aa r X i v : . [ m a t h . OA ] A ug A RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE
BEN HAYES
Abstract.
The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free groupfactor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related resultsin Popa’s deformation/rigidity theory and Peterson-Thom’s results on L -Betti numbers. We present anapproach to this conjecture in terms of so-called strong convergence of random matrices by formulating aconjecture which is a natural generalization of the Haagerup-Thorbjørnsen theorem whose validity wouldimply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud [15]. Introduction
Amenability is arguably the central concept in von Neumann algebra theory. Celebrated and fundamentalwork of Connes [17] shows that amenable von Neumann algebras are precisely the hyperfinite ones and alsogives several equivalent forms of amenability. Because of this famous work, amenable algebras are wellunderstood and completely classified.Given our understanding of amenable von Neumann algebras, it is natural to try to bootstrap our knowl-edge of arbitrary von Neumann algebras from our knowledge of the amenable ones. This naturally motivatesthe study of maximal amenable subalgebras of a von Neumann algebra, those subalgebras which are amenableand are maximal with respect to inclusion among amenable subalgebras. In a landmark discovery [59], Popashowed that L ( Z ) is a maximal amenable subalgebra of L ( F r ) = L ( Z ∗ F r − ) for any r ∈ N (here andthroughout the paper F r denotes the free group on r letters). This provided the first example of a maximalamenable subalgebra that was abelian (a phenomenon that was unexpected at the time), and it gave anegative answer to a related problem of Kadison stated during the Baton Rouge conference in 1967. In fact,Popa’s work establishes the more general result that L ( Z ) is maximal Gamma in L ( F r ). The foundationalinsight of Popa was the usage of his asymptotic orthogonal property to establish maximal amenability. Manyauthors have used Popa’s asymptotic orthogonal property to establish maximal amenability in various cases,see [25, 65, 22, 11, 24, 8, 34, 35, 47].Popa’s deformation/rigidity theory also allows one to deduce strong ”malnormality” properties of freegroup factors. For example, primeness of all nonamenable subfactors [56], as well as the celebrated strongsolidity of free group factors [53]. Much of the developments in deformation/rigidity theory go beyondfree group factors and apply to von Neumann algebras associated to groups which have a combination ofapproximation properties and nontrivial cohomology [61, 60, 62, 63, 56, 54, 14], as well as crossed productalgebras associated to actions of such groups. See [64, 67, 68, 38, 39] for further results, including resolutions Date : August 28, 2020.The author gratefully acknowledges support from NSF Grants DMS-1600802 , DMS-1827376, and DMS-2000105. of long-standing open problems. These developments parallel, and frequently require input from, the theoryof L -Betti numbers for groups (developed in [2]) as well as equivalence relations (developed in [23]). Giventhese connections, we should expect in general that results from the theory of L -Betti numbers will havenatural analogues for von Neumann algebras. In [57], Peterson-Thom proved various indecomposability andmalnormality results for groups with positive first L -Betti number. Based on their work, and previous workof Ozawa-Popa, Peterson, and Jung [53, 56, 44], they conjectured the following for von Neumann algebras. Conjecture 1.
Fix r > . If Q is a von Neumann subalgebra of L ( F r ) which is both diffuse and amenable,then there is a unique maximal amenable von Neumann subalgebra P of L ( F r ) with Q ⊆ P. For the rest of the article, if M is a von Neumann algebra, we will use the notation N ≤ M to meanthat N is a unital, von Neumann subalgebra of M. Given N ≤ M with M a finite von Neumann algebraand N diffuse, we say that N has the absorbing amenability property (see [36, Theorem 4.1]) if whenever Q ≤ M is amenable and Q ∩ N is diffuse, we have Q ⊆ N. An equivalent way of phrasing Conjecture 1is to say that if r > , then any maximal amenable N ≤ L ( F r ) has the absorbing amenability property.For many examples of maximal amenable subalgebras of free group factors this has been verified [74, 9, 55],and typically uses a generalization of Popa’s asymptotic orthogonality property, called the strong asymptoticorthogonality property implicitly defined in [36, Theorem 3.1]. Many exciting recent works [6, 52, 7] applyan alternative method using an analysis of states.The fact that we can show the absorbing amenability property for many examples of maximal amenablesubalgebras of free group factors is strong evidence for the Peterson-Thom conjecture, but as of yet themethods of proof for these examples have not led to a general approach to the problem. The goal ofthis paper is to provide such an approach through Voiculescu’s free entropy dimension theory and randommatrices. Free entropy dimension theory was initiated by Voiculescu in a series of papers [70, 71], and providesa powerful method to deduce indecomposability and malnormality results for free group factors (amongother algebras). For example, Voiculescu used free entropy dimension, in combination with his previouslyestablished random matrix results [69, 72], to give the first proof of absence of Cartan subalgebras in freegroup factors [71]. Shortly after this work, primeness and thinness of free group factors were first establishedby Ge, Ge-Popa using free entropy dimension theory [26, 27]. See [18, 44, 37] for other applications. Popa’sdeformation/rigidty theory and asymptotic orthogonality techniques apply to a wider range of algebrasthan free group factors/amalgamated free products, and do not require the Connes approximate embeddingproperty for their applicability. However, some indecomposability results for free group factors shown usingfree entropy dimension theory cannot currently be approached by deformation/rigidity theory or the (strong)asymptotic orthogonal property [18, 27, 31, 33].We now present our result which reduces the Peterson-Thom conjecture to a natural random matrixproblem. It requires usage of the 1 -bounded entropy , which was implicitly defined in [44] and explicitly in[31]. If N ≤ M are diffuse, tracial von Neumann algebras the 1 -bounded entropy of N in the presence of M (denoted h ( N : M )) is some sort of measurement of “how many” ways there are to “simulate” N by matriceswhich have an extension to a “simulation” of M by matrices (see Definition 2.5 for the precise definition). RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 3
Throughout the paper, we say that a random self-adjoint matrix X ∈ M k ( C ) s.a. is GUE distributed if { X ii : i = 1 , · · · , k } ∪ {√ X ij : 1 ≤ i < j ≤ k } ∪ {√ X ij : 1 ≤ i < j ≤ k } is an independent family of Gaussian random variables each with mean 0 and variance k . For a naturalnumber k, we use C h T , T , · · · , T k i for the C -algebra of noncommutative polynomials in k -variables (i.e. thefree C -algebra in k indeterminates). Theorem 1.1.
Fix an integer r ≥ . Consider the following statements.(i) If Q ≤ L ( F r ) is diffuse and amenable, then there is a unique maximal amenable P ≤ L ( F r ) with Q ≤ P. (ii) If Q ≤ L ( F r ) is nonamenable, then h ( Q : L ( F r )) > . (iii) For k ∈ N , let X ( k )1 , · · · , X ( k ) r , Y ( k )1 , · · · , Y ( k ) r be random, self-adjoint k × k matrices which are in-dependent and are each GUE distributed. Set X ( k ) ⊗ M k ( C ) = ( X ( k ) i ⊗ M k ( C ) ) ri =1 , M k ( C ) ⊗ Y =(1 M k ( C ) ⊗ Y ( k ) i ) ri =1 . Let s = ( s , · · · , s r ) be r free-semicirculars each with mean zero and variance . Let s ⊗ C ∗ ( s ) = ( s i ⊗ C ∗ ( s ) ) ri =1 , C ∗ ( s ) ⊗ s = (1 C ∗ ( s ) ⊗ s i ) ri =1 ∈ ( C ∗ ( s ) ⊗ min C ∗ ( s )) r . Then with highprobability the law of ( X ( k ) ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ Y ( k ) ) tends (as k → ∞ ) to the law of ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) strongly. Namely, for every polynomial P ∈ C h T , · · · , T r , T r +1 , · · · , T r i we have (1) k P ( X ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ Y ( k ) ) k ∞ → k →∞ k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k C ∗ ( s ) ⊗ min C ∗ ( s ) in probability.The (iii) implies (ii) implies (i). As we remark explicitly in Sections 3, 4 the GUE ensemble can be replaced by the Haar unitary ensemble(with the limit distribution being free Haar unitaries) and many of the other random matrix ensembles fromrandom matrix theory (this is implied by the more general Theorem 1.2). We state the results for the GUEensemble mostly for convenience. Strictly speaking, so-called strong convergence is the conjunction of (1)with(2) 1 k Tr ⊗ Tr( P ( X ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ Y ( k ) )) → k →∞ τ ⊗ τ ( P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ))for every noncommutative polynomial P ∈ C h T , · · · , T r , T r +1 , · · · , T r i , where τ is the underlying tracialstate on C ∗ ( s , · · · , s r ). However, the fact that (2) holds for every P ∈ C h T , · · · , T r , T r +1 , · · · , T r i isalready a consequence of Voiculescu’s asymptotic freeness theorem [69]. The concept of strong convergencearises from groundbreaking work of Haagerup-Thorbjørnsen [29] who showed that the law of an r -tuple ofindependent, k × k GUE distributed matrices converges strongly to the law of an r -tuple of freely independentsemicircular variables which each have mean zero and variance 1 . This work opened up an array of powerfultools which have been used in combination with delicate analytic and combinatorial arguments to establishstrong convergence for many other ensembles, and also for a mixture of random and deterministic ensembles[49, 16, 4]. In particular, recent work [58, 15] shows that if m k is a sequence of positive integers with | m k | ≤ Ck / for some constant C ≥ , and if X ( k ) is an r -tuple of independent, k × k GUE distributedmatrices, and Y ( m k ) is an r -tuple of independent, m k × m k GUE distributed matrices chosen independentof X ( k ) , then (in the notation of the above theorem) the law of ( X ( k ) ⊗ M mk ( C ) , M k ( C ) ⊗ Y ( m k ) ) converges BEN HAYES strongly to the law of ( x ⊗ C ∗ ( x ) , C ∗ ( x ) ⊗ x ) . While this work does not quite resolve our conjecture (weneed m k = k ), it nevertheless lends positive evidence to the validity of our approach.Our work actually establishes something slightly more general than the above. To state this more gen-eral result requires as input the notion of exponential concentration of measure, a well-established tool inprobability and geometric functional analysis (see Definition 2.12 for the precise definition). It also uses thehighly general noncommutative functional calculus of Jekel initiated in [40, 41, 33]. We recall the preciseconstruction in Section 2.4, but for now the reader should simply know that for r ∈ N and R ∈ [0 , ∞ ) thereis a space F R,r, ∞ of “noncommutative functions” defined on the R -ball of any von Neumann algebra whichare uniformly L -continuous in an appropriate sense and have the property that if ( M, τ ) is any tracialvon Neumann algebra, and x ∈ M r has k x j k ≤ R, j = 1 , · · · , r then given any y ∈ W ∗ ( x ) , there is an f ∈ F R,r, ∞ with f ( x ) = y. For natural numbers k, r, we define a pseudometric d orb on M k ( C ) r as follows.For A ∈ M k ( C ) , we set k A k = k Tr( A ∗ A ) . We then define d orb ( A, B ) = inf U ∈ U ( k ) r X j =1 k U A j U ∗ − B j k / . We also use the notational convention that if C ∈ M k ( C ), D = ( D , D , · · · , D r ) ∈ M k ( C ) r then CD =( CD , CD , · · · , CD r ) ∈ M k ( C ) r and similarly for DC.
For A ∈ M K ( C ), we let A t be its transpose. Thefollowing is our more general random matrix result. Theorem 1.2.
Let X ( k ) = ( X ( k ) j ) lj =1 be a tuple of n ( k ) × n ( k ) -random matrices. Suppose that • There is an ( R j ) lj =1 ∈ [0 , ∞ ) l so that for all j lim k →∞ P ( k X ( k ) j k ∞ ≤ R j ) = 1 . • The law of X ( k ) converges in probability to the law of a tuple x = ( x j ) lj =1 in a tracial von Neumannalgebra ( M, τ ) with M = W ∗ ( x , · · · , x l ) . • The probability distribution of X ( k ) exhibits exponential concentration of measure at scale n ( k ) as k → ∞ . (i) Suppose that Q ≤ M is finitely generated, diffuse, and h ( Q : M ) ≤ . Suppose y ∈ Q r with Q = W ∗ ( y ) and write y = f ( x ) for some f ∈ F R,r, ∞ . Then there exists A ( k ) ∈ M n ( k ) ( C ) l such that A ( k ) convergesto x in law and so that d orb ( f ( X ( k ) ) , f ( A ( k ) )) → , in probability. Namely, for every ε > P ( d orb ( f ( X ( k ) ) , f ( A ( k ) )) < ε ) → k →∞ . (ii) Assume that C ∗ ( x ) is locally reflexive. Let Y ( k ) = ( Y ( k ) j ) lj =1 be an independent copy of ( X ( k ) j ) lj =1 . If ( X ( k ) ⊗ M n ( k ) ( C )) , M n ( k ) ( C ) ⊗ ( Y ( k ) ) t ) converges strongly in probability to ( x ⊗ C ∗ ( x ) op , C ∗ ( x ) ⊗ x op ) , then any diffuse Q ≤ M with h ( Q : M ) ≤ is necessarily amenable. In particular, given any diffuse,amenable Q ≤ M there is a unique maximal amenable P ≤ M with Q ⊆ P. RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 5
Let us comment a bit on the intuition for (i). The assumption that the law of X ( k ) converges in probabilityto the law of x means that the randomly chosen matrix X ( k ) “simulates” x with high probability. Interminology introduced in Section 2.2, we say that X ( k ) are microstates for x. The general properties of Jekel’snoncommutative functional calculus then guarantee that the random matrix f ( X ( k ) ) are also microstatesfor f ( x ) = y. The conclusion of (i) then asserts that the random microstates for y produced by the randommatrix f ( X ( k ) ) are all approximately unitarily equivalent to each other. If one passes to the ultraproductframework, then the picture becomes much clearer. These randomly chosen microstates then turn into(random) honest embeddings into a ultraproduct of matrices, and the conclusion of (i) is then the assertionthat, with high probability, these different embeddings are all unitarily conjugate when restricted to Q. Viewed through this lens, part (ii) is connected with Jung’s theorem [43] that a tracial von Neumannalgebra which satisfies Connes approximate embeddability property is amenable if and only if any twoembeddings into an ultraproduct of matrices are unitarily conjugate (see [3] for a recent generalization ofthis fact to conjugation by unital, completely positive maps). In fact, Jung’s argument shows the followingmore general fact: if (
M, τ ) is a tracial von Neumann algebra which embeds into an ultraproduct of matricesthen given any nonamenable N ≤ M there are two embeddings of M into an ultraproduct of matrices whichare not unitarily conjugate when restricted to N. A consequence of (ii) is that, under the assumption ofstrong convergence, given a nonamenable N ≤ M a randomly chosen pair of embeddings of M into anultraproduct of matrices are not unitarily conjugate when restricted to N . We refer the reader to Section 4for a more precise discussion of parts (i),(ii) in an ultraproduct framework.We close with a discussion of organization of the paper. Section 2 is a discussion of background for thepaper. In Section 2.1 we state our conventions and notation from von Neumann algebra and operator spacetheory. In Section 2.2 we recall the notion of noncommutative laws, and their use in defining 1-boundedentropy. Here we also recall the definitions of the weak ∗ and strong topologies on the space of laws. In Section2.3, we discuss (sequences of) measures on microstates spaces and the two important conditions on them wewill use: being asymptotically supported on microstates spaces, and exponential concentration. Section 2.4describes Jekel’s noncommutative functional calculus, as well as the modification we will need for the non-self-adjoint case. Strictly speaking, the usage of this general functional calculus is not necessary for the proofsof the main results and earlier versions of this paper did not use it. However, its usage drastically simplifiesboth the conception and the deduction of the main results and so we think its inclusion is worthwhile. Section3 contains the proofs of Theorems 1.1,1.2. Specifically, in Section 3.1 we prove Theorem 1.2 (i) which statesthat under an assumption of exponential concentration there is a “collapse” of the microstates space wherethe vast majority of the measure lives near a single unitary conjugation orbit. The methods of proof hereare similar to those in [33]. Section 3.2 contains a proof of Theorem 1.2 (ii), and it is here that both strongconvergence and local reflexivity play a crucial role. In Section 3.3 we deduce Theorem 1.1 from Theorem1.2. In Section 4 we explain how the results are related to Jung’s theorem, and give reformulations of themain results in an ultraproduct framework. In Section 4 we also introduce several conjectures related to thePeterson-Thom conjecture and Theorem 1.1, and we explicitly explore their relative strength. Finally, weclose in Section 5 with a few comments on the approach. In particular, we discuss the discontinuity in the BEN HAYES strong topology of taking tensors, and how exactness of free group factors may allow one to follow previousapproaches to proving strong convergence in probability.
