Ct. The main use of this estimate will be to give a lower bound for the microstates free entropy dimension of X,..., X N see Section 5). 3.. An Otto Villani type estimate on Wasserstein distance via free SDEs. Proposition 8. Let M N N R)), M = W X,..., X N ) and let : L 2 M) L 2 M M) be derivations densely defined on polynomials in X,..., X N and determined by X i ) = i X,..., X N ). Assume that for all, domain i and that there exist ζ,..., ζ N R) so that ζ X,..., X N ) = ), =, 2,..., N. Then there exists a II factor = M L ) and a t 0 > 0 so that for all 0 t < t 0 there exists an embedding α t : M = W X,..., X N ) and a free 0, )-semicircular family S,..., S N, free from M and satisfying the inequality αt X ) X + t X,..., X N )#S) 2 Ct, 3-)

15 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 3 where C is a fixed constant. Furthermore, α t X ) W X,..., X N, S,..., S N, {S } = ), where {S } = are a free semicircular family, free from X,..., X N, S,..., S N ). If A can be embedded into R ω, so can. In particular, the noncommutative Wasserstein distance of Biane Voiculescu satisfies d W X ) N =, X + t X,..., X N )#S) N ) = Ct. Proof. By Lemma 6, we can find ξ,..., ξ N R) so that ξ X,..., X N ) = X ). Let = W X,..., X N, {S s),..., S N s) : 0 s t} ), where S t) is a free semicircular Brownian motion. Let X t) be a stationary solution to the SDE 2-) see Lemma 7). The map that takes a polynomial in X,..., X N to a polynomial in X t),..., X N t) preserves traces and so extends to an embedding α t : M. By the free Burkholder Gundy inequality [Biane and Speicher 998], it follows that for 0 t < t 0 < X t) X 0) C t + C2 t C 3 t, where C = sup t<t0 X t),..., X N t) <, C 2 = max sup t<t0 ξ X,..., X N t). Furthermore, t t X t) X 0) = X s),..., X N s))#d S s) ξ X s),..., X Nn s))ds 0 0 = t 0 t X 0),..., X N 0))#d S s) 0 t 0 [ X 0),... X N 0)) X s),..., X N s)) ] #d S s) ξ X s),..., X N s))ds. By the Lipschitz property of the coefficients of the SDE 2-), we see that X s),..., X N s)) X 0),..., X N 0)) K max X s) X 0) K s. Combining this with the estimate ξ X t),..., X N t)) < K we may apply the free Burkholder Gundy inequality to deduce that t X t) X 0) + X 0),..., X N 0))#S t)) K s) 2 /2 t ds + K ds Ct. Thus it is enough to notice that 2 and to take S = t S t), which is a 0, ) semicircular element. If M is R ω -embeddable, we may choose to be R ω -embeddable as well, since it can be chosen to be a free product of M and a free group factor. Finally, note that X t) W X,..., X N, {S s) : 0 s t} N =) by construction. But the algebra W {S s) : 0 s t} ) can be viewed as the algebra of the Free Gaussian functor applied to the space L 2 [0, ], in such a way that S s)= S[0, s]). Then W {S s):0 s t} ) W S,..., S N, {S k } k I )), where {S k : k I )} are free semicircular elements corresponding to the completion of the singleton set {t /2 χ [0,t] } to an orthonomal basis of L 2 [0, ]. 0 0

