Much of my mathematical research presented in this dissertation was inspired by the groundbreaking work of Dan-Virgil Voiculescu. By studying his extraordinary work on the Veena, I got a fundamental purpose in my life.
Main results
A subsequent breakthrough by Jung and Shlyakhtenko [JS07] showed that the finitely generated property (T) of the von Neumann algebra has the largest free entropy dimension1. Furthermore, if M is a finite von Neumann algebra with property (T), then there exists some faithful normal tracking stateτ on M such that (M,τ) is strongly1-bounded.
Discussion of Theorem 1.0.1
Thus, the remaining case is a property (T) von Neumann algebra that is a countable direct sum of factors. By [Pop06a, Proposition 4.7], a direct sum of trace von Neumann algebras has Property (T) if and only if every direct summan has Property (T).
Discussion of Theorem 1.0.5
In this article we generalize and streamline Jung's strategy from [Juna], which uses iteration to bound numbers for smaller and smaller numbers, with errors controlled by the integral (1.0.1). Much of the technical challenge in Jung's work had to do with converting between coverage numbers with respect to various noncommutative Lp norms in von Neumann algebra (and in fact Lp quasinorms for p∈(0,1)) .
Tracial von Neumann algebras and non-commutative laws
If (M,τ) is considered a noncommutative probability space, its elements can be considered noncommutative random variables. In fact, ad-tuplex= (x1, . . ,xd)∈Msad is the non-commutative analogue of an Rd-valued random variable.
Microstate spaces and 1-bounded entropy
Therefore, for any finitely generated tracial von Neumann algebra M, we can define h(M) as h(x) for some generating tuplex. A tracial von Neumann algebra M is strongly 1- bounded in Jung's sense [Jun07b] if and only ifh(M)<∞.
Properties and applications of 1-bounded entropy
If M is not a factor, the 1-bounded entropy depends on the track we choose on M. We will use h(M,τ) for a tracial von Neumann algebra (M,τ) if we want to emphasize the dependence of the 1-bounded entropy on τ.
Property (T)
Unlike the group case, it is not necessary that every traceable von Neumann algebra with property (T) admits Kazhdan tuples (the finite set F in the definition of property (T) may depend a priori on ε). If (M,τ) is a trace von Neumann algebra and x= (x1, . . . ,xd) is a Kazhdanov tuple, then x generates M as a von Neumann algebra.
Ultraproducts of matrix algebras
As a result, we can heuristically think of the microstate spacesΓ(n)R (O)asO→ℓx,n→∞ as the parameterization of the space of embeddings of M in an ultraproduct of matrices. See [Jun07a] and [AKE21, Sections 1.2, 1.3] for further discussion of the connections between ultraproducts and microstate spaces.
Proof of Theorem 1.0.1
Now that we know that microstates are conjugate to a small projection, to control the coverage number of the microstate space, we need to estimate the coverage numbers of the space of these projections. We rely on an estimate by Szarek [Sza98] on the coverage numbers of Grassmen, which we indicate in the following form. The coverage number of each of the individual sets can be bounded by (C/ε)2ℓ(n−ℓ)≤(C/ε)2n2t sinceℓ≤nt and n−ℓ≤n.
Next, we combine Lemma 3.0.1 and Lemma 3.0.2 to obtain the following estimate, which bounds the η-covering numbers for microstate spaces in terms of theε-covering numbers for η≤ε. Let O be enn0 as in Lemma 3.0.1 for the constantsδ and2ε (rather thanδ and ε), and assume that n≥n0.
Direct sums and strong 1-boundedness
Since the sum converges on the right-hand side, we have h(M)<∞, and so M is strongly 1-bounded. It is also convenient for the proof to use the description of 1-bounded entropy in terms of relative microstates rather than unitary orbits. In [Hay18] this is only shown if a is a single element that generates a diffuse, abelian von Neumann algebra, but the same argument works for any tuple that generates a diffuse, hyperfinite von Neumann algebra.
The above definition using relative microstates can be intuitively thought of as describing the1-bounded entropy as a measure of how many embeddings N has in an ultraproduct of matrices both bounded to a given embedding of W∗(a) and ' has an extension to M. Since{ ⌊τ(z1)n⌋:n∈N}and{⌈τ(z2)n⌉:n∈N}have finite complements inN, and since limit infinums are superadditive, the above inequality shows that for allε>0 we have .
Amplification and strong 1-boundedness
Since N≤r is an increasing sequence of subalgebras of N and WN≤r=N, we have that. ii) If (M,τ) is Connes embedding and N≤M is hyperfinite, then the inclusion N≤M has the extension property of the microstates. iii) If (Mj,τj),j=1,2is Connes embedding, then M1∗1≤M1∗M2 has the microstate expansion property. If Q≤M has the microstate extension property, then every embedding of Q extends to M, and thus the quantities h(N:Q),h(N:M) should be the same. ii): Without loss of generality, we can and will assume that N,M has separable predual. With Lemma 3.0.5 on direct sums and Proposition 3.0.9 on amplifications in hand, we are ready to finish the proof of Proposition 1.0.2 showing that strong1-boundedness of all traces of von Neumann algebras with property (T ) is equivalent to h( M)≤0 for allII1factors with property (T).
If any von Neumann tracial algebra with property (T) satisfies(M)≤0, then it is strongly 1-restricted since strongly1-restricted is equivalent to toh(M)<∞ by [Hay18, Proposition A.16]. Finally, we show that for every finite von Neumann algebra M with property (T), there exists a faithful tracial state τ such that (M, τ) is strongly bounded 1.