Acknowledgements
The initial stages of this work were carried out at the Hausdorff Research Institutefor Mathematics during the 2016 trimester program “Von Neumann Algebras.” I thank the Hausdorff insti-tute for their hospitality. Conversations during the “Quantitative Linear Algebra” program at the Instituteof Pure and Applied Mathematics at the University of California, Los Angeles were also insightful. I thankIPAM for its hospitality. I would like to thank Roy Araiza, Benoit Collins, Yoann Dabrowski, David Jekel,and Thomas Sinclair for inspirational conversations related to this work.2.
Background
General convention and notation.
For k ∈ N , we let M k ( C ) be the space of k × k matrices over C , and M k ( C ) s.a. be the space of k × k self-adjoint matrices over C . We also use U ( k ) for the unitaries in M k ( C ) . We define tr : M k ( C ) → C by tr( A ) = 1 k k X j =1 A jj . We define a Hilbert space inner product on M k ( C ) by h A, B i = tr( B ∗ A ) , and we let k · k be the norminduced by this inner product. We use S ( n, tr) for M n ( C ) equipped with this Hilbertian structure. For anindex set J, a finite F ⊆ J, and A ∈ M k ( C ) J we set k A k ,F = X j ∈ F k A j k / . If J itself is finite, and F = J we will often use k · k instead of k · k J . The pair ( M k ( C ) , tr) is an importantexample of a more general concept. Definition 2.1. A tracial von Neumann algebra is a pair ( M, τ ) where M is a von Neumann algebra, and τ : M → C is a faithful, normal, tracial state.For a von Neumann algebra M we use M ∗ for the normal linear functionals M → C . We call M ∗ the predual of M , it is a Banach space under the operator norm. For von Neumann algebras, we will adoptsimilar conventions as in the case of matrices. For example, U ( M ) , M s.a. will refer to the unitaries andself-adjoints in M. For reasons that will become clear shortly, for x ∈ M we use k x k ∞ for the operator normof x. Given a von Neumann algebra M we shall use N ≤ M to mean that N is a unital von Neumannsubalgebra of M. If M is a von Neumann algebra, J an index set and x = ( x j ) j ∈ J ∈ M J , and y ∈ M, wewill use yx, xy for ( yx j ) j ∈ J , ( x j y ) j ∈ J ∈ M J , respectively. Similarly, if M , M are von Neumann algebrasand π : M → M is a ∗ -homomorphism, then for an index set J and x = ( x j ) j ∈ J ∈ M J we will use π ( x ) for( π ( x j )) j ∈ j ∈ M J . While a von Neumann algebra is assumed to come with an ambient embedding into bounded operatorson a Hilbert space, one significant advantage of a tracial von Neumann algebra is that there is a naturalrepresentation of the algebra we can build from the trace. Given a tracial von Neumann algebra (
M, τ ) , wedefine an inner product on M by h a, b i = τ ( b ∗ a ) . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 7
We use k · k for the norm induced by this inner product, and we let L ( M, τ ) be the Hilbert space which isthe completion under this inner product. It is direct to show (see [1, Section 7.1.1]) that for all x, y ∈ M k xy k ≤ k x k ∞ k y k , k xy k ≤ k y k ∞ k x k . Thus the operators y xy, y yx extend continuously to bounded operators on L ( M, τ ) whose operatornorms are the same as the operator norm of x as an element of B ( H ) . For ξ ∈ L ( M, τ ) , we use xξ, ξx forthe image of ξ under these operators. Given a tracial von Neumann algebra ( M, τ ) the above allows us toview it as a von Neumann algebra of operators on L ( M, τ ) by left multiplication. We will essentially alwaysview a tracial von Neumann algebra in this manner and ignore whatever other Hilbert space it arises from.We will need to use tensor products at various points in the paper. For vector spaces
V, W we use V ⊗ alg W for their algebraic (i.e. not completed) tensor product . If H , H are Hilbert spaces, we let H ⊗ H denotetheir Hilbert space tensor product. Given T ∈ B ( H ), S ∈ B ( H ), we let T ⊗ S be the unique operator in B ( H ⊗ H ) given by ( T ⊗ S )( ξ ⊗ η ) = T ξ ⊗ Sη for ξ ∈ H , η ∈ H .For von Neumann algebras M j ⊆ B ( H j ) , j = 1 , M ⊗ M = span { T ⊗ S : T ∈ M , S ∈ M } SOT . At various important points in the paper, we will need to use approximation properties in terms of completely bounded/completely positive maps . These are the appropriate morphisms for what are now calledoperator spaces/operator systems.
Definition 2.2. A (concrete) operator space is a closed, linear subspace of B ( H ) for some Hilbert space H . A (concrete) operator system is a closed, linear subspace of B ( H ) which is closed under adjoints andcontains the identity operator.If E ⊆ B ( H ) is an operator space, then we may view M n ( E ) ⊆ B ( H ⊕ n ) in a natural way. So if we aregiven A ∈ M n ( E ) then the embedding M n ( E ) ⊆ B ( H ⊕ n ) allows us to make sense of k A k M n ( E ) . Properlyspeaking, an operator space is really a Banach space E together with the data of these norms on M n ( E )and one can give an axiomatic description for such norms to arise from an embedding into B ( H ) (see [21,Theorem 2.3.5]). We will stick to concrete operator spaces (i.e. given as a subspace of B ( H )) for thepurposes of this paper. Given operator spaces E, F and a bounded, linear map T : E → F we define for n ∈ N , T ⊗ id M n ( C ) : M n ( E ) → M n ( F ) by [( T ⊗ id M n ( C ) )( A )] ij = T ( A ij ) for A ∈ M n ( E ) . We say that T is completely bounded if sup n k T ⊗ id M n ( C ) k < ∞ , the norm in question being the operator norm. If T is completely bounded, we set k T k cb = sup n k T ⊗ id M n ( C ) k . We say that T is completely contractive if k T k cb ≤ . We let CB ( E, F ) be the completely bounded maps E → F and will often use CB ( E ) instead of CB ( E, E ) . If E , E are operator spaces and E j ⊆ B ( H j ) , j = BEN HAYES , , then we let E ⊗ min E be the operator space given byspan { A ⊗ B : A ∈ E , B ∈ E } k·k ∞ ⊆ B ( H ⊗ H ) . If E j , F j , j = 1 , T j : E j → F j , j = 1 , T ⊗ T : E ⊗ alg E → F ⊗ alg F extends continuously to a completely bounded map E ⊗ min E → F ⊗ min F which we still denote T ⊗ T . We also have k T ⊗ T k cb = k T k cb k T k cb . This is decidedly not true if we consider bounded maps instead of completely bounded maps, and indeedarguably the main motivation for completely bounded maps and operator spaces is to provide a context inwhich one can extend bounded maps to tensor products.For operator systems there is a natural order structure at play. Suppose E ⊆ B ( H ) is an operatorsystem, and consider the embeddings M n ( E ) ⊆ B ( H ⊕ n ) . We can then define the positive elements in M n ( E ) to be those which are positive as operators on B ( H ⊕ n ) . Since E is an operator system, it has anabundance of positive elements, e.g. every element of E is a linear combination of 4 positive elements.Given operator systems E, F a map T : E → F is positive if T ( x ) ≥ x ∈ E with x ≥ . It is completely positive if T ⊗ M n ( C ) is positive for all n. We say T is unital if T (1) = 1 . We use CP ( E, F )and
U CP ( E, F ) for the completely positive and unital, completely positive maps E → F respectively. It isa fact that for T ∈ CP ( E, F ) we have k T k cb = k T (1) k (see [21, Lemma 5.1.1]). As in the operator spacecase, if E j , F j , j = 1 , T j : E j → F j , j = 1 , T ⊗ T : E ⊗ min E → F ⊗ min F . As in the operator space case, the analogous statement is false for positive maps .Since any C ∗ -algebra can be embedded in bounded operators on a Hilbert space, we may view any closedsubspace of a C ∗ -algebra as an operator space. Similarly, we may view any closed subspace which is closedunder adjoints and contains the unit as an operator system.If M j , j = 1 , , N j , j = 1 , T j : M j → N j , j = 1 , T ⊗ T has a unique, normal extension to a map M ⊗ M → N ⊗ N which we stilldenote T ⊗ T . Moreover, k T ⊗ T k CB ( M ⊗ M ,N ⊗ N ) = k T k cb k T k cb . Further, if each T j is completely positive, then so is T ⊗ T . Laws, Microstates, and -Bounded Entropy. Recall that a ∗ -algebra is an algebra A over C together with a conjugate linear involution A A, a a ∗ which is anti-multiplicative: ( ab ) ∗ = b ∗ a ∗ . Givenan index set J, we let C ∗ h ( T j ) j ∈ J i be the ∗ -algebra of noncommutative ∗ -polynomials in the abstract variables( T j ) j ∈ J . We may think of C ∗ h ( T j ) j ∈ J i as the (algebraically) free ∗ -algebra indexed by J. If J = { , · · · , n } , we typically use C ∗ h T , · · · , T n i for C ∗ h ( T j ) nj =1 i . If we are given a ∗ -algebra A, and a tuple x ∈ A J , thenby algebraic freeness there is a unique ∗ -homomorphism ev x : C ∗ h ( T j ) j ∈ J i → A such that ev x ( T j ) = x j . For P ∈ C ∗ h ( T j ) j ∈ J i , we denote ev x ( P ) by P (( x j ) j ∈ J ) . Again, if J = { , · · · , n } , we usually use P ( T , · · · , T n ) . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 9
Definition 2.3.
Let J be an index set. A linear functional ℓ : C ∗ h ( T j ) j ∈ J i → C is called a tracial law ifthere is a R : J → [0 , ∞ ) so that • ℓ ( P ∗ P ) ≥ P ∈ C ∗ h ( T j ) j ∈ J i , • ℓ (1) = 1 , • ℓ ( P Q ) = ℓ ( QP ) for all P, Q ∈ C ∗ h ( T j ) j ∈ J i . • for all n ∈ N , all j , j , · · · , j n ∈ J and all σ , · · · , σ n ∈ { , ∗} , | ℓ ( T σ j T σ j · · · T σ n j n ) | ≤ R j R j R j · · · R j n . We let Σ J be the space of tracial laws indexed by J. If J = { , · · · , n } , we typically use Σ n instead of Σ J . Given a function R : J → [0 , ∞ ) , we let Σ R,J be the set of all laws ℓ satisfying the fourth item above for thisspecific R. If R ∈ [0 , ∞ ) we will frequently use Σ R,J for Σ b R,J where b R : J → [0 , ∞ ) is the function which isconstantly R. As above, if J = { , · · · , n } we will frequently use Σ R,n (in both the case that R is a functionand the case that it is a constant).The above may be regarded as an abstract definition of a law. If we are concretely given a tracial vonNeumann algebra ( M, τ ) and a tuple x ∈ M J for some indexing set J, we define the law of x to be the linearfunctional ℓ x : C ∗ h ( T j ) j ∈ J i → C given by ℓ x ( P ) = τ ( P (( x j ) j ∈ J )) . We always equip M k ( C ) with its unique tracial state tr given bytr( A ) = 1 k k X j =1 A jj . So if A ∈ M k ( C ) J , we have a notion of its law ℓ A . In fact, every abstract law arises as a concrete law for some tuple in a tracial von Neumann algebra. Thisfollows from the GNS (Gelfand-Naimark-Segal) construction, which we sketch here. Let J be an index setand ℓ ∈ Σ J . Define a semi-inner product on C ∗ h ( T j ) j ∈ J i by h P, Q i = ℓ ( Q ∗ P ) . For P ∈ C ∗ h ( T j ) j ∈ J i we set k P k L ( ℓ ) = ℓ ( P ∗ P ) / , and we define W = { P ∈ C ∗ h ( T j ) j ∈ J i : k P k L ( ℓ ) = 0 } , and V = C ∗ h ( T j ) j ∈ J i /W. The semi-inner product C ∗ h ( T j ) j ∈ J i descends to a genuine inner product on V, and we let L ( ℓ ) be theHilbert space which is the completion of V under the norm coming from this inner product. From the fourthbullet point in Definition 2.3, one can deduce that there is a C : C ∗ h ( T j ) j ∈ J i → [0 , ∞ ] so that k P Q k L ( ℓ ) ≤ C ( P ) k Q k L ( ℓ ) for all P, Q ∈ C ∗ h ( T j ) j ∈ J i . So we may proceed as in the tracial von Neumann algebra case to deduce thatthere is a well-defined ∗ -homomorphism π ℓ : C ∗ h ( T j ) j ∈ J i → B ( L ( ℓ )) satisfying(3) π ℓ ( P )( Q + W ) = P Q + W for all P, Q ∈ C ∗ h ( T j ) j ∈ J i . Set(4) W ∗ ( ℓ ) = π ℓ ( C ∗ h ( T j ) j ∈ J i ) SOT , and let x = ( π ℓ ( T j )) j ∈ J ∈ M J . We then have a faithful, normal, tracial state τ ℓ : M → C given by(5) τ ℓ ( a ) = h a (1 + W ) , W i , for a ∈ M and by construction the law of x with respect to τ ℓ is ℓ. It is an exercise using the spectral theorem to showthat(6) k π l ( P ) k ∞ = sup k ℓ (( P ∗ P ) k ) / k = lim k →∞ ℓ (( P ∗ P ) k ) / k . for all P ∈ C ∗ h ( T j ) j ∈ J i . So if R ∈ [0 , ∞ ) J and ℓ ∈ Σ R,J , then k π ℓ ( x j ) k ∞ ≤ R j for all j ∈ J. Laws may be viewed as a natural noncommutative extension of probability measures. If (
M, τ ) is a tracialvon Neumann algebra and x ∈ M is normal , we let µ x ∈ Prob( C ) be the spectral measure of x defined by µ x ( E ) = τ (1 E ( x )) for all Borel E ⊆ C. Then, by definition, for all P ∈ C ∗ h T i we have(7) τ ( P ( x )) = Z P ( z ) dµ x ( z ) . Here we are using P ( z ) for the image of T under the unique ∗ -homomorphism C ∗ h T i → C given by T z. Of course, C ∗ h T i is noncommutative, whereas z P ( z ) is given by a (different) commutative polynomial in z and z. Since µ x is compactly supported, the Stone-Weierstrass theorem tells us that equation (7) uniquely determines µ x . This equation may be read as ℓ x ( P ) = Z P ( z ) dµ x ( z )and so we see that the law of x encodes the same information as the spectral measure of x. Fix a set J. Since Σ J is a subset of the algebraic dual of C ∗ h ( T j ) j ∈ J i it can be naturally endowed withthe weak ∗ -topology. So a basic neighborhood of ℓ ∈ Σ J is given by U F,ε ( l ) = \ Q ∈ F { φ ∈ Σ J : | φ ( Q ) − ℓ ( Q ) | < ε } for a finite F ⊆ C ∗ h ( T j ) j ∈ J i and an ε > . We leave it as an exercise to verify that for every R ∈ [0 , ∞ ) J wehave that Σ R,J is compact in the weak ∗ -topology.We recall the notion of freely independent random variables , which forms the basis for Voiculescu’s freeprobability. Let ( M, τ ) be a tracial von Neumann algebra, and let ( A j ) j ∈ J be ∗ -subalgebras of M. We saythat ( A j ) j ∈ J are freely independent (or free ) if for all n ∈ N , and all j ∈ J n with j = j , j = j , j = j , · · · , j n − = j n , and for all a ∈ M n with a i ∈ A j i and τ ( a i ) = 0 we have τ ( a a · · · a n ) = 0 . Say that ( x j ) j ∈ J ∈ M J are freely independent (or free ) if the ∗ -algebras they generate are free as a J -tuple.This necessarily forces ( W ∗ ( x j )) j ∈ J to be free. Given any collection ( M j , τ j ) j ∈ J , one may find (see [73,Chapter 1]) another tracial von Neumann algebra ( M, τ ) for which there are trace-preserving embeddings M j ֒ → M so that if we identify M j with its image under this embedding, then M = W ∗ (cid:16)S j M j (cid:17) , and( M j ) j ∈ J are free. If ( f M , e τ ) is another such algebra, then there is a unique trace-preserving isomorphism RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 11 ( M, τ ) ∼ = ( f M , e τ ) which respects the embeddings of ( M j , τ j ) into ( M, τ ) , ( f M , e τ ) for each j ∈ J. So we maydefine the free product of ( M j , τ j ), denoted ∗ j ∈ J ( M j , τ j ) , to be any such algebra. Given index sets ( J i ) i ∈ I , and ℓ i ∈ Σ J i define ℓ = ∗ i ∈ I ℓ i ∈ Σ ⊔ i J i to be the law of x = ( π ℓ i ( T j )) j ∈ J i ,i ∈ I in ∗ i ∈ I ( W ∗ ( ℓ i ) , τ ℓ i ) . By itsvery nature, freeness of noncommutative variables depends only upon their joint law. So we will often omitreference to the underlying von Neumann algebra. For example, we will often say “suppose x = ( x , · · · , x r )is a free tuple”. Provided we specify ℓ x j for all j, this unambiguously gives ℓ x . Since many of our results onlyrequire knowledge of the law of x, this will suffice for our purposes. One case of utmost importance is thefollowing. A tuple s = ( s , · · · , s r ) is a free semicircular family if it is a free family, each s j is self-adjoint,and for each j we have that dµ s j = 12 πσ p σ − ( x − µ ) [ µ − σ,µ +2 σ ] dx for some µ ∈ R , σ ∈ (0 , ∞ ) . Definition 2.4.