16 32 DIMITRI SHLYAKHTENKO The estimate for the Wasserstein distance now follows if we note that the law of α t X )) N = is the same as that of X ) N = ; thus X t)) N = X + t #S) N = is a particular 2N-tuple with marginal distributions the same as those of X ) N = and X + t #S) N =, so that the estimate 3-) becomes an estimate on the Wasserstein distance. Remark 9. Although we do not need this in the rest of the paper, we note that the estimate in Proposition 8 also holds in the operator norm. We should mention that an estimate similar to the one in Proposition 8 was obtained by Biane and Voiculescu [200] in the case N = under the much less restrictive assumptions that = and domain that is, the free Fisher information X) is finite). Setting i = δ i we have proved an analog of their estimate in the N-variable case), but under the very restrictive assumption that the conugate variables ) are analytic functions in X,..., X N. The main technical difficulty in removing this restriction lies in the question of existence of a stationary solution to 2-) in the case of very general drifts ξ. 4. Applications to q-semicircular families 4.. Estimates on certain operators related to q-semicircular families Background on q-semicircular elements. Let H be a finite-dimensional real Hilbert space, H its complexification H = H, and let F q H) be the q-deformed Fock space on H [Bożeko and Speicher 99]. Thus F q H) = n H n, with the inner product given by ξ ξ n, ζ ζ m = δ n=m π S n q iπ) n ξ, ζ π )), where iπ) = # { i, ) : i < and πi) > π ) }. We write H S for the space of Hilbert Schmidt operators on F q H). We denote by H S the operator = q n P n, where P n is the orthogonal proection onto the subspace H n F q H). For h H, let lh) : F q h) F q H) be the creation operator, lh)ξ ξ n ) = h ξ ξ n, and for h H, let sh) = lh) + lh). We denote by M the von Neumann algebra W sh) : h H ). It is known [Ricard 2005; Śniady 2004] that M is a II factor and that τ =, is a faithful tracial state on M. Moreover, F q H) = L 2 M, τ) and H S = L 2 M, τ) L 2 M, τ). Fix an orthonormal basis {h i } N i= H and let X i = sh i ). Thus M = W X,..., X N ), N = dim H. Lemma 0 [Shlyakhtenko 2004]. For =,..., N, let : [X,..., X N ] H S be the derivation given by X i ) = δ i=. Let : [X,..., X N ] H S N be given by = N and regard as an unbounded operator densely defined on L 2 M). Then: i) is closable. =

17 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 33 ii) If we denote by Z the vector 0 P 0 H S N nonzero entry in -th place, P is the orthogonal proection onto F q H)), then Z is in the domain of and Z ) = h. As a consequence of ii), if we let be as in the lemma, we have ξ = Z ) [X,, X N ] R) for any R as an analytic function of X,..., X n. We now claim that for small values of q, the element L 2 M) 2 defined in Lemma 0 can be thought of as an analytic function of the variables X,..., X N. Recall that h i H is a fixed orthonormal basis and X = sh ), =,..., N thus form a q-semicircular family. Lemma. Let W i,...,i n be noncommutative polynomials so that W i,...,i n X,..., X N ) = h i h in. Then the degree of W i,...,i n is n, and the maximal absolute value c n) k X X k, k n, in W i,...,i n satisfies of a coefficient of a monomial Furthermore, W i,...,i n 2 L 2 M) 2n q ) n. ) c n) n k k 2 n k. q Proof. Clearly, c n) n =. Moreover compare [Effros and Popa 2003]) W i,...,i n = X i W i2,...,i n 2 q 2 δ i =i W i2,..., ˆ i,...,i n where ˆ denotes omission). So the degree of W i,...,i n is n and the coefficient c n of a monomial of degree k in W i,...,i n is at most the sum of a coefficient of a degree k monomial in W i2,...,i n and 2 q 2 k, where k is a coefficient of a degree k monomial in W i2,...,î,...,i n. By induction, we see that c n) k c n ) k + n 2 q 2 c n 2) k 2 n k 2 ) n k ) + 2 n k 2 n k 2 q q q 0 [ ) = 2 n k 2 n k ) ] n k 2 + q q q ) 2 n k 2 n k ) 2 2 n k n k. q q as claimed. The upper estimate on W i,...,i n 2 follows in a similar way. L 2 M)

18 34 DIMITRI SHLYAKHTENKO Lemma 2. Let {ξ k : k K } be a finite set of vectors in an inner product space V. Let Ɣ be the matrix Ɣ k,l = ξ k, ξ l. Assume that Ɣ is invertible and let B = Ɣ /2. Then the vectors ζ l = k B k,l ξ k form an orthonormal basis for the span of {ξ k : k K }. Moreover, if λ denotes the smallest eigenvalue of Ɣ, then B k,l λ /2 for each k, l. Proof. We have, using the fact that B is symmetric and BƔB = I : ζ l, ζ l = k,k B k,lξ k, B k,l ξ k = k,k B k,l B k,l Ɣ k,k = BƔB) l,l = δ l=l. Lemma 3. There exist noncommutative polynomials p i,...,i n in X,..., X N so that the vectors {p i,...,i n X,..., X n ) } N i,...,i n = are orthonormal and have the same span as {W i,...,i n } N i,...,i n =. Moreover, these can be chosen so that p i,...,i n is a polynomial of degree at most n and the coefficient of each degree k monomial in p is at most 2 q ) n/2 2N) n q ) k 2 k. Proof. Consider the inner product matrix Ɣ n = [ W i,...,i n, W,..., n ] N i,...,i n,,..., n =. Dykema and Nica [993, Lemma 3.] proved that one has the following recursive formula for Ɣ n. Consider an N n -dimensional vector space W with orthonormal basis e i,...,i n, i,..., i n {,..., N}, and consider the unitary representation π n of the symmetric group S n given by σ e i,...,i n = e iσ ),...,i σ n). Denote by ) the action via π n ) of the permutation that sends to, k to k for 2 k, and l to l for l > on W. Let M n = n = q ) EndW ). Then Ɣ is the identity N N matrix, and Ɣ n = Ɣ n )M n, where Ɣ n acts on the basis element e,..., n by sending it to k 2,...,k n Ɣ n ) 2,..., n, k 2,...,k n e,k 2,...,k n and Ɣ acts on the basis elements by sending e,..., n to k,...,k n Ɣ n ),..., n, k,...,k n e k,...,k n. They then proceeded to prove that the operator M n is invertible and derive a bound for its inverse in the course of proving [Dykema and Nica 993, Lemma 4.]. Combining this bound and the recursive formula for Ɣ n yields the following lower estimate for the smallest eigenvalue of Ɣ n : c n = q q q k= q k ) n + q k = q )) n q k2 k 0 q )) n = q k= q ) n/2 ) 2 q n q ) 2. ) k q k2 ) n ) n q k k