Direct sums and free entropy dimension
The following lemma, based on Jung's earlier work, describes the free entropy dimension of behavior under direct sums. Since being strongly bounded 1 implies the free entropy dimension of the microstates at most 1 with respect to any set of generators, the previous lemma automatically implies the following. We note that if one works carefully with the free entropy dimension in the presence of Lemma 3.0.11, then one can give a proof of Corollary 3.0.12 even for infinite tuples.
After modifying the results in [Jun06, Jun03b] to work for the free entropy dimension in the presence, the proof of (3.0.3) in the case where x is a finite set and J={1,2} proceeds exactly as in Lemma 3.0 .11 . The proof of the general case (3.0.3) also follows from this special case exactly as in Lemma 3.0.11.
Proof of Theorem 1.0.6
Background on non-commutative derivatives and Taylor expansion
This matrix plays a similar role to the derivative of the function Rd→Cm, providing a first-order term in the noncommutative Taylor expansion for the evaluation of f on the elements of the tracking von Neumann algebra. Then there exists a constant Af,Bf,Cf that depends only on f and R, such that for every trace von Neumann algebra (M,τ) and x,y∈Msad z. 1Mop is an algebra with the same addition and*-operation, but the order of multiplication is reversed; note that Mopis is a tracking von Neumann algebra.
The following lemma will be needed to show that the spectral measures of certain operators on Mn(C)d associated with matrix microstates forx∈Msad asn→∞ converge to the spectral measures of corresponding operators from a trace von Neumann algebra. In the following, for a trace von Neumann algebraM, we denote by M⊗Mop the trace von Neumann algebraic tensor product of M, endowed with the trace τM⊗τMop.
Covering the microstate space
By the setup of Theorem 1.0.6 there is a constant C>0 (depending only on f and R), so that we have it for allε,η>0. Recall that if a set can be covered by a certain number of ε-balls with centers not necessarily in that set, it can be covered by the same number of 2ε-balls with centers in the set. If Y is in the orbital (2ε,∥·∥2) ball around at
Of course, the number of η,∥·∥2)-balls needed to cover the set of eZs obtained in this way is the maximum. Because all the covering numbers are monotone in the variable "O", taking the infimum over all O gives the same result whether or not we intersect with U first.
Covering the approximate kernel
The second ingredient to estimate ΨR,η,δ,ε(x,f) is the following standard estimate for covering the number of approximate kernels of operators on a Hilbert space. Of course, we will apply this lemma to the operator Df(X)# from the Hilbert space Mn(C)d with the normalized Hilbert-Schmidt norm. Now note that as O shrinks to {ℓx}, measures µ|Df(X)| forX∈Γ(n)(O)converges uniformly in distribution to µ|Df(x)| using Lemma 4.0.2.
Iteration of the estimates
Although we have stated Theorem 1.0.6 only for the polynomial f for simplicity, the same argument works for more general non-commutative functions. More generally, it holds for the non-commutative C2 functions of [JLS21] (as well as those of [DGS16]). Expanding this trace tod×d matrices over Ctrk−1(R∗d,M1) allows us to understand the spectral measure of ∂f(x)∗∂f(x).
In this section, we recall the connection between ℓ2cohomology and the noncommutative difference quotient (§5.0.1) used by Shlyakhtenko [Shl21], as well as his argument why Theorem 1.0.6 implies Theorem. We then show how the argument for Theorem 1.0.5, together with Shalom's result [Sha00], provides an alternative proof of strong boundedness1 for von Neumann algebras associated with sofic groups Properties (T) (§5.0.3).
Cocycles, derivations, and the free difference quotient
Let G be a countable discrete group, let τ be the canonical trace and set M= L(G). ii) Suppose Gi is finitely generated, and suppose g1,· · ·,gr is a finite current aggregate. Then Gi is finitely presented if and only if J is finitely produced as a two-sided ideal. ii): Let Frbe be the free group lettersa1, · · ·,ar.Consider the surjective homomorphismq:Fr→. First suppose that Gi is finitely presented, and let F be a finite subset of the kernel of q:Fr → G such that ker(q) is the smallest normal subgroup containing F. It can be directly verified that the kernel of q: C[Fr] → C [G ]is the smallest ideal inC[Fr]that contains{w−1 :w∈F}.Forw∈F,letQw∈C⟨t1, · · ·,t2k⟩ be an element such that π(Qw) =φ(w) . We leave it as an exercise to show that J is generated as a two-sided ideal by.
Then Qj∈I for all j, and so I=ker(q:C[Fr]→C[G]). If Ne is the normal subgroup of G generated by R, then I is the core of the natural quotient map. Let (Fj)∞j=1 be a sequence that generates a two-sided ideal in C⟨t1, · · ·,tk⟩.
Strong 1-boundedness from vanishing ℓ 2 -Betti numbers
Strong 1-boundedness of Property (T) sofic groups from Theorem 1.0.5
AKE21] Scott Atkinson and Srivatsav Kunnawalkam Elayavalli, On ultraproduct embeddings and adaptability for von Neumann tracian algebras, Int. GS02] Liming Ge and Junhao Shen, On the free entropy dimension of finite von Neumann algebras, Geom. L¨uc98] Wolfgang L¨uck, Dimension theory of arbitrary modules over finite von Neumann algebras and L2-Betti numbers.
Pet09a] Jesse Peterson, A 1-cohomological characterization of property (t) in von neumann algebras, Pacific Journal of Math. PP05] Jesse Peterson and Sorin Popa, On the notion of relative property (T) for inclusions of von Neumann algebras, J.