Let J be an index set, and fix R : [0 , ∞ ) → J. Given a set O ⊆ Σ R,J with nonempty interior(relative to Σ
R,J ) and an k ∈ N , we define Voiculescu’s space of ( O , k ) microstates to beΓ ( k ) R ( O ) = { A ∈ M k ( C ) J : ℓ A ∈ O } ∩ Y j ∈ J { B ∈ M k ( C ) : k B k ∞ ≤ R j } . The reader may be more familiar with the following case. Let (
M, τ ) be a tracial von Neumann algebra,and let J be an index set. Let x ∈ M J and choose R ∈ [0 , ∞ ) J with k x j k < R j for all j ∈ J. Then itis typical to consider Γ ( k ) R ( O ) for O a weak ∗ -neighborhood of x. Indeed, it is common to denote this byΓ ( k ) R ( x ; O ) even though it does not require x for its definition. If W ∗ ( x ) = M, then we have that M embedsinto an ultrapower of the hyperfinite II -factor if and only if for every neighborhood O of the law of x inΣ R,J there is an integer k ∈ N so that Γ ( k ) R ( O ) = ∅ . Because of this, the spaces Γ ( k ) R ( O ) are often regarded as spaces of “finitary approximations” of x, and theyform the basis for microstates free entropy, microstates free entropy dimension, and the 1-bounded entropyof x. There is a mild, but very important, variant of this which takes into account microstates which have an“extension” to a larger algebra. Suppose that (
M, τ ) is a tracial von Neumann algebra, and that
I, J areindex sets. Let y ∈ M I , and fix R ∈ [0 , ∞ ) I ⊔ J with k y i k ∞ ≤ R i for all i ∈ I. Let O ⊆ Σ R,I ⊔ J and assumethat { ℓ (cid:12)(cid:12) C ∗ h ( T i ) i ∈ I i : ℓ ∈ O } is a neighborhood of ℓ y . For an integer k ∈ N we define Voiculescu’s microstates space for y in the presenceof O , denoted Γ ( k ) R ( y : O ) , byΓ ( k ) R ( y : O ) = { A ∈ M k ( C ) I : there exists a B ∈ M k ( C ) J with ℓ A,B ∈ O } . Typically, one takes y ∈ M k ( C ) J and O to be a neighborhood of ℓ y,x . In this case, one thinks of Γ ( k ) R ( y : O )as “microstates for y which have an extension to microstates for ( y, x )”.Given a set Ω , a pseudometric on Ω is a function d : Ω × Ω → [0 , ∞ ) satisfying • d ( x, y ) = d ( y, x ) for all x, y ∈ Ω, • d ( x, z ) ≤ d ( x, y ) + d ( y, z ) for all x, y, z ∈ Ω . Given a pseudometric d on Ω , an r > , and an x ∈ Ω , we let B r ( x, d ) = { y ∈ Ω : d ( x, y ) < r } . For E ⊆ X and r > , we let N r ( E, d ) = [ x ∈ E B r ( x, d ) . We call N r ( E, d ) the r -neighborhood of E. For ε > E ⊆ Ω , we let K ε ( E, d ) be the minimal cardinalityof a set F ⊆ E which has N ε ( F, d ) ⊇ E. If there is no such finite set F, then by convention N ε ( F, d ) = ∞ . If F ⊆ C , then N ε ( F ) will refer to the ε -neighborhood of F with respect to the Euclidean distance on C andwe will not make reference to the fact that we are using the Euclidean distance.The most important pseudometric for our purposes is the following. Given an index set J, a finite set F ⊆ J, and a natural number k, we define a pseudometric d orb F on M k ( C ) J by d orb F ( A, B ) = inf U ∈ U ( k ) k A − U BU ∗ k ,F . As in the case of k · k ,F if J itself is finite we will usually use d orb instead of d orb J . Definition 2.5.
Let (
M, τ ) be a tracial von Neumann algebra, and y ∈ M I , x ∈ M J . Fix R ∈ [0 , ∞ ) I ∪ J with k x j k ∞ ≤ R j for all j ∈ J, and k y i k ∞ ≤ R i for all i ∈ I. For a weak ∗ -neighborhood O of ℓ y,x and a finite F ⊆ I, we set K orb ε,F ( y : O , k · k ) = lim sup k →∞ k log K ε (Γ ( k ) R ( y : O ) , d orb F ) . We then define K orb ε,F ( y : x ) = inf O K orb ε,F ( y : O , k · k ) ,h ( y : x ) = sup ε,F K orb ε,F ( y : x ) , where the infimum is over all weak ∗ -neighborhoods O of ℓ y,x and the supremum is over all ε > F of J. We call h ( y : x ) the y in the presence of x .It follows from [32, Theorem A.9] that if J ′ , I ′ are other index sets, and if y ′ ∈ M J ′ , x ′ ∈ M I ′ , and W ∗ ( y ) = W ∗ ( y ′ ) , W ∗ ( x, y ) = W ∗ ( x ′ , y ′ ) , then h ( y : x ) = h ( y ′ : x ′ ) . Suppose N ≤ M, and that N is diffuse. If y ∈ N J , x ∈ M I with W ∗ ( y ) = N, W ∗ ( x, y ) = M, we may definethe N in the presence of M by h ( N : M ) = h ( y : x ) . We think of h ( N : M ) as some precise measurement of the “size of the space of microstates for N which havean extension to M ”. Note that since we allow arbitrary index sets, the quantity h ( N : M ) is always defined(provided N is diffuse), though it may be −∞ . We set h ( M ) = h ( M : M ), and call h ( M ) the 1 -boundedentropy of M . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 13
We now turn to permanence properties the 1-bounded entropy enjoys. We say that a von Neumannalgebra M is hyperfinite if there is an increasing net ( M α ) α of finite-dimensional von Neumann subalgebrasof M with M = [ α M αW OT . By a celebrated result of Connes’ [17], this is equivalent to several other properties of M such as beingamenable ([1, Chapter 10]). Because of Connes’ famous and deep work we will use hyperfinite and amenableinterchangeably. We use R for the (unique modulo isomorphism) hyperfinite II -factor. For a tracial vonNeumann algebra ( M, τ ) and x ∈ M J for some set J, we let δ ( x ) be the microstates free entropy dimensionof x. We will not need the precise definition, and refer the reader to [70, Definition 6.1] for the details. Weassume that all our von Neumann algebras are diffuse for all the properties listed below.
P1: h ( N : M ) ≥ N ≤ M and every von Neumann subalgebra of M with separable predual embedsinto an ultrapower of R , and h ( N : M ) = −∞ if there exists a von Neumann subalgebra of M withseparable predual which does not embed into an ultrapower of R . (Exercise from the definitions.) P2: h ( N : M ) ≤ h ( N : M ) if N ≤ N ≤ M ≤ M . (Exercise from the definitions.) P3: h ( N : M ) = 0 if N ≤ M and N is diffuse and hyperfinite. (Exercise from the definitions). P4:
For M diffuse, h ( M ) < ∞ if and only if M is strongly 1-bounded in the sense of Jung. (See [32,Proposition A.16]). P5: h ( M ) = ∞ if M = W ∗ ( x , · · · , x n ) where x j ∈ M sa for all 1 ≤ j ≤ n and δ ( x , · · · , x n ) >
1. Forexample, this applies if M = L ( F n ) , for n >
1. (This follows from Property 4 and [44, Corollary3.5]).
P6: h ( N ∨ N : M ) ≤ h ( N : M ) + h ( N : M ) if N , N ≤ M and N ∩ N is diffuse. (See [32, LemmaA.12] ). P7:
Suppose that ( N α ) α is an increasing chain of diffuse von Neumann subalgebras of a von Neumannalgebra M . Then h _ α N α : M ! = sup α h ( N α : M ) . (See [32, Lemma A.10]). P8: h ( N : M ) = h ( N : M ω ) if N ≤ M is diffuse, and ω is a free ultrafilter on an infinite set. (See [32,Proposition 4.5]). P9: h (W ∗ ( N M ( N )) : M ) = h ( N : M ) if N ≤ M is diffuse. Here N M ( N ) = { u ∈ U ( M ) : uN u ∗ = N } . (This is a special case of [32, Theorem 3.8]).These properties are sufficient by themselves to deduce the landmark results of Voiculescu [71], Ge [26] thatfree group factors do not have Cartan subalgebras, and are prime, as well as the fact that a von Neumannalgebra generated by a family with free entropy dimension bigger than 1 does not have Property Gamma.We refer the reader to [33, Section 1.2] for a more detailed discussion on this.The 1-bounded entropy allows us to single out a particularly nice set of von Neumann subalgebras of afixed tracial von Neumann algebra. Definition 2.6.
Suppose (
M, τ ) is a tracial von Neumann algebra, and that P ≤ M. We say that P is a Pinsker algebra in M if h ( P : M ) ≤ P ≤ Q ≤ M with Q = P we have h ( Q : M ) > . Recall that if M is a von Neumann algebra, then Q ≤ M is maximal amenable if Q is amenable and forevery N ≤ M with Q ⊆ N and N amenable, we have N = Q. It follows from Property P6 that if Q ≤ M is diffuse, and h ( Q : M ) = 0 , then there is a unique Pinsker P ≤ M with Q ≤ P. If P ≤ M is Pinsker and amenable, it is necessarily maximal amenable by Property P3. Moreover, by P6, P3 it has the absorbingamenability property . Namely, if Q ≤ M is amenable and Q ∩ P is diffuse, then Q ≤ P. It also has thefollowing
Gamma stability property (in the sense of [36]): if Q ≤ M is such that Q ′ ∩ M ω and Q ∩ P arediffuse, then Q ≤ P. Of relevance to the Peterson-Thom conjecture is the following.
Proposition 2.7.
Let ( M, τ ) be a tracial von Neumann algebra. Suppose that every Pinsker algebra in M is amenable. Then given any diffuse, amenable Q ≤ M there is a unique, maximal amenable P ≤ M with Q ⊆ P. Proof.
Suppose that Q ≤ M is diffuse and amenable. Let P ≤ M be the unique Pinsker algebra in M with Q ⊆ P. By assumption, P is amenable. Since P is Pinsker, it is necessarily maximal amenable. Suppose b P ≤ M is another maximal amenable subalgebra of M with Q ⊆ b P .
Then b P ∩ P ⊇ Q, and since Q is diffuse this forces b P ∩ P to be diffuse. So h ( P ∨ b P : M ) ≤
0, by Property P6. Since P isPinsker, b P ∨ P ≤ P , which forces b P ⊆ P. Since b P is maximal amenable, we have b P = P and this completesthe proof. (cid:3) The weak ∗ -topology on laws is what allows us to define Voiculescu’s microstates, and by extension the1-bounded entropy. We will also need another topology on the space of laws. Fix an index set J. Recall thedefinition of π ℓ for ℓ ∈ Σ J discussed after Definition 2 . . For P ∈ C ∗ h ( T j ) j ∈ J i , ℓ ∈ Σ J , set k P k L ∞ ( ℓ ) = k π ℓ ( P ) k ∞ . It is not true that the map Σ J × C ∗ h ( T j ) j ∈ J i → [0 , ∞ ]given by ( ℓ, P )
7→ k P k L ∞ ( ℓ ) is continuous in the first variable. However, from (6) we have the followingsemi-continuity: if ℓ α is a net in Σ J and ℓ α → ℓ weak ∗ , then for all P ∈ C ∗ h ( T j ) j ∈ J ik P k L ∞ ( ℓ ) ≤ lim inf α k P k L ∞ ( ℓ α ) . This motivates the definition of a different topology on Σ J . Definition 2.8.