19 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 35 Thus if we set B = Ɣn /2, then all entries of B are bounded from above by cn /2. Thus if we apply the previous lemma with K = {,..., N} n to the vectors ξ i,...,i n = W i,...,i n, we obtain that the vectors ζ i = K B,i ξ, i K form an orthonormal basis for the subspace of the Fock space spanned by tensors of length n. Now for i = i,..., i n ) K, let p i X,..., X N ) = K B,i W X,..., X N ). Then ζ i = p i X,..., X N ) are orthonormal and because the vacuum vector is separating), the polynomials {p i : i K } have the same span as {W i : i K }. Furthermore, if a is the coefficient of a degree k monomial r in p i, then a is a sum of at most N n terms, each of the form the coefficient of r in W )B,i. Using Lemma, we therefore obtain the estimate a N n c /2 n 2 n k q ) n k) = 2N 2 q ) /2 ) n 2 k q ) k. We now use the terminology of Section 2.. in dealing with noncommutative power series. Let R 0 = 2 q ) 2 q) X. Then if α >, p = p i,...,i n is as in Lemma 3, and φ p is as in Section 2.., then the coefficient of z k, k n in φ p is bounded by 2N 2 q ) /2 ) n R k 0 2Nα 2 q ) /2 In particular for any ρ < αr 0, ) 2Nα n n p i,...,i n ρ 2 q ) /2 αn R 0 ) k N k ρ k k=0 Lemma 4. Let q be such that q < 4N 3 + 2). Then: ) n αn R 0 ) k. 2Nα 2 q ) /2 ) n ρ/αr 0 ). a) The formula Y,..., Y N ) = q n p i,...,i n Y,..., Y N ) p i,...,i n Y,..., Y N ) n i,...,i n defines a noncommutative power series with values in Y,..., Y N 2 with radius of convergence strictly bigger than the norm of a q-semicircular element, X 2 q). b) If X,..., X N are q-semicircular elements and is as in Lemma 0, then = X,..., X N ) convergence in Hilbert Schmidt norm, identifying H S with L 2 M) L 2 M)). Proof. Clearly, ) p i,...,i n p i,...,i n ρ p i,...,i n 2 ρ 2Nα 2n 4N 2 2 q ) /2 ρ/αr 0 )) 2 = K α 2 ) n ρ 2 q for any ρ < αr 0, where R 0 = 2 q ) X.