Let J be an index set. The strong topology on Σ J is the coarsest topology finer thanthe weak ∗ -topology which makes the map Σ J → [0 , ∞ ) given by P
7→ k P k L ∞ ( ℓ ) continuous for each P ∈ C ∗ h ( T j ) j ∈ J i . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 15
Given ℓ ∈ Σ J , a neighborhood basis at ℓ in the strong topology may be given by O V,F,ε ( ℓ ) = V ∩ \ P ∈ F { φ ∈ Σ J : |k P k L ∞ ( ℓ ) − k P k L ∞ ( φ ) | < ε } ranging over weak ∗ -neighborhoods V of ℓ, finite sets F ⊆ C ∗ h ( T j ) j ∈ J i , and ε ∈ (0 , ∞ ) . In fact, by sem-incontinuity, the sets V V,F,ε ( ℓ ) = V ∩ \ P ∈ F { φ ∈ Σ J : k P k L ∞ ( φ ) < k P k L ∞ ( ℓ ) + ε } ranging over weak ∗ -neighborhoods V of ℓ, finite sets F ⊆ C ∗ h ( T j ) j ∈ J i , and ε ∈ (0 , ∞ ) form a neighborhoodbasis of ℓ ∈ Σ J in the strong topology.Suppose we are given a sequence ( M k , τ k ) k of tracial von Neumann algebras, x k ∈ M Jk , another vonNeumann algebra ( M, τ ) and x ∈ M J . We will then say that the distribution of x k converges strongly to thedistribution of x if ℓ x k → ℓ x in the strong topology. Concretely, this is just the conjunction of the followingtwo properties: • τ k ( P ( x k )) → τ ( P ( x )) for all P ∈ C ∗ h ( T j ) j ∈ J i , and • k P ( x k ) k ∞ → k P ( x ) k ∞ for all P ∈ C ∗ h ( T j ) j ∈ J i . So our notion of strong convergence agrees with that already discussed in [49, 16]. To further illustrate themeaning of strong convergence, we close with the following Lemma relating strong convergence to Hausdorffconvergence of spectra. This is a well-known result, and we mainly prove it to give the reader some insightand practice as to what strong convergence is and why it is important. Recall that the Hausdorff metric onnonempty, compact subsets of C is given by d Haus ( E, F ) = inf { r > E ⊆ N r ( F ) and F ⊆ N r ( E ) } . Lemma 2.9.
Fix an index set J. Let ( M k , τ k ) , k ∈ N be a sequence of tracial von Neumann algebras and x k ∈ M Jk . Let ( M, τ ) be a tracial von Neumann algebra and x ∈ M J . Assume that sup k k x k,j k ∞ < ∞ for all j ∈ J. Suppose that ℓ x k → ℓ x weak ∗ . Then ℓ x k → ℓ x strongly if and only if for every P ∈ C ∗ h ( T j ) j ∈ J i with P = P ∗ we have that σ ( P ( x k )) → σ ( P ( x )) in the Hausdorff metric.Proof. First, suppose that for every self-adjoint P ∈ C ∗ h ( T j ) j ∈ J i we have that σ ( P ( x k )) → σ ( P ( x )) inthe Hausdorff metric. Now fix Q ∈ C ∗ h ( T j ) j ∈ J i , and let ε > . Then for all sufficiently large k, we havethat σ (( Q ∗ Q )( x k )) ⊆ N ε ( σ (( Q ∗ Q )( x ))) . Since the norm of a self-adjoint element is given by its spectralradius, it follows that k Q ( x k ) k ∞ = k ( Q ∗ Q )( x k ) k ∞ ≤ ε + k ( Q ∗ Q )( x ) k ∞ = ε + k Q ( x ) k ∞ for large k. Since ε > k →∞ k Q ( x k ) k ∞ ≤ k Q ( x ) k ∞ . The fact that k Q ( x ) k ∞ ≤ lim inf k →∞ k Q ( x k ) k ∞ is already a consequence of weak ∗ convergence of ℓ x k to ℓ x . For the reverse direction, assume that ℓ x k → ℓ x strongly. First choose a M > k P ( x k ) k ∞ ≤ M for all k. This is possible as sup k k x k,j k ∞ < ∞ for all j ∈ J. Note that we have k f ( P ( x k )) k ∞ → k f ( P ( x )) k ∞ for all f ∈ C ([ − M, M ]) . Indeed, the set of f ∈ C ([ − M, M ]) for which k f ( P ( x k )) k ∞ → k →∞ k f ( P ( x )) k ∞ can be directly shown to be a closed subset of C ([ − M, M ]) , and by our assumption of strong convergence itcontains all polynomials. So k f ( P ( x k )) k ∞ → k f ( P ( x )) k ∞ for all f ∈ C ([ − M, M ]) by the Stone-Weierstrasstheorem. Let ε > , and apply Urysohn’s Lemma to find a continuous function φ ∈ C ([ − M, M ]) which is σ ( P ( x )) and is 1 on N ε ( σ ( P ( x )) c ∩ [ − M, M ] . Then k φ ( P ( x k )) k ∞ → k →∞ k φ ( P ( x )) k ∞ = 0 , and so forall large k we have k φ ( P ( x k )) k ∞ < . By the spectral mapping theorem, σ ( φ ( P ( x k ))) = φ ( σ ( P ( x k ))) . Since φ = 1 on N ε ( σ ( P ( x ))) c ∩ [ − M, M ] , and k φ ( P ( x k )) k is the supremum of | φ | over σ ( P ( x k )) , it follows that σ ( P ( x k )) ⊆ N ε ( σ ( P ( x ))) for all large k. So it just remains to show that σ ( P ( x )) ⊆ N ε ( σ ( P ( x k ))) for all large k. For every t ∈ σ ( P ( x )) , choosea ψ t ∈ C ([ − M, M ]) with ψ t ( t ) = 1 and ψ t (cid:12)(cid:12) B ε/ ( t ) c ∩ [ − M,M ] = 0 . As above, there is a K t ∈ N so that for all k ≥ K t we have k ψ t ( P ( x k )) k ∞ ≥ / . As in the above paragraph, this implies that B ε/ ( t ) ∩ σ ( P ( x k )) = ∅ and so t ∈ N ε/ ( σ ( P ( x k ))) for all k ≥ K t . Since σ ( P ( x )) is compact, we can choose t , · · · , t n ∈ σ ( P ( x )) sothat σ ( P ( x )) ⊆ N ε/ ( { t , · · · , t n } ) . Set K = max( K t , · · · , K t n ) . Then for all k ≥ K,σ ( P ( x )) ⊆ N ε/ ( { t , · · · , t n } ) ⊆ N ε ( σ ( P ( x k ))) . (cid:3) Note that the weak ∗ -convergence of ℓ x k → ℓ x implies the weak ∗ -convergence of µ P ( x k ) to µ P ( x ) for allself-adjoint P ∈ C ∗ h ( T j ) j ∈ J i . The above lemma than asserts that strong convergence of ℓ x k → ℓ x means thatthere one does not have any “outliers” in the spectrum of P ( x k ) . In random matrices, this is often calledhaving a “hard edge.”We phrased Lemma 2.9 in terms of strong convergence of laws of specific elements because it is more naturalfor the reader who might have some experience with random matrices. For example, if A ( k ) ∈ M n ( k ) ( C ) J and if there is a tracial von Neumann algebra ( M, τ ) and a tuple x ∈ M J with ℓ A ( k ) → ℓ x weak ∗ , then strongconvergence of A ( k ) to x just asserts that for any self-adjoint P ∈ C ∗ h ( T j ) j ∈ J i the spectral distribution of P ( A ( k ) ) converges weak ∗ to the spectral measure of P ( x ) , and the spectrum of P ( A ( k ) ) converges to thespectrum of P ( x ) in the Hausdorff sense. However, one can phrase Lemma 2.9 without referring to anyambient tracial von Neumann algebra. Suppose we have a sequence ℓ n ∈ Σ R,J for some R ∈ [0 , ∞ ) J with ℓ n → ℓ weak ∗ . Lemma 2.9 is then equivalent to saying that ℓ n → ℓ strongly if and only if for all self-adjoint P ∈ C ∗ h ( T j ) j ∈ J i we have σ ( π ℓ k ( P )) → σ ( π ℓ ( P )) in the Hausdorff sense.2.3. Measures on microstates and concentration thereof.Definition 2.10.
Let (
M, τ ) be a tracial von Neumann algebra, I an index set, and x ∈ M J . Supposewe have a sequence n ( k ) ∈ N with n ( k ) → ∞ , and µ ( k ) ∈ Prob( M n ( k ) ( C ) I ) . Then we say that µ ( k ) is asymptotically supported on microstates for x if there exists an R ∈ [0 , ∞ ) I with • k x i k ∞ ≤ R i for all i ∈ I, • we have µ ( k ) (Γ ( n ( k )) R ( O )) → k →∞ ∗ -neighborhood O of ℓ x . Suppose J is another index set, y ∈ M J , and ν ( k ) ∈ Prob( M k ( C ) J ) . Then we say that ν ( k ) is asymptoticallysupported on microstates for y in the presence of x if there is an R ∈ [0 , ∞ ) J ⊔ I so that • k x i k ∞ ≤ R i for all i ∈ I, • k y j k ∞ ≤ R j for all j ∈ J, • ν ( k ) (Γ ( n ( k )) R ( y : O )) → ∗ -neighborhood O of ℓ y,x . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 17
Recall that a
Polish space is a topological space X which is separable and completely metrizable. Such aspace is naturally equipped with its Borel σ -algebra, which is the σ -algebra generated by the open subsetsof X. For a Polish space X, we let Prob( X ) be the space of Borel probability measures on X. Definition 2.11. A pseudometric measure space is a triple ( X, µ, d ) where • X is a Polish space, • µ ∈ Prob( X ) , • d is a continuous pseudometric on X (giving X × X the product topology).Given a pseudometric measure space ( X, µ, d ) we define its concentration function α µ,d : (0 , ∞ ) → [0 ,
1] by α µ,d ( ε ) = inf { µ ( N ε ( E, d ) c ) : E ⊆ X is Borel and µ ( E ) ≥ / } . An alternative (and typically more useful) way to view the concentration function is as follows. If E ⊆ X is Borel, and µ ( E ) ≥ / , then µ ( N ε ( E, d )) ≥ − α µ,d ( ε ) . Typically one is interested in sequences of pseudometric measure spaces ( X k , µ k , d k ) so that α µ k ,d k ( ε ) decaysrapidly for each fixed ε > . Definition 2.12.
Let ( X k , µ k , d k ) be a sequence of pseudometric measures spaces, µ ( k ) ∈ Prob( X k ) , and( r k ) k ∈ (0 , ∞ ) N with r k → ∞ . We say that ( X k , µ ( k ) , d k ) exhibits exponential concentration at scale r k if forevery ε > , lim sup k →∞ r k log α µ ( k ) ,d k ( ε ) < . Suppose that n ( k ) ∈ N , with n ( k ) → ∞ , and that J is a set. Suppose that we have a sequence µ ( k ) ∈ Prob( M k ( C ) J ) . Then we say that µ ( k ) exhibits exponential concentration at scale n ( k ) if for every finite F ⊆ J the sequence ( M n ( k ) ( C ) J , µ ( k ) , d orb F ) exhibits exponential concentration at scale n ( k ) . If it is clear form the context, we will often drop reference to X k , d k and say “ µ k exhibits exponentialconcentration at scale r k .” As we shall show shortly (see Lemma 3.1) exponential concentration implies thatif E k ⊆ X k are Borel and asymptotically have “nontrivial” size, i.e.lim k →∞ µ ( k ) ( E k ) /r k = 1 , then lim k →∞ µ ( k ) ( N ε ( E k , d )) = 1for every ε > . So for every sequence of Borel subsets of X k which are “not exponentially small”, then,no matter how small ε is, expanding E k to its ε -neighborhood makes it “nearly everything.” This is thereason for the name “concentration function”, it gives precise control over the rate at which the measuresmust concentrate near sets that are not “exponentially small”. Despite being a very strong concentrationproperty, there are nevertheless many examples of natural sequences of metric measure spaces which satisfyexponential concentration, and this concept is of frequent use in probability theory and functional analysis. L -Continuous Functional Calculus. We discuss a generalization L -continuous functional calculusof Jekel developed in [41, Section 3], and developed further in [33, Section 2]. That functional calculus wasdefined for self-adjoint noncommutative variables and we will need the version for general variables definedin [42, Section 13.7]. We start by recalling the construction and general properties in the self-adjoint case,and then explain how to give the appropriate definition in generality and derive the corresponding resultsfrom the self-adjoint case.We will need to introduce some notation for the self-adjoint case. Given an index set J, we let C h ( T j ) j ∈ J i be the algebra of noncommutative polynomials ( not ∗ -polynomials) in the abstract variables ( T j ) j ∈ J . Wemay view C h ( T j ) j ∈ J i as the free C -algebra indexed by J. We turn C h ( T j ) j ∈ J i into a ∗ -algebra, by giving itthe unique ∗ -structure which makes T j self-adjoint for all j ∈ J. When viewed as a ∗ -algebra, we may thinkof C h ( T j ) j ∈ J i as the universal ∗ -algebra generated by self-adjoint elements indexed by J. We will also needa space of self-adjoint laws. We adopt similar notational conventions as in the non-self-adjoint case, e.g. if J = { , · · · , n } we will typically use C h T , · · · , T n i instead of C h ( T j ) j ∈ J i . Definition 2.13.
Let J be an index set, and R ∈ [0 , ∞ ) J . Let ev T : C ∗ h ( S j ) j ∈ J i → C h ( T j ) j ∈ J i be theunique ∗ -homomorphism satisfying ev T ( S j ) = T j for all j ∈ J. Let Σ ( s ) J be the set of all linear functionals ℓ : C h ( T j ) j ∈ J i → C so that ℓ ◦ ev T ∈ Σ J . We letΣ ( s ) R,J = { ℓ ∈ Σ ( s ) J : ℓ ◦ ev T ∈ Σ R,J } . Concretely, a linear functional ℓ : C h ( T j ) j ∈ J i → C is in Σ ( s ) J if and only if it satisfies the following axioms: • ℓ ( P Q ) = ℓ ( QP ) for all Q, P ∈ C h ( T j ) j ∈ J i , • ℓ ( P ∗ P ) ≥ P ∈ C h ( T j ) j ∈ J i , • ℓ (1) = 1 , • there is a R ∈ [0 , ∞ ) J so that for all n ∈ N , and all j , j , · · · , j n ∈ J , | ℓ ( T j T j · · · T j n ) | ≤ R j R j · · · R j n . Moreover, given ℓ ∈ Σ ( s ) J and R ∈ [0 , ∞ ) J , we have that ℓ ∈ Σ ( s ) R,J if and only if it satisfies the fourth bulletpoint for this R. Definition 2.14.
Fix an index set J and R ∈ (0 , + ∞ ) J . Consider the space A ( s ) R,J = C (Σ ( s ) R,J ) ⊗ alg C h ( T j ) j ∈ J i Given a tracial von Neumann algebra (
M, τ ) and x ∈ M Jsa with k x j k ≤ R j , we define the evaluation map tobe the linear map ev x : A ( s ) R,J → M satisfyingev x ( φ ⊗ P ) = φ ( ℓ x ) P ( x ) , for φ ∈ C (Σ ( s ) R,J ), P ∈ C h ( T j ) j ∈ J i .We then define a semi-norm on A ( s ) R,J by k f k R, = sup ( M,τ ) ,x k ev x ( f ) k L ( M,τ ) , where the supremum is over all tracial W ∗ -algebras ( M, τ ) and all x ∈ M Isa with k x j k ≤ R j for all j ∈ J .Denote by F ( s ) R,J, the completion of A ( s ) R,J / { f ∈ A R,J : k f k R, = 0 } . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 19
In [33], the superscripts ( s ) are not there, so for example A ( s ) R,J, is denoted by A R,J, etc. We have electedto use the superscript ( s ) here to reference the fact that spaces Σ ( s ) R,J , A ( s ) R,J, , F ( s ) R,J, are noncommutativefunction spaces of self-adjoint variables, in contrast to the function spaces for non-self-adjoint variables wewill discuss imminently.By construction, for every tracial von Neumann algebra ( M, τ ), and for every x ∈ M Is.a. with k x j k ≤ R j ,the evaluation map ev x : A ( s ) R,J → M extends to a well-defined F ( s ) R,J, → L ( M, τ ), which we continue todenote by ev x , and we will also write f ( x ) = ev x ( f ). If ( M, τ ) is a tracial von Neumann algebra, and ξ ∈ L ( M, τ ) ∩ M c we set k ξ k ∞ = ∞ . For f ∈ F ( s ) R,J, we set k f k R, ∞ = sup x, ( M,τ ) k f ( x ) k ∞ ∈ [0 , + ∞ ]where the supremum is over all tracial von Neumann algebras and all x ∈ M Js.a. . We now recall the mainproperties of this construction, with pointers to [33] where the relevant details are shown. We let F ( s ) R,J, ∞ = { f ∈ F R,J, : k f k R, ∞ < ∞} . P1:
The natural multiplication, addition, and ∗ -algebra operations on C (Σ ( s ) R,J ) ⊗ alg C h ( T j ) j ∈ J i have aunique extension to F ( s ) R,J, ∞ which satisfies k f k R, ∞ = k f ∗ k R, ∞ , k f k R, = k f ∗ k R, k f g k R, ∞ ≤ k f k R, ∞ k g k R, ∞ , k f g k R, ≤ k f k R, ∞ k g k R, . These operations together with the norm k · k R, ∞ turn F ( s ) R,J, ∞ into a C ∗ -algebra. [33, Lemma 2.3] P2:
For any tracial von Neumann algebra (
M, τ ) , and any x ∈ M Js.a. with k x j k ∞ ≤ R j for all j ∈ J, theevaluation map ev x : F ( s ) R,J, ∞ → M is surjective [33, Proposition 2.4]. In fact, given any a ∈ M thereis an f ∈ F ( s ) R,J with k f k R, ∞ ≤ k a k ∞ so that ev x ( f ) = a. P3:
Every f ∈ F ( s ) R,J, ∞ is k · k -uniformly continuous in the following sense. For every ε > , there is a δ > F ⊆ J so that if ( M, τ ) is a tracial von Neumann algebra and x, y ∈ M Js.a. with k x j k ∞ , k y j k ∞ ≤ R j for all j ∈ J and k x j − y j k < δ for all j ∈ F, then k f ( x ) − f ( y ) k < ε. [33,Proposition 2.8].We now wish to define an analogous space of “noncommutative functions” when the variables are not self-adjoint, and we will want it to satisfies analogues of the above 3 properties. So fix an index set J, and R ∈ [0 , ∞ ) J . Define A R,J = C (Σ R,J ) ⊗ alg C ∗ h ( T j ) j ∈ J i . Given a tracial von Neumann algebra (
M, τ ) and x ∈ M J with k x k ∞ ≤ R j for all j ∈ J, we let ev x : A R,J → M be the linear map satisfying ev x ( φ ⊗ P ) = φ ( ℓ x ) P ( x ) for φ ∈ C (Σ R,J ), P ∈ C ∗ h ( T j ) j ∈ J i . For f ∈ A R,J, we will use f ( x ) for ev x ( f ) . Define a seminorm k · k R, on A R,J by k f k R, = sup x, ( M,τ ) k f ( x ) k , where the supremum is over all tracial von Neumann algebras ( M, τ ) and all x ∈ M J . We then let F R,J, bethe completion of A R,J, / { f ∈ A R,J, : k f k R, = 0 } under the norm induced by k · k R, . For f ∈ F R,J, we let k f k R, ∞ = sup x, ( M,τ ) k f ( x ) k ∞ ∈ [0 , + ∞ ] , and we set F R,J, ∞ = { f ∈ F R,J, : k f k R, ∞ < ∞} . For a tracial von Neumann algebra (
M, τ ) and x ∈ M J , we then have that f ( x ) ∈ M. The algebras F R,J , F ( s ) R,J are both examples of algebras which are completions of a C ∗ -algebra with respectto uniform 2-norm coming from a family of traces. These are now known as uniformly tracially complete C ∗ -algebras . Ozawa defined such a completion when the family consisted of all traces (see [51, p. 351-352]),the special case of convex subsets of the trace space appeared recently in the study of classification of nuclear C ∗ -algebras and their homomorphisms (see [5, 13, 12]).The following is proved exactly as in [33, Lemma 2.3]. Proposition 2.15.
Let J be an index set and R ∈ [0 , ∞ ) J . Then the product and ∗ -operation have a uniqueextension to product and ∗ -operations on F R,J, ∞ which satisfy the axioms of a ∗ -algebra as well as thefollowing estimates k f k R, ∞ = k f ∗ k R, ∞ , k f k R, = k f ∗ k R, k f g k R, ∞ ≤ k f k R, ∞ k g k R, ∞ , k f g k R, ≤ k f k R, ∞ k g k R, . Under these extended operations and the norm k · k R, ∞ , the ∗ -algebra F R,J, ∞ is a C ∗ -algebra. We now turn to the other two main properties of F R,J, ∞ we will want. Theorem 2.16.
Let J be an index set and R ∈ [0 , ∞ ) J . We then have the following properties of thenoncommutative function space F R,J, ∞ . (i) Let ( M, τ ) be a tracial von Neumann algebra and x ∈ M J with k x k ∞ ≤ R j for all j ∈ J. Then the map F R,J, ∞ → W ∗ ( x ) given by f f ( x ) is surjective. In fact, for all a ∈ W ∗ ( x ) , there is an f ∈ F R,J, ∞ with k f k R, ∞ ≤ k a k ∞ and so that f ( x ) = a. (ii) Every f ∈ F R,J, is k · k -uniformly continuous in the following sense. For every ε > , there is a δ > and a finite F ⊆ J so that if ( M, τ ) is any tracial von Neumann algebra and x, y ∈ M J with k x j − y j k < δ for all j ∈ J, we have k f ( x ) − f ( y ) k < ε. (iii) Suppose ( M k , τ k ) , k = 1 , are tracial von Neumann algebras and x ∈ Q j ∈ J { a ∈ M : k a k ∞ ≤ R j } , andthat Θ : M → M is a trace-preserving, unital, normal ∗ -homomorphism. Then f (Θ( x )) = Θ( f ( x )) for all f ∈ F R,J, . Proof.
Let K = J × { , } , and define self-adjoint elements of C ∗ h ( T j ) j ∈ J i indexed by K as follows: A ( j, = T j + T ∗ j , A ( j, = T j − T ∗ j i , for all j ∈ J. Let π : C h ( S k ) k ∈ K i → C ∗ h ( T j ) j ∈ j i be the unique ∗ -homomorphism satisfying π ( S k ) = A k forall k ∈ K. Then π is surjective. Define a continuous map Ψ : Σ R,J → Σ ( s ) R,K byΨ( ℓ )( P ) = ℓ ( π ( P )) , RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 21 and let b Ψ : C (Σ ( s ) R,K ) → C (Σ R,J ) be the induced map defined by b Ψ( φ ) = φ ◦ Ψ . Suppose (
M, τ ) is any tracial von Neumann algebra and x ∈ M J satisfies k x k ∞ ≤ R j for all j ∈ J. Define y ∈ M K by y ( j, = x j + x ∗ j , y ( j, = x j − x ∗ j i for all j ∈ J. Direction calculations show that for all f ∈ A ( s ) R,K we have ev y ( f ) = ev x [( b Ψ ⊗ π )( f )] . Indeed,since ev y , ev x , b Ψ , and π are all ∗ -homomorphisms, it suffices to check this equation on an element of theform φ ⊗ S k for some k ∈ K. In this case, the desired equality follows from the fact that ev y ( S k ) = y k ,ev x ( π ( S k )) = y k , and Ψ( ℓ x ) = ℓ y . It follows that for all f ∈ A ( s ) R,K we have k [ b Ψ ⊗ π ]( f ) k R, ≤ k f k R, , k [ b Ψ ⊗ π ]( f ) k R, ∞ ≤ k f k R, ∞ . From the above two inequalities it follows that b Ψ ⊗ π uniquely extends to maps, still denoted b Ψ ⊗ π , from F ( s ) R,K, → F R,J, , F ( s ) R,K, ∞ → F R,J, ∞ , which are k · k R, – k · k R, , k · k R, ∞ – k · k R, ∞ contractions. Moreover, westill have that ev x ◦ b Ψ ⊗ π = ev y . (i): Given x ∈ M J , let y ∈ M Ks.a. be defined as above. Then by [33, Proposition 2.4] for any a ∈ W ∗ ( x ) = W ∗ ( y ) , there is an g ∈ F ( s ) R,K, ∞ with g ( y ) = a and k g k R, ∞ ≤ k a k ∞ . Set f = [ b Ψ ⊗ π ]( g ) , then as ev x ◦ b Ψ ⊗ π = ev y we know f ( x ) = a. Since b Ψ ⊗ π is k · k R, ∞ – k · k R, ∞ contractive, it follows that k f k R, ∞ ≤ k g k R, ∞ ≤ k a k ∞ . (ii): Let V the set of all f ∈ F R,J, which satisfy the conclusion of (ii). As elements of F ( s ) R,K, areuniformly continuous, it follows that V contains [ b Ψ ⊗ π ]( F ( s ) R,K, ) . In particular, V is dense. It then sufficesto show that V is k · k R, –closed. For every f ∈ F R, , and every tracial von Neumann algebra ( M, τ ) and all x ∈ Q j ∈ J { y ∈ M : k y k ∞ ≤ R j } we have k f ( x ) k ≤ k f k R, . From the above estimate, it is a standard argument to show that V is closed.(iii): First, observe that because Θ is trace-preserving, we know that ℓ Θ( x ) = ℓ x . From here, the conclusionis direct to check for the case that f ∈ A R,J . For f ∈ F R,J, we have the estimatemax( k f (Θ( x )) k , k f ( x ) k ) ≤ k f k R, . The above estimate allows to deduce the conclusion for a general element of F R,J, from the case of elementsof A R,J by approximation. (cid:3)
Given an index set J, and an R ∈ [0 , ∞ ) J , for any tracial von Neumann algebra ( M, τ ) , we may abusenotation and view f as a map f : Y j ∈ J { a ∈ M : k a k ∞ ≤ R j } → M via x f ( x ) . Given another index set J ′ and R ′ ∈ [0 , ∞ ) J ′ , we define F R,R ′ ,J,J ′ = { f = ( f j ′ ) j ′ ∈ J ′ ∈ ( F R,J, ∞ ) J ′ : k f j ′ k R, ∞ ≤ R ′ j ′ for all j ′ ∈ J } . Then f also determines a map f : Y j ∈ J { a ∈ M : k a k ∞ ≤ R j } → Y j ′ ∈ J ′ { a ∈ M : k a k ∞ ≤ R ′ j ′ for all j ′ ∈ J ′ } by x ( f j ′ ( x )) j ′ ∈ J ′ . In particular, all of this makes sense for M = M k ( C ) . So given a µ ∈ Prob( M k ( C ) J ) , with µ Y j ∈ J { a ∈ M : k a k ∞ ≤ R j } = 1 , we can make sense of f ∗ µ ∈ Prob( M k ( C ) J ′ ) . A nice consequence of k · k -uniform continuity is that takingpushforwards of measures preserves exponential concentration. Proposition 2.17.
Let J , J ′ be countable index set, and R ∈ [0 , ∞ ) J , R ′ ∈ [0 , ∞ ) J ′ , and f ∈ F R,R ′ ,J,J ′ . Suppose we are given a sequence µ ( k ) ∈ Prob( M n ( k ) ( C ) J ) with µ ( k ) Y j ∈ J { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R j } . If µ ( k ) has exponential concentration at scale n ( k ) , then so does f ∗ µ ( k ) . Proof.
Fix a finite F ′ ⊆ J ′ , and an ε > . By Theorem 2.16 (ii), we may choose a finite F ⊆ J and a δ > M, τ ) is any tracial von Neumann algebra, and x, y ∈ M J satisfy k x j k ∞ , k y j k ∞ ≤ R j for all j ∈ J and k x j − y j k < δ for all j ∈ F, then k f ( x ) − f ( y ) k ,F ′ < ε. By Theorem 2.16 (iii) for every g ∈ F R,J , every tracial von Neumann algebra (
M, τ ), every u ∈ U ( M ) , and every x ∈ Q j ∈ J { a ∈ M : k a k ∞ ≤ R j } we have g ( uxu ∗ ) = ug ( x ) u ∗ . Hence, for all k ∈ N , and all A, B ∈ Q j ∈ J { C ∈ M k ( C ) : k C k ∞ ≤ R j } with d orb F ( A, B ) < δ, we have d orb F ′ ( f ( A ) , f ( B )) < ε. So suppose Ω ⊆ M n ( k ) ( C ) J ′ and f ∗ µ ( k ) (Ω) ≥ / . Then µ ( k ) ( f − (Ω)) ≥ / , so µ ( k ) ( N δ ( f − (Ω) , d orb F )) ≥ − α µ ( k ) ,d orb F ( δ ) . Our choice of δ implies that N δ ( f − (Ω) , d orb F ) ⊆ f − ( N ε (Ω , d orb F ′ )) , and thus f ∗ µ ( k ) ( N ε (Ω , d orb F ′ )) ≥ − α µ ( k ) ,d orb F ( δ ) . So lim sup k →∞ n ( k ) log α f ∗ µ ( k ) ,d orb F ′ ( ε ) ≤ lim sup k →∞ n ( k ) log α µ ( k ) ,d orb F ( δ ) < . (cid:3) We also need the following analogues of [33, Propostion 2.6 (1) and (2)], whose proofs are identical.
Lemma 2.18.
Let
J, J ′ be index sets and R ∈ [0 , ∞ ) J , R ′ ∈ [0 , ∞ ) J ′ . Suppose f ∈ F R,R ′ ,J,J ′ . (i) Suppose ( M k , τ k ) , k = 1 , are two tracial von Neumann algebras, and x k ∈ Q j ∈ J { a ∈ M k : k a k ∞ ≤ R j } , k = 1 , . If ℓ x = ℓ x , then ℓ f ( x ) = ℓ f ( x ) . (ii) Define a map f ∗ : Σ R,J → Σ R ′ ,J ′ as follows. Given ℓ ∈ Σ R,J , let π ℓ , W ∗ ( ℓ ) be as in (3),(4) andequip W ∗ ( ℓ ) with the trace τ l given by (5). Set x = ( π ℓ ( T j )) j ∈ J , and define f ∗ ℓ = ℓ f ( x ) . Then f ∗ isweak ∗ -weak ∗ continuous. Recall that the point of the construction in (3) was that ℓ = ℓ x . So by (i), for any tracial von Neumannalgebra (
M, τ ) , and any y ∈ M J with ℓ y = ℓ, we have f ∗ ℓ = ℓ f ( y ) . So, for example, if we have a sequence
RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 23 ( x n ) n of tuples in tracial von Neumann algebras and if ℓ x n → weak ∗ n →∞ ℓ x for some other tuple x, then by (ii)we know ℓ f ( x n ) → weak ∗ n →∞ ℓ f ( x ) provide x n , x satisfy the appropriate norm bounds to define f ( x n ) , f ( x ) . We also need a simple consequence of the above, which is that microstates behave well with respect tothe noncommutative function spaces F R,R ′ ,J,J ′ . The proof is the same as in [33, Corollary 2.7].