20 36 DIMITRI SHLYAKHTENKO Thus 4N 2 α 2 ) n 4N ρ K ρ q n N n 3 ) α q n K ρ, 2 q 2 q n which is finite as long as ρ < αr 0 and the fraction in the sum in the right is less than. Thus as long as 4N 3 q < 2 q, that is, q < 4N 3 + 2), we can choose some α > so that the series defining has a radius of convergence of at least αr 0 > X. For part b), we note that because L 2 M) M and because of the definition of the proective tensor product, we see that H S M ˆ M on M ˆ M. Thus convergence in the proective norm on M ˆ M guarantees convergence in Hilbert Schmidt norm. Furthermore, by definition of orthogonal proection onto a space, = q n P n, where P n = i,...,i n p i,...,i n p i,...,i n = n) X,..., X N ) are the partial sums of X,..., X N ) here we again identify H S and L 2 L 2 ). Hence = X,..., X N ). 5. An estimate on free entropy dimension We now show how an estimate of the form -3) can be used to prove a lower bound for the free entropy dimension δ 0. Recall from [Voiculescu 996; 994] that if X,..., X n M, τ) is an n-tuple of self-adoint elements, then the set of microstates Ɣ R X,..., X n ; l, k, ε) is defined by { Ɣ R X,..., X n ; l, k, ε) = x,..., x n ) M sa n k k )n : x < R and τpx,..., X n )) /k) Trpx,..., x n )) < ε for any monomial p of degree l If R is omitted, the value R = is understood. The free entropy is defined by χx,..., X n ) = sup R inf l,ε lim sup k k 2 log VolƔ RX,..., X n ; l, k, ε)) + n 2 log k The set of microstates for X,..., X n in the presence of Y,..., Y m is defined by Ɣ R X,..., X n : Y,..., Y m ; l, k, ε) = { x,..., x n ) : y,..., y m ) The corresponding free entropy in the presence is then defined as by χx,..., X n : Y,..., Y m ) = sup R inf l,ε s.t. x,..., x n, y,..., y m ) Ɣ R X,..., X n, Y,..., Y m ; l, k, ε) }. lim sup k k 2 log VolƔ RX,..., X n : Y,..., Y m ; l, k, ε)) + n ) 2 log k. ). }.

21 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 37 The sup R is attained [Belinschi and Bercovici 2003]; in fact, once R > max i, { X i, Y }, we have χx,..., X n : Y,..., Y m ) = χ R X,..., X n : Y,..., Y m ). Finally, the free entropy dimension δ 0 is defined by δ 0 X,..., X n ) = n + lim sup t 0 χx + t S,..., X n + t S n : S,..., S n ), log t where S,..., S n are a free semicircular family, free from X,..., X n. Equivalently [Jung 2003a] one sets δ X,..., X n ) = inf lim sup ε,l k k 2 log K δɣ X,..., X n ; k, l, ε)), where K δ X) is the covering number of a set X the minimal number of δ-balls needed to cover X). Then δ 0 X,..., X n ) = lim sup t 0 t X,..., X n ). log t Lemma 5. Assume that X,..., X n M, τ), T k W X,..., X n ) W X,..., X n ) op are given. Set S T = k T k#s k. Let η = dim M M o span M S T M + + M ST n M L2 M M o ) ). Then there exists a constant K depending only on T so that for all R > 0, α > 0, t > 0, there are ε > 0, l > 0, and k > 0 so that for all 0 < ε < ε, k > k, and l >l, and any x,..., x n ) ƔX,..., X n ; k, l, ε) the set Ɣ R t S I T,..., t Sn I T x,..., x n ) : S,..., S n ; k, l, ε) = { y,..., y n ) : s,..., s n ) s.t. y,..., y n, x,..., x n, s,..., s n ) Ɣ R t S I T,..., t Sn I T, X,..., X n, S,..., S n ; k, l, ε) } can be covered by K/t) n η+α)k2 balls of radius t 2. Proof. By considering the diffeomorphism of Mk k sa )n given by a,..., a n ) /t)a,..., /t)a n ), we may reduce the statement to showing that the set Ɣ R S I T,..., Sn I T x,..., x n ) : S,... S n ; k, l, ε) can be covered by C/t) n η+α)k2 balls of radius t. Note that η is the Murray von Neumann dimension over M M o of the image of the map ζ,..., ζ n ) ζ T,..., ζ T n ), where ζ L 2 M) L 2 M), M = W X,..., X n ). Thus if we view T as a matrix in M n n M M o ), then τ τ TrE {0} I T ) I T ))) = η here E X denotes the spectral proection corresponding to the set X ). Fix α > 0. Then there exists Q M n n [X,..., X n ] 2 ) depending only on t so that Q i I T ) i 2 <t/2n) here we view Q as a matrix whose entries are noncommutative functions in n indeterminates; the entries of Q are in the space ) in the notation of Section 2..).