Lemma 2.19.
Let
J, J ′ be index sets, and R ∈ [0 , ∞ ) J and R ′ ∈ [0 , ∞ ) J ′ . Fix an f ∈ F R,R ′ ,J,J ′ , a tracialvon Neumann algebra ( M, τ ) and x ∈ Q j ∈ J { a ∈ M : k a k ∞ ≤ R j } . Then for any weak ∗ -neighborhood V of ℓ f ( x ) ,x in Σ R ′ ⊔ R,J ′ ⊔ J there is a weak ∗ -neighborhood O of ℓ x so that f (Γ ( k ) R ( O )) ⊆ Γ ( k ) R ′ ⊔ R ( f ( x ) : V ) . Proofs of the main theorems
Microstates Collapse and the proof of Theorem 1.2 (i).
Intuitively, Theorem 1.2 (i) asserts thatif we sample microstates for M according to the sequence of measures µ ( k ) , and use them to “induce” (viathe function f ) microstates for N, then “most” of these microstates for N are unitarily conjugate to eachother. This will be proved in a manner entirely similar to the proof of [33, Proposition 3.3], with only minorchanges in place to take care of the fact that we are dealing with unitary conjugation orbits of microstatesinstead of relative microstates as in [33, Section 3.3]. Lemma 3.1.
Let ( X, µ, d ) be a pseudometric measure space. If Ω ⊆ X and µ (Ω) > α µ,d ( ε ) , then µ ( N ε (Ω , d )) ≥ − α µ,d ( ε ) . Proof.
Suppose µ (Ω) > α µ,d ( ε ) , and set Θ = N ε (Ω , d ) c . Then µ ( N ε (Θ) c ) ≥ µ (Ω) > α µ,d ( ε ) . The definition of α implies that µ (Θ) < / . So µ ( N ε (Ω , d )) > / , and this in turn implies that µ ( N ε (Ω , d )) ≥ − α µ,d ( ε ) . (cid:3) It will frequently be useful to note the following facts about sequences of measures which are asymptoticallyconcentrated on microstates spaces and have exponential concentration.
Lemma 3.2.
Let ( M, τ ) be a tracial von Neumann algebra, I an index set, R ∈ [0 , ∞ ) I and x ∈ Q i ∈ I { a ∈ M : k a k ∞ ≤ R i } . Assume we are given integers n ( k ) for k ∈ N with n ( k ) → ∞ and µ ( k ) ∈ Prob( M k ( C ) J ) with µ ( k ) (Γ ( n ( k )) R ( O )) → for all weak ∗ neighborhoods O of ℓ x in Σ R,I . Further assume that µ ( k ) has exponential concentration withscale n ( k ) .(i) Assume that lim k →∞ µ ( k ) Y i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } ! n ( k ) 2 = 1 . Define ν ( k ) ∈ Prob( M n ( k ) ( C ) I ) by ν ( k ) ( E ) = µ ( k ) (cid:0) E ∩ Q i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } (cid:1) µ ( k ) (cid:0)Q i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } (cid:1) . Then ν ( k ) still has exponential concentration.(ii) Assume that lim k →∞ µ ( k ) Y i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } ! = 1 , and define ν ( k ) as in (i). For every weak ∗ neighborhood O of ℓ x in Σ R,I we have lim sup k →∞ n ( k ) log ν ( k ) (Γ ( n ( k )) R ( O ) c ) < . Proof. (i):To see that ν ( k ) still exhibits exponential concentration fix ε >
0, and suppose that E k ⊆ M n ( k ) ( C ) J has ν ( k ) ( E k ) ≥ for all k. Then µ ( k ) ( E k ) ≥ µ ( k ) Y i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } ! . By our assumptions and Lemma 3.1, we have µ ( k ) ( N ε ( E k , d orb F ) c ) ≤ α µ ( k ) ,d orb F ( ε )for all large k. But then for all large k we have ν ( k ) ( N ε ( E k , d orb F ) c ) ≤ α µ ( k ) ,d orb F ( ε ) µ ( k ) (cid:0)Q i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } (cid:1) . Hence α ν ( k ) ,d orb F (2 ε ) ≤ α µ ( k ) ,d orb F ( ε ) µ ( k ) (cid:0)Q i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } (cid:1) for all large k. This estimate and our hypotheses on µ ( k ) (cid:0)Q i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } (cid:1) are enough toshow that ν ( k ) still has exponential concentration.(ii): We may choose a weak ∗ -neighborhood V of the law of x, a δ > F ⊆ I so that N δ (Γ n ( k ) R ( V ) , k · k ,F ) ∩ Y i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } ⊆ Γ n ( k ) R ( O )for all k ∈ N . Since Γ n ( k ) R ( V ) is conjugation invariant, it follows that N δ (Γ n ( k ) R ( V ) , d orb F ) ∩ Y i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } ⊆ Γ n ( k ) R ( O ) . Our assumptions on ν ( k ) guarantee that for all large k , we have ν ( k ) (Γ n ( k ) R ( V )) ≥ . So ν ( k ) (Γ n ( k ) R ( O ) c ) ≤ α µ ( k ) ,d orb F ( δ ) + µ ( k ) for all large k. Taking n ( k ) log of both sides and letting k → ∞ completes the proof. (cid:3) We now prove Theorem 1.2 (i), and in fact show a more general result.
RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 25
Theorem 3.3.
Let ( M, τ ) be a tracial von Neumann algebra and N ≤ M with h ( N : M ) = 0 . Fix index sets
I, J with J countable, and let x ∈ M I , y ∈ N J be given. Suppose that b R ∈ [0 , ∞ ) I ⊔ J with k x i k ∞ ≤ b R i forall i ∈ I, and k y j k ∞ ≤ b R j for all j ∈ J and so that M = W ∗ ( x ) . Set R = b R (cid:12)(cid:12) I , R ′ = b R (cid:12)(cid:12) J . Write y = f ( x ) for some f ∈ F R,R ′ ,I,J . Assume that n ( k ) ∈ N is a sequence of integers with n ( k ) → ∞ , and that µ ( k ) ∈ Prob( M n ( k ) ( C ) I ) satisfies µ ( k ) (Γ ( n ( k )) R ( O )) → k →∞ for every weak ∗ -neighborhood O of ℓ x in Σ R,I . Further assume that lim k →∞ µ ( k ) Y i ∈ I { A ∈ M k ( C ) : k A k ∞ ≤ R i } ! = 1 . If ( µ ( k ) ) k has exponential concentration at scale n ( k ) , then there is a sequence Ω k ⊆ Q i ∈ I { C ∈ M k ( C ) : k C k ∞ ≤ R i } with the following properties: • µ ( k ) (Ω k ) → , • for every weak ∗ -neighborhood O of ℓ x we have Ω k ⊆ Γ ( n ( k )) R ( O ) for all sufficiently large k, • for every finite F ⊆ J lim k →∞ sup A ,A ∈ Ω k d orb F ( f ( A ) , f ( A )) = 0 . Proof.
By Lemma 3.2 we may, and will, assume that µ ( k ) Y i ∈ I { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R i } ! = 1 . We first start with the following claim.
Claim: For every finite F ⊆ J, for every ε > , and for every weak ∗ -neighborhood O of ℓ x , there is asequence Ω k ⊆ Γ ( n ( k )) R ( O ) (depending upon ε, F, O ) satisfying • lim k →∞ µ ( k ) (Ω k ) = 1 , and • lim sup k →∞ sup A ,A ∈ Ω k d orb F ( f ( A ) , f ( A )) ≤ ε. To prove the claim, let ν ( k ) = f ∗ µ ( k ) as defined in the discussion preceding Proposition 2.17. Set η = − lim sup k →∞ n ( k ) α ν ( k ) ,d orb F (2 ε ) . By Proposition 2.17, we know η > . Since h ( N : M ) ≤ , we may choose a weak ∗ -neighborhood V of ℓ ( y,x ) so that K orb ε,F ( y : V , k · k ) ≤ η . Let Ξ k ⊆ Γ ( n ( k )) b R ( y : V ) be ε -dense with respect to d orb F and so that | Ξ k | = K ε (Γ ( n ( k )) b R ( y : V ) , d orb F ) . Let φ k : Γ ( n ( k )) b R ( y : V ) → Ξ k be Borel maps which satisfy d orb F ( A, φ k ( A )) < ε for all A ∈ Γ ( n ( k )) b R ( y : V ).Set Θ k = { A ∈ Γ ( n ( k )) b R ( y : V ) : ν ( k ) ( N ε ( A, d orb F )) ≥ exp( − n ( k ) η / } , and ∆ k = Θ k \ Γ ( n ( k )) ( y ; V ) . Observe that for every A ∈ ∆ k , we have ν ( k ) ( N ε ( φ k ( A ) , d orb F )) < exp( − n ( k ) η / . So ν ( k ) (∆ k ) ≤ X B ∈ φ k (∆ k ) ν ( k ) ( N ε ( B, d orb F )) < exp( − n ( k ) η / | Ξ k | . Thus ν ( k ) (∆ k ) ≤ exp( − n ( k ) /
4) for all large k , and so ν ( k ) (∆ k ) → . So ν ( k ) (Θ k ) = ν ( k ) (Γ ( n ( k )) R ( y : O )) − ν ( k ) (∆ k ) → , as ν ( k ) is asymptotically supported on the microstates space for y in the presence of x. Suppose B , B ∈ Θ k . If k is sufficiently large, then by Lemma 3.1 N ε ( B , d orb F ) ∩ N ε ( B , d orb F ) = ∅ , and thus d orb F ( B , B ) ≤ ε. By definition of ν ( k ) , we have µ ( k ) ( f − (Θ k )) = ν ( k ) (Θ k ) → . So if we setΩ k = f − (Θ k ) ∩ Γ ( n ( k )) R ( O ) , then it is direct to show that Ω k has the desired properties with ε replaced by8 ε. Since ε > F m ) m be an increasing sequence of finite subsets of J with J = S m F m , andlet O m be a decreasing sequence of weak ∗ -neighborhoods of ℓ x in Σ R,J with T m O m = { ℓ x } . By the claim,for every positive integer m, we may choose a sequence Ω k,m ⊆ Γ ( n ( k )) b R ( y : O m ) withlim sup k →∞ sup A ,A ∈ Ω k,m d orb F m ( f ( A ) , f ( A )) < − m , lim k →∞ µ ( k ) (Ω k,m ) = 1 . We may thus find a strictly increasing sequence 1 < K < K < · · · of integers so that for every positiveinteger m sup k ≥ K m ,A ,A ∈ Ω k,m d orb F m ( f ( A ) , f ( A )) < − m , and inf k ≥ K m µ ( k ) (Ω k,m ) ≥ − − m . Define Ω k as follows. For k < K , set Ω k = ∅ , and for k ≥ K let m the unique integer so that K m ≤ k Proof of Theorem 1.2 (ii). The Peterson-Thom conjecture is inherently a question about von Neu-mann algebras, whereas strong convergence of laws is inherently a question about C ∗ -algebras. For example,strong convergence can be reformulated in terms of trace-preserving embeddings into C ∗ -ultraproducts . So asignificant aspect of Theorem 1.2 is the assertion that we can reduce the von Neumann question of validity ofthe Peterson-Thom conjecture to a C ∗ question about strong convergence. In order to do this, we will needto assume that our given von Neumann algebra can be approximated by any “nice enough” weak ∗ -dense ∗ -subalgebra in a manner robust enough to preserve some key structure of the von Neumann algebra. Inparticular, we will make use of the Connes-Haagerup characterization of nonamenability of a von Neumannalgebra in terms of norms of “Laplace-like” operators in the tensor of the algebra with its opposite. Thus,we will need to assume that our approximation process keeps norms under control when we pass to tensor RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 27 products. Maintaining control over norms when passing to tensor products is the raison d’ˆetre for the no-tions of completely bounded/completely positive maps. So the above discussion naturally leads us to theconsideration of approximation properties formulated via completely positive and completely bounded maps.Given a C ∗ -algebra A , there is a canonical way to view A ∗∗ as a von Neumann algebra. Moreover thenatural inclusion A ֒ → A ∗∗ allows us to view A ∗∗ as the universal enveloping von Neumann algebra of A .Namely, given any ∗ -representation π : A → B ( H ) with H a Hilbert space, there is a unique, normal ∗ -representation e π : A ∗∗ → B ( H ) with e π (cid:12)(cid:12) A = π . Moreover, e π ( A ) = π ( A ) SOT (see [66, Theorem 2.4] for a proofof all of this). Definition 3.4. We say that a (unital) C ∗ -algebra A is locally reflexive if given any finite-dimensionaloperator system E ⊆ A ∗∗ , there is a net φ α : E → A of completely positive maps with k φ α k cb ≤ φ α ( x ) → α x in the weak ∗ -topology.An alternate way to phrase this is as follows. Let E, F be operator systems. If F is an operator systemconcretely embedded in B ( H ) with H a Hilbert space, then we can give CP ( E, F ) the point-WOT topology.So a basic neighborhood of φ ∈ CP ( E, F ) is given by O G ,G ,ε ( φ ) = \ x ∈ G ,ξ,η ∈ G { ψ ∈ CP ( E, F ) : |h φ ( x ) ξ, η i − h ψ ( x ) ξ, η i| < ε } for finite sets G ⊆ E , G ⊆ H , and an ε ∈ (0 , ∞ ) . Let A be a C ∗ -algebra. For an operator space E ⊆ A ∗∗ , we use ι E for the inclusion map E ֒ → A ∗∗ . Locally reflexivity is then just the assertion that ι E ∈ { φ ∈ CP ( E, A ∗∗ ) : φ ( E ) ⊆ A, k φ k cb ≤ } point − W OT , for every finite dimensional E ⊆ A ∗∗ . The main result on locally reflexivity that we need is that everyexact C ∗ -algebra is locally reflexive (see [45, 46] and also [10, Theorem 9.3.1]). Since exact C ∗ -algebras areubiquitous in free probability, this provides us with an adequate source of examples. E.g., the reduced freegroup C ∗ -algebra is locally reflexive, as is the C ∗ -algebra generated by a free semicircular family. Indeed,given any free tuple ( x , · · · , x k ) ∈ M k in a tracial von Neumann algebra ( M, τ ) , with each x j being normal,we have that C ∗ ( x , · · · , x k ) is exact by [19, 20].It should be emphasized that A ∗∗ is a very large von Neumann algebra. For example, it is only in veryrare circumstances that A ∗ is separable (e.g. this does not occur if A contains a copy of C ( X ) where X is anuncountable compact Hausdorff space). Consequently, it is rare that A ∗∗ can be represented on a separableHilbert space. However, the fact that A ∗∗ is the universal enveloping von Neumann algebra allows us todeduce more concrete approximations for other von Neumann algebras associated to A. Recall that if H isa Hilbert space, and M ⊆ B ( H ) is a von Neumann algebra, then M is a von Neumann completion of A ifthere is a faithful ∗ -representation π : A → B ( H ) with M = π ( A ) SOT . Suppose A is locally reflexive and M is a von Neumann completion of A, and view A ⊆ M. By universality of A ∗∗ it follows that if E ⊆ M is afinite-dimensional operator system, then ι E ∈ { φ ∈ CP ( E, M ) : φ ( E ) ⊆ A, k φ k