22 38 DIMITRI SHLYAKHTENKO Set S Q = k Q k#s k. Then In particular, S QX,...,X n ) so that S QX,...,X n ) S I T < t 2. S I T 2 < t/2. We may moreover choose Q again, depending only on t) τ τ TrE [0,t/2[ Q Q) /2 X,..., X n )) τ τ TrE {0} I T ) I T )) = η 2 α. Thus for l sufficiently large and ε > 0 sufficiently small, we have that if y,..., y n ) Ɣ R S I T,..., Sn I T x,..., x n ) : S,..., S n ; k, l, ε), then there exist s,..., s n such that and s,..., s n, x,..., x n ) Ɣ R S,..., S n, X,..., X n ; k, l, ε) s Qx,...,x n ) y 2 < t. By approximating the characteristic function χ [0,t/2] with polynomials on the interval [0, Qx,..., x n ) ] which is compact, since x < R), we may moreover assume that l is large enough and ε is small enough that k 2 Tr Tr Tr E [0,t/2] Q Q) /2 x,..., x n ) ) η α. Denote by φ the map s,..., s n ) s Qx,...,x n ),..., s Qx ),...,x n ) n. Let R = max S I T 2 +. Assume that ε <. Then φ : Mk k sa )n Mk k sa )n is a linear map, and since s ε < 2, we have the inclusion Ɣ R S I T,..., S I T n x,..., x n ) : S,..., S n ; k, l, ε) N t φb2)) BR )), where BR) the a ball of radius R in Mk k sa )n endowed with the L 2 norm) and N t denotes a t- neighborhood. The matrix of φ is precisely the matrix Qx,..., x n ) M n n M k k ) 2. Let β be such that βnk 2 eigenvalues of φ φ) /2 are less than R 0. Then the t-covering number of φb2)) BR ) is at most ) β)nk 2 ) βnk 2 R 2R0. t t Let R 0 = t/2, so β = η α)/n. We conclude that the t-covering number of Ɣ R S I T,..., Sn I T x,..., x n ) : S,..., S n ; k, l, ε ) is at most K/t) n η+α)k2, for some constant K depending only on R, which itself depends only on T.

23 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 39 Theorem 6. Assume that X,..., X n M, τ), S,..., S n, {S : J} is a free semicircular family, free from M, T k W X,..., X n ) W X,..., X n ) op are given, and that for each t > 0 there exist W X,..., X n, S,..., S n, {S } J ) so that: Y t) the oint law of Y t),..., Y t) n ) is the same as that of X,..., X n ), = k T k#s k and Z t) if we set S T independent of t, we have Z t) Let M = W X,..., X n ) and let η = dim M M o = X + t S T, then for some t 0 > 0 and some constant C < Y t) 2 Ct 2 for all t < t 0. span M S T M + + M ST n M L2). Assume finally that W X,..., X n ) embeds into R ω. Then δ 0 X,..., X n ) η. Proof. Let T : M M o ) n M M o ) n be the linear map given by T Y,..., Y n ) = T k #Y k,..., ) T nk #Y k k k here, as before, we identify M M o ) n with the linear span of M S M + + M S n M via the map T,..., T n ) S T,..., S T n )). Then η is the Murray von Neumann dimension of the image of T, and consequently η = n dim M M o ker T. Let t be fixed. Since Y t) can be approximated by noncommutative polynomials in X,..., X n, S,..., S n and {S : J}, for any k 0, ε 0, l 0 sufficiently large we may find k > k 0, l > l 0, ε < ε 0 and J 0 J finite so that whenever z,..., z n ) Ɣ R X + t S T,..., X n + t S T n : S,..., S n, {S } J 0 ; k, l, ε ), there exists so that y,..., y n ) Ɣ R X,..., X n ; k, l 0, ε 0 ) y z 2 Ct 2. 5-) For a set X Mk k sa )n we ll write K r for its covering number by balls of radius r. Consider a covering of Ɣ R X + t S,..., X n + t S n : S,..., S n, {S } J 0 ; k, l, ε) by balls of radius C + 2)t 2 constructed as follows. First, let B α ) α I be a covering of Ɣ R X + t S T,..., X n + t Sn T : S,..., S n, {S } ) J 0 ; k, l 0, ε 0 by balls of radius C + )t 2. Because of 5-), we may assume that I K t 2Ɣ R X,..., X n ; k, l, ε)). Next, for each α I, let x α),..., x n α) ) B α be the center of B α. Consider a covering C α) β : β J α ) of Ɣ R t S I T,..., t Sn I T x α),..., x n α) ) : S,..., S n ; k, l, ε ) by balls of radius t 2. By Lemma 5, this

24 40 DIMITRI SHLYAKHTENKO covering can be chosen to contain J α K/t) n η balls, for any η < η. Thus the sets B α + C α) β : α I, β J α ), each of which is contained in a ball of radius at most C + 2)t 2, cover the set The cardinality of this covering is at most Ɣ R X + t S,..., X n + t S n : S,..., S n ; k, l 0, ε 0 ). f t 2, k) I sup J α K t 2Ɣ R X,..., X n ; k, l, ε) K t) η n. α It follows that if we denote by V R, d) the volume of a ball of radius R in d, we find that VolƔ R X + t S,..., X n + t S n : S,..., S n, {S } J 0 )) f t 2, k) V C + 2)t 2, nk 2 ), so that if we denote by t 2X,..., X n ) the expression and set C = logc + 2), we obtain the inequality inf ɛ,l lim sup k inf lim sup ε,l k k 2 log K t 2ƔX,..., X n ; k, l, ε)) k 2 log VolƔ RX + t S,..., X n + t S n : S,..., S n, {S } J 0 ; k, l, ε) lim sup k log f t 2, k) + 2n log t + logc + 2) t 2X,..., X n ) + η n) log K t + 2n log t + C = t 2X,..., X n ) + η + n) log t + η n) log K + C. By the freeness of {S } J and {S,..., S n, X,..., X n }, the lim sup on the right-hand side remains the same if we take J 0 =. Thus χ R X + t S,..., X n + t S n : S,..., S n ) t 2X,..., X n ) + η + n) log t + C. If we divide both sides by log t and add n to both sides of the resulting inequality, we obtain n + χ RX + t S,..., X n + t S n : S,..., S n ) log t t 2X,..., X n ) log t = 2 t 2X,..., X n ) log t 2 + η + n) log t log t + n + η + n) log t log t + n. Taking sup R and lim sup t 0 and noticing that log t < 0 for t <, we get the inequality δ 0 X,..., X n ) 2δ 0 X,..., X n ) η + n) + n = 2δ 0 X,..., X n ) η. Solving this inequality for δ 0 X,..., X n ) gives finally δ 0 X,..., X n ) η. Since η < η was arbitrary, we conclude that δ 0 X,..., X n ) η as claimed.

25 LOWER ESTIMATES ON MICROSTATES FREE ENTROPY DIMENSION 4 Corollary 7. Let A, τ) be a finitely-generated algebra with a positive trace τ and generators X,..., X n, and let Der a A; A A) denote the space of derivations from A to L 2 A A, τ τ) which are L 2 closable and so that for some R), ξ R), R > max X, ) = ξx,..., X n ) and X ) = X,..., X n ). Consider the A,A-bimodule V = { δx ),..., δx n )) : δ Der a A; A A) } L 2 A A, τ τ) n. Assume finally that M = W A, τ) R ω. Then δ 0 X,..., X n ) dim M M o V L2 A A,τ τ) n. Proof. Let P : L 2 A A, τ τ) n V be the orthogonal proection, and set v = P0,...,,..., 0) with in the -th position. Let v k) = v k),..., vk) n ) L2 A A) n be vectors approximating v and having the property that the derivations defined by δx ) = v k) i lie in Der a. Then η k = dim M M o span Av k) A + + Av n k) A dim M M o V as k. Now for each k, the derivations δ : A L 2 A A) so that δ X i ) = v k) i belong to Der a. Applying Lemma 6 and Proposition 8 to T i = v k) i and combining the conclusion with Theorem 6 gives δ 0 X,..., X n ) η k. Taking k we get as claimed. δ 0 X,..., X n ) dim M M o V, Corollary 8. For a fixed N, and all q < 4N 3 + 2), the q-semicircular family X,..., X N satisfies ) δ 0 X,..., X N ) > and δ 0 X,..., X N ) N q2 N q 2. N In particular, M = W X,..., X N ) has no Cartan subalgebra. Moreover, for any abelian subalgebra M, L 2 M), as an, -bimodule, contains a copy of the coarse correspondence. Proof. Let i be a derivation as in Lemma 0; thus i X ) = δ i=, as defined in Lemma 0. Then for q < 4N 3 + 2), Lemma 4 shows that i Der a. Then Theorem 6 implies that δ 0 X,..., X N ) dim M M o M i M, M = W X,..., X n ). It is known [Shlyakhtenko 2004] that for q 2 < /N which is the case if we make the assumptions about q as in the hypothesis of the corollary), this dimension is strictly bigger than and is no less than N q 2 N q 2 N) ). The facts about M follow from [Voiculescu 996]. For N = 2, 4N 3 + 2) = /34. Thus the theorem applies for 0 q /34 = Our estimate also shows that δ 0 X,..., X N ) N as q 0.