cb ≤ } point − W OT , where ι E : E → M is the inclusion map. This is the precise manner in which we shall use local reflexivity toapproximate elements of M by a prescribed weak ∗ -dense ∗ -subalgebra. We also need to recall some notation and a result of Haagerup. We have an action M k ( C ) ⊗ M k ( C )on S ( k, tr) defined on elementary tensors by( A ⊗ B ) C = ACB t . It is direct to check that this gives a ∗ -isomorphism M k ( C ) ⊗ M k ( C ) ∼ = B ( S ( k, tr)) . Since ∗ -isomorphisms between C ∗ -algebras are isometric, it follows that k x k ∞ = k x k B ( S ( k, tr)) for all x ∈ M k ( C ) ⊗ M k ( C ) . For a tracial von Neumann algebra ( M, τ ) and x ∈ M, we let M op be the vonNeumann algebra which as a set is { x op : x ∈ M } . The vector space operation, and the ∗ -operation is thesame as in M, but the product is the opposite: x op y op = ( yx ) op . For x ∈ M, we let x = ( x ∗ ) op . Note that we have a canonical identification M k ( C ) ∼ = M k ( C ) op given by A ( A t ) op . For A ∈ M k ( C ) , we let A = ( A ∗ ) t . Technically, this means we have two different notionsof A for A ∈ M k ( C ) . One as an element of M k ( C ) and one as an element of M k ( C ) op . However, underthe identification M k ( C ) ∼ = M k ( C ) op given above, these two notations coincide. Since we always identify M k ( C ), M k ( C ) op via the map A ( A t ) op , this will not cause confusion. The way we shall use nonamenabilityis in the following characterization of nonamenability of tracial von Neumann algebras, due to Haagerup. Theorem 3.5 (Haagerup, Lemma 2.2 in [28]) . Let ( M, τ ) be a tracial von Neumann algebra. Then M isnonamenable if and only if there is a nonzero central projection f ∈ M and u , · · · , u r ∈ U ( M f ) so that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r r X j =1 u j ⊗ u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < . In order to prove Theorem 1.2 (ii), we need to reduce the validity of the Peterson-Thom conjecture to the C ∗ -question of strong convergence. We begin with the following Proposition, which gives a general resultalong these lines. We remark that the argument for the proof of this Proposition is analogous with a methodof proof of Chifan-Sinclair (see [14, Theorem 3.2]) in the context of Popa’s deformation/rigidity theory. Proposition 3.6. Let ( M, τ ) be a tracial von Neumann algebra, I an index set, and x ∈ M I with W ∗ ( x ) = M. Fix R ∈ [0 , ∞ ) I with k x i k ∞ ≤ R i for all i ∈ I. Suppose that C ∗ ( x ) is locally reflexive and that Q ≤ M is nonamenable. Then there is an r ∈ N , an F ∈ ( F R,I ) r with F ( x ) ∈ Q r and an ε > which satisfies thefollowing property. Assume we are given • positive integers ( n ( k )) ∞ k =1 with n ( k ) → ∞ , • A ( k ) , B ( k ) ∈ Q i ∈ I { C ∈ M n ( k ) ( C ) : k C k ∞ ≤ R i } such that the law of ( A ( k ) ⊗ M n ( k ) ( C ) , M n ( k ) ( C ) ⊗ ( B ( k ) ) t ) converges strongly to the law of ( x ⊗ C ∗ ( x ) op , C ∗ ( x ) ⊗ x op ) . Then lim inf k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) )) ≥ ε. RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 29 Proof. By [28, Lemma 2.1], we may find a nonzero projection p ∈ Z ( Q ) and u , · · · , u r ∈ U ( Qp ) so that C = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r r X j =1 u j ⊗ u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < . Fix any C ′ ∈ ( C, . Choose P ∈ F R,I with k P k R, ∞ ≤ P ( x ) = p, and F j ∈ F R,J , j = 1 , · · · , r with F j ( x ) = u j and k F j k R, ∞ ≤ . Set F = ( F , · · · , F r ) . Suppose we have • positive integers ( n ( k )) ∞ k =1 with n ( k ) → ∞ , • A ( k ) , B ( k ) ∈ Q i ∈ I { C ∈ M n ( k ) ( C ) : k C k ∞ ≤ R i } so that the law of ( A ( k ) ⊗ M n ( k ) ( C ) , M n ( k ) ( C ) ⊗ ( B ( k ) ) t ) converges strongly to the law of ( x ⊗ C ∗ ( x ) op , C ∗ ( x ) ⊗ x op ). Choose unitaries U ( k ) ∈ U ( k ) so that d orb ( F ( A ( k ) ) , F ( B ( k ) )) = k F ( A ( k ) ) − U ( k ) F ( B ( k ) )( U ( k ) ) ∗ k . Then, d orb ( F ( A ( k ) ) , F ( B ( k ) )) = r X j =1 ( k F j ( A ( k ) ) k + k F j ( B ( k ) ) k ) − r X j =1 Re tr( F j ( A ( k ) ) U ( k ) F j ( B ( k ) ) ∗ ( U ( k ) ) ∗ ) . By weak ∗ convergence of laws,lim inf k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) )) ≥ rτ ( p ) − k →∞ r X j =1 Re tr( F j ( A ( k ) ) U ( k ) F j ( B ( k ) ) ∗ ( U ( k ) ) ∗ ) . Since F j ( x ) P ( x ) = F j ( x ) , and k F j k R, ∞ ≤ k P k R, ∞ ≤ j = 1 , · · · , r, we have k F j ( B ( k ) ) − P ( B ( k ) ) F j ( B ( k ) ) P ( B ( k ) ) ∗ k → . So using once again that k F j k R, ∞ ≤ j = 1 · · · , r, it follows thatlim inf k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) )) ≥ rτ ( p ) − k →∞ r X j =1 Re tr( F j ( A ( k ) ) U ( k ) P ( B ( k ) ) F j ( B ( k ) ) ∗ P ( B ( k ) ) ∗ ( U ( k ) ) ∗ ) ≥ rτ ( p ) − k →∞ k P ( B ( k ) ) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 F j ( A ( k ) ) U ( k ) P ( B ( k ) ) F j ( B ( k ) ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Since k P ( B ( k ) ) k → k P ( x ) k = p τ ( p ) , we obtain:(8) lim inf k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) )) ≥ rτ ( p ) − p τ ( p ) lim sup k →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 F j ( A ( k ) ) U ( k ) P ( B ( k ) ) F j ( B ( k ) ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) To bound the second term in this expression, let E = span( { u j } rj =1 ∪ { } ∪ { u ∗ j } rj =1 ) . By local reflexivitywe have ( u j ) rj =1 ∈ { ( φ ( u j )) rj =1 : φ ∈ CP ( E, C ∗ ( x )) , k φ k cb ≤ } W OT . So, by convexity, ( u j ) rj =1 ∈ { ( φ ( u j )) rj =1 : φ ∈ CP ( E, C ∗ ( x )) , k φ k cb ≤ } SOT . Hence we may find a sequence φ m : E → C ∗ ( x ) of contractive, completely positive maps with k φ m ( u j ) − u j k → j = 1 , · · · , r. Choose Q j,m ∈ C ∗ h ( T j ) j ∈ J i with k Q j,m ( x ) − φ m ( u j ) k ∞ ≤ min (cid:18) C ′ − C , − m (cid:19) for all j = 1 , · · · , r , k Q j,m k ∞ ≤ j = 1 , · · · , r .Since u j = F j ( x ) , and k P k R, ∞ , k Q j,m k R, ∞ , k F j k R, ∞ ≤ j = 1 , · · · , r , we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 F j ( A ( k ) ) U ( k ) P ( B ( k ) ) F j ( B ( k ) ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ r X j =1 k F j ( A ( k ) ) − Q j,m ( A ( k ) ) k + r X j =1 k F j ( B ( k ) ) − Q j,m ( B ( k ) ) k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 Q j,m ( A ( k ) ) U ( k ) P ( B ( k ) ) Q j,m ( B ( k ) ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ r X j =1 k F j ( A ( k ) ) − Q j,m ( A ( k ) ) k + r X j =1 k F j ( B ( k ) ) − Q j,m ( B ( k ) ) k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 Q j,m ( A ( k ) ) ⊗ Q j,m ( B ( k ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ k P ( B ( k ) ) k . Using strong convergence,lim sup k →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 F j ( A ( k ) ) U ( k ) P ( B ( k ) ) F j ( B ( k ) ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ r X j =1 k u j − Q j,m ( x ) k + p τ ( p ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 Q j,m ( x ) ⊗ Q j,m ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ By our choice of Q j,m , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 Q j,m ( x ) ⊗ Q j,m ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ ( C ′ − C ) r + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 φ m ( u j ) ⊗ φ m ( u j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ C ′ r, where in the last step we use the definition of C and the fact that k φ m k cb ≤ k φ m ⊗ φ opm k cb ≤ . Soaltogether we have shown thatlim sup k →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 F j ( A ( k ) ) U ( k ) P ( B ( k ) ) F j ( B ( k ) ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C ′ r p τ ( p ) + 2 r X j =1 k u j − Q j,m ( x ) k . Inserting this into (8),lim inf k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) )) ≥ rτ ( p )(1 − C ′ ) − r X j =1 k u j − Q j,m ( x ) k . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 31 Letting m → ∞ and then C ′ → C shows thatlim inf k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) )) ≥ p τ ( p )(1 − C ) . So setting ε = p τ ( p )(1 − C ) completes the proof. (cid:3) We will give a cleaner way to state the above Proposition in terms of ultraproducts in Section 4 (seeProposition 4.5). For now, we proceed to the proof of Theorem 1.2. Proof of Theorem 1.2 (ii). Set R = ( R , · · · , R l ) ∈ [0 , ∞ ) l . Let µ ( k ) ∈ Prob( M k ( C ) l ) be the distribution of( X ( k ) j ) lj =1 . Suppose, for the sake of contradiction, that Q ≤ M is nonamenable and h ( Q : M ) ≤ . Sinceour hypotheses necessarily imply that M embeds into an ultrapower of R , it follows that h ( Q : M ) = 0 . Let r ∈ N and F ∈ ( F R,l ) r with F ( x ) ∈ Q r and ε > k ⊆ Q lj =1 { A ∈ M k ( C ) : k A k ∞ ≤ l } with µ ( k ) (Ω k ) = 1 and so thatlim k →∞ sup A,B ∈ Ω k d orb ( F ( A ) , F ( B )) = 0 . By strong convergence in probability, we may choose a sequence Θ k ⊆ M k ( C ) l × M k ( C ) l with µ ( k ) × µ ( k ) (Θ k ) → , and so that for any sequence ( A ( k ) , B ( k ) ) ∈ Θ k the law of ( A ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ ( B ( k ) ) t ) converges stronglyto the law of ( x ⊗ , ⊗ x op ) . Since µ ( k ) (Ω k ) → µ ( k ) ⊗ µ ( k ) (Θ k ) → , for all large k we may choose a( A ( k ) , B ( k ) ) ∈ (Ω k × Ω k ) ∩ Θ k . Then Proposition 3.6 showslim inf k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) ) ≥ ε, whereas our choice of Ω k implies lim k →∞ d orb ( F ( A ( k ) ) , F ( B ( k ) ) = 0 . This gives a contradiction, which completes the proof. (cid:3) Deduction of Theorem 1.1 from Theorem 1.2. In this section, we shall deduce Theorem 1.1from Theorem 1.2. We also state a version for free families of Haar unitaries instead of free semicirculars .Moreover, it is not hard to see that our proof applies equally well to many other families of random matriceswhich model L ( F r ), provided that they exhibit exponential concentration with the correct rate. We startwith the proof of Theorem 1.1 Proof of Theorem 1.1. The fact that (ii) implies (i) is the content of Proposition 2.7. So we focus on provingthat (iii) implies (ii).Let s = ( s , s , · · · , s r ) be a free semicircular family with mean zero and variance 1 . So W ∗ ( s ) ∼ = L ( F r ) . We let µ ( k ) ∈ Prob( M k ( C ) rs.a. ) be the distribution of ( X ( k ) , Y ( k ) ) . It is well known (see [30, Proof of Lemma3.3]) that there is a C > k →∞ k log µ ( k ) (( { A ∈ M k ( C ) rs.a : k A k ∞ ≤ C } ) c ) < , and Voiculescu’s asymptotic freeness theorem (specifically [69, Theorem 2.2]) implies that µ ( k ) is asymp-totically concentrated on microstates for s (using R as the constant function C ). Further, exponentialconcentration of measure with scale k is well known and follows, e.g., from [48, Equation (2.10)]. Itis direct to see that the coordinate-wise transpose map M k ( C ) rs.a. → M k ( C ) rs.a. preserves µ ( k ) . Further, s op = ( s op , s op , · · · , s opr ) is also a tuple of r free semicircular elements each with mean 0 and variance 1 , and so s op has the same distribution as s. So the strong convergence in probability of ( X ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ Y ( k ) ) to( s ⊗ C ∗ ( x ) , C ∗ ( x ) ⊗ s ) is equivalent to the strong convergence in probability of ( X ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ ( Y ( k ) ) t )to ( s ⊗ C ∗ ( s op ) , C ∗ ( s ) ⊗ s op ) . Thus we may apply Theorem 1.2 (ii), and the conclusion of that Theorem givesus exactly what we want. (cid:3) We also state a version of Theorem 1.1 using independent Haar unitaries instead of the GUE ensemble. Theorem 3.7. Fix an integer r ≥ . For each k ∈ N , let U ( k )1 , · · · , U ( k ) r , V ( k )1 , · · · , V ( k ) r be random k × k unitary matrices which are independent and are each distributed according to Haar measure on U ( k ) . Set U ( k ) ⊗ M k ( C ) = ( U ( k ) j ⊗ M k ( C ) ) rj =1 , M k ( C ) ⊗ V ( k ) = (1 M k ( C ) ⊗ V ( k ) j ) rj =1 . Let F r be the free group on r letters a , · · · , a r , and let λ ( a ) ⊗ λ ( a j ) ⊗ C ∗ λ ( F r ) ) rj =1 , ⊗ λ ( a ) = (1 ⊗ λ ( a j )) rj =1 . If the distribution of ( U ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ U ( k ) ) converges (as k → ∞ ) to the distribution of ( λ ( a ) ⊗ C ∗ λ ( F r ) , C ∗ λ ( F r ) ⊗ U ( k ) ) strongly in probability, then for any Q ≤ L ( F r ) with h ( Q : L ( F r )) ≤ we have that Q is amenable.Proof. It is well known that the distribution of ( U ( k ) j ) rj =1 satisfies exponential concentration of measurewith scale k (for example this follows from [48, Theorem 5.3] and [50, Theorem 15]). By compactness, theHaar measure on U ( k ) is invariant under right multiplication, and thus under anti-automorphisms. So thedistribution of ( U ( k ) j ) t is the same as the distribution of U ( k ) j for all j = 1 , · · · , r. Additionally, it is directto show that the unique homomorphism F r → F r sending a j to a − j is bijective, and so λ ( a ) op has the samelaw as λ ( a ) . The proof now proceeds exactly as in the proof of (iii) implies (ii) of Theorem 1.1. (cid:3) Intermediate conjectures and relation to Jung’s Theorem In this section, we collect various conjectures which imply the Peterson-Thom conjecture, and discusstheir relative strength. We start by stating the conjectures already discussed in the introduction. Conjecture 2. Let X ( k )1 , X ( k )2 , · · · , X ( k ) r , Y ( k )1 , Y ( k )2 , · · · , Y ( k ) r be random, self-adjoint k × k matrices whichare independent and are each GUE distributed. Set X ( k ) = ( X ( k ) j ) rj =1 , Y ( k ) = ( Y ( k ) j ) rj =1 . Let s = ( s , · · · , s r ) be a tuple of free semicircular elements which have mean zero and variance . Then for every P ∈ C ∗ h ( T j ) j ∈ J i we have k P ( X ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ Y ( k ) ) k ∞ → k →∞ k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ in probability. RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 33 Conjecture 3. Let Q ≤ L ( F r ) be diffuse and nonamenable, then h ( Q : L ( F r )) > . We now explain some intermediate conjectures, the first of which is formulated in an ultraproduct frame-work. Definition 4.1. Let ω be a free ultrafilter on N , and let ( M k , τ k ) ∞ k =1 be a sequence of tracial von Neumannalgebras. We define their tracial ultraproduct with respect to ω by Y k → ω ( M k , τ k ) = { ( x k ) k ∈ Q k M k : sup k k x k k ∞ < ∞}{ ( x k ) k ∈ Q k M k : sup k k x k k ∞ < ∞ , and lim k → ω k x k k L ( τ k ) = 0 } . If ( x k ) k ∈ Q k M k and sup k k x k k ∞ < ∞ , we let ( x k ) k → ω be the image of ( x k ) k under the quotient map. If J is an index set, and ( x k ) k ∈ Q k M Jk andsup k k x k,j k ∞ < ∞ for all j ∈ J, then we let ( x k ) k → ω ∈ ( Q k → ω ( M k , τ k )) J be the tuple whose j th coordinate is ( x k,j ) k → ω . As is well known, Q k → ω ( M k , τ k ) is a tracial von Neumann algebra with the ∗ -algebra operations definedpointwise and the trace given by τ ω (( x k ) k → ω ) = lim k → ω τ k ( x k ) (this follows from the same argument as [10,Lemma A.9]). It will helpful to know that the noncommutative functional calculus described in Section 2.4commutes with passing to the ultraproduct. Lemma 4.2. Let ( M k , τ k ) k be a sequence of tracial von Neumann algebras and let ω be a free ultrafilter onthe natural numbers. Fix an index set J, R ∈ [0 , ∞ ) J and suppose ( x k ) k ∈ Y k M Jk with k x k,j k ∞ ≤ R j for all k ∈ N , j ∈ J . Then for any f ∈ F R,J, ∞ f (( x k ) k → ω ) = ( f ( x k )) k → ω . Proof. First, note that the conclusion of the lemma is true for f ∈ A R,J . For the general case, fix f ∈ F R,J, ∞ . Given ε > , choose a g ∈ A R,J with k f − g k R, < ε. Then k f (( x k ) k → ω ) − ( f ( x k )) k → ω k ≤ k ( f − g )(( x k ) k → ω ) k + k (( f − g )( x k )) k → ω k ≤ k f − g k R, + lim k → ω k ( f − g )( x k ) k ≤ k f − g k R, < ε. (cid:3) Theorem 4.3. Suppose we are given a tracial von Neumann algebra ( M, τ ) , a countable index set J, andan x ∈ M J with W ∗ ( x ) = M. Suppose R ∈ [0 , ∞ ) J satisfies k x j k ∞ ≤ R j for all j ∈ J. Assume we are givena sequence of natural numbers n ( k ) → ∞ , and a sequence µ ( k ) ∈ Prob( M ) k ( C ) J ) such that • P k µ ( k ) (cid:16)(cid:16)Q j ∈ J { A ∈ M k ( C ) J : k A j k ∞ ≤ R j (cid:17) c (cid:17) < ∞ . • µ ( k ) (Γ ( n ( k )) R ( O )) → for all weak ∗ neighborhoods O of ℓ x in Σ R,J , • µ ( k ) has exponential concentration with scale n ( k ) . Then:(i) there is a conull subset Ω ⊆ Q k M n ( k ) ( C ) J so that for any A = ( A ( k ) ) k ∈ Ω , and for every free ultra-filter ω on N , there is a unique trace-preserving ∗ -homomorphism Θ A,ω : M → Q k → ω ( M n ( k ) ( C ) , tr n ( k ) ) so that Θ A,ω ( P ( x )) = ( P ( A ( k ) )) k → ω for all P ∈ C ∗ h ( T j ) j ∈ J i . (ii) If P ≤ M satisfies h ( P : M ) ≤ , then there is a conull subset Ω ⊆ Ω so that for all A, B ∈ Ω and forevery free ultrafilter ω on N , we have that Θ A,ω (cid:12)(cid:12) P , Θ B,ω (cid:12)(cid:12) P are unitarily conjugate.Proof. Let ν ( k ) be the measure on M n ( k ) ( C ) J given by ν ( k ) ( E ) = µ ( k ) (cid:16) E ∩ (cid:16)Q j ∈ J { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R j } (cid:17)(cid:17) µ ( k ) (cid:16)Q j ∈ J { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R j } (cid:17) . (i): It suffices to find a N k µ ( k ) -conull Ω ⊆ Q k M n ( k ) ( C ) J so that for every A = ( A ( k ) ) k ∈ Ω , we have ℓ A k → ℓ x . Fix a decreasing sequence O m ⊆ Σ R,J of weak ∗ -neighborhoods of ℓ x with ∞ \ m =1 O m = { } , this is possible as J is countable. By Lemma 3.2 (ii), X k (cid:16) ⊗ s ν ( s ) (cid:17) (cid:16) { ( A ( s ) ) s : A ( k ) ∈ Γ ( n ( k )) R ( O m ) c } (cid:17) < ∞ for every m ∈ N . For every k ∈ N , we have that µ ( k ) (Γ ( n ( k )) R ( O m ) c )) ≤ µ ( k ) Γ ( n ( k )) R ( O m ) c ∩ Y j ∈ J { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R j } + µ ( k ) Y j ∈ J { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R j } c = ν ( k ) (cid:16) Γ ( n ( k )) R ( O m ) c (cid:17) µ ( k ) Y j ∈ J { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R j } + µ ( k ) Y j ∈ J { A ∈ M n ( k ) ( C ) : k A k ∞ ≤ R j } c . So X k (cid:16) ⊗ s µ ( s ) (cid:17) (cid:16) { ( A ( s ) ) s : A ( k ) ∈ Γ ( n ( k )) R ( O m ) c } (cid:17) < ∞ . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 35 Hence Ω = \ m [ k \ l ≥ k { ( A ( s ) ) s : A ( l ) ∈ Γ ( n ( l )) R ( O m ) } is a conull subset of Q k M n ( k ) ( C ) . By construction, for every A ∈ Ω we have ℓ A ( k ) → ℓ x . (ii): Fix a countable set J ′ and a tuple y ∈ M J ′ with W ∗ ( y ) = P. Choose an R ∈ [0 , ∞ ) J with k x j k ∞ ≤ R j for all j ∈ J and an f ∈ ( F R,J, ∞ ) J ′ with f ( x ) = y. By Lemma 4.2 and Theorem 2.16 (iii), for any freeultrafilter ω, and any A = ( A ( k ) ) ∈ Ω we haveΘ A,ω ( y ) = Θ A,ω ( f ( x )) = f (Θ A,ω ( x )) = ( f ( A ( k ) )) k → ω . So it suffices to find a conull subset Ω of Q k M n ( k ) ( C ) J so that for all A = ( A ( k ) ) , B = ( B ( k ) ) ∈ Ω there is asequence of unitaries U ( k ) ∈ M n ( k ) ( C ) so that k U ( k ) f j ′ ( A ( k ) )( U ( k ) ) ∗ − f j ′ ( B ( k ) ) k → k →∞ , for all j ′ ∈ J ′ .Since J is countable, by a diagonal argument it is sufficient to show that for every ε > , and for every finite F ′ ⊆ J, there is a conull subset Υ F ′ ,ε of Q k Prob( M k ( C ) J ) so that for all A = ( A ( k ) ) k , B = ( B ( k ) ) k ∈ Υ F ′ ,ε we have lim sup k →∞ d orb F ′ ( f ( A ( k ) ) , f ( B ( k ) )) ≤ ε. So fix an ε > F ′ ⊆ J ′ . Then by Theorem 2.16 (ii), we may find a δ > F ⊆ J so that if ( M, τ ) is any tracial von Neumann algebra, and if a, b ∈ Q j ∈ J { c ∈ M : k c k ∞ ≤ R j } satisfy k a − b k ,F < δ, then k f ( a ) − f ( b ) k ,F ′ < ε/ . Since f commutes with unitary conjugation by Theorem 2.16(iii), it follows that for every n ∈ N , and all A, B ∈ Q j ∈ J { C ∈ M n ( C ) : k C k ∞ ≤ R j } with d orb F ( A, B ) < δ, we have d orb F ′ ( f ( A ) , f ( B )) < ε/ . By Theorem 3.3, we may choose a sequence e Υ k ⊆ Q j ∈ J { C ∈ M n ( k ) ( C ) : k C k ∞ ≤ R j } with ν ( k ) ( e Υ k ) → k →∞ sup A ,A ∈ e Υ k d orb F ( f ( A ) , f ( A )) = 0 . Now choose K so that for all k ≥ K we have ν ( k ) ( e Υ k ) ≥ / A ,A ∈ e Υ k d orb F ( f ( A ) , f ( A )) < ε/ . Then, by exponential concentration, X k ≥ K ν ( k ) ( N δ ( e Υ k ) c ) < ∞ . As in part (i) we have X k ≥ K µ ( k ) ( N δ ( e Υ k ) c ) < ∞ . If A, B ∈ N δ ( e Υ k ) , choose A , B ∈ e Υ k with d orb F ( A, A ) , d orb F ( B, B ) < δ. Then, d orb F ′ ( f ( A ) , f ( B )) < ε/ d orb F ′ ( f ( A ) , f ( B )) < ε. So Υ F,ε = [ k ≥ K \ l ≥ k { ( A ( m ) ) m : A ( l ) ∈ N δ ( e Υ l ) } is N k µ ( k ) -conull and for all ( A ( k ) ) k , ( B ( k ) ) k ∈ Υ F,ε we havelim sup k →∞ d orb F ′ ( f ( A ( k ) ) , f ( B ( k ) )) ≤ ε. This completes the proof. (cid:3) The conclusion of Theorem 4.3 (ii) is interesting in light of the following theorem of Jung. Theorem 4.4 (Jung, [43]) . Let ( M, τ ) be a tracial von Neumann which admits an embedding into a tracialultraproduct of matrix algebras. Then given any nonamenable Q ≤ M and any free ultrafilter ω on N , thereare trace-preserving, normal ∗ -homomorphisms Θ j : M → Y k → ω M k , j = 1 , so that Θ (cid:12)(cid:12) Q is not unitarily equivalent to Θ (cid:12)(cid:12) Q . Conversely, if Q ≤ M is amenable, then any two embeddingsof Q into an ultraproduct of matrix algebras are unitarily equivalent. Strictly speaking, Jung only proved the case Q = M and when M is finitely generated of Theorem 4.4.However, by analyzing his proof and replacing microstates spaces with microstates spaces in the presence, itis not hard to prove the case Q is nonamenable and finitely generated of Theorem 4.4. Since any nonamenablevon Neumann algebra has a finitely generated nonamenable von Neumann subalgebra, this is sufficient tohandle the general case of Theorem 4.4.Under the hypotheses of Theorem 4.3, if for every nonamenable P ≤ M, almost every ( A, B ) ∈ Ω andevery free ultrafilter ω on the natural numbers, we had that Θ A,ω (cid:12)(cid:12) P and Θ B,ω (cid:12)(cid:12) P are not unitarily conjugate,then it would follow that any Q ≤ M with h ( Q : M ) ≤ M must have a maximal amenable extension. We can think of the statement that almost surelyΘ A,ω (cid:12)(cid:12) P is not unitarily equivalent to Θ B,ω (cid:12)(cid:12) P as a “randomized Jung theorem”. It would mean that not onlycan we find a pair of homomorphisms satisfying the conclusion of Jung’s theorem, but that a randomlychosen pair satisfies Jung’s theorem. This motivates the following conjecture. Conjecture 4. Fix an integer r ≥ , and let µ ( k ) ∈ Prob( M k ( C ) rs.a. ) be the r -fold product of the GUEdistribution. Set µ = Q k µ ( k ) , and choose a µ -conull Ω ⊆ Q k M k ( C ) rs.a. as in Theorem 4.3. Then for every P ≤ L ( F r ) nonamenable and for every free ultrafilter ω on N , there is a µ ⊗ µ -conull subset Ω ⊆ Ω × Ω sothat Θ A,ω (cid:12)(cid:12) P , Θ B,ω (cid:12)(cid:12) P are not unitarily conjugate for all ( A, B ) ∈ Ω . Related to Jung’s theorem, we can use strong convergence and local reflexivity to give criteria so that aconcrete pair of embeddings into ultraproducts of matrices are not unitarily conjugate when restricted toany nonamenable subalgebra. Proposition 4.5. Let ( M, τ ) be a tracial von Neumann algebra, I an index set and x ∈ M I with W ∗ ( x ) = M. Suppose we are given positive integers n ( k ) → ∞ and ( A ( k ) , B ( k ) ) ∈ M n ( k ) ( C ) I so that the law of ( A ( k ) ⊗ M n ( k ) ( C ) , M n ( k ) ( C ) ⊗ ( B ( k ) ) t ) converges strongly to the law of ( x ⊗ C ∗ ( x ) op , C ∗ ( x ) ⊗ x op ) . For a free RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 37 ultrafilter ω, let Θ A,ω : M → Q k → ω M n ( k ) ( C ) , Θ B,ω : M → Q k → ω M n ( k ) ( C ) be the unique trace-preserving,normal ∗ -homomorphisms which satisfy Θ A,ω ( x ) = ( A ( k ) ) k → ω , Θ B,ω ( x ) = ( B ( k ) ) k → ω . If C ∗ ( x ) is locally reflexive, then for any nonamenable Q ≤ M we have that Θ A,ω (cid:12)(cid:12) Q and Θ B,ω (cid:12)(cid:12) Q are notunitarily conjugate.Proof. Note that strong convergence implies that for all i ∈ I,R i = sup k max( k A ( k ) i k ∞ , k B ( k ) i k ∞ ) < ∞ . From here, it is an exercise to derive this from Proposition 3.6. (cid:3) Recall that if x ∈ M n ( C ) ⊗ M n ( C ) then we have an operator x M n ( C ) → M n ( C ) defined on elementarytensors by ( A ⊗ B ) C = ACB t . Moreover, x x ∗ -homomorphism M n ( C ) ⊗ M n ( C ) → B ( S ( n, tr)) and as such it isisometric. So k x k M n ( C ) ⊗ M n ( C ) = k x k B ( S ( n, tr)) and this is precisely what we used in our reduction to strong convergence. However, it is natural to view x L p -spaces. Recall that if 1 ≤ p < ∞ , then we have a norm k · k p on M n ( C ) by k A k p = tr( | A | p ) /p , with | A | = ( A ∗ A ) / .As usual, we let k A k ∞ be the operator norm of A ∈ M n ( C ) . We let S p ( n, tr) be M n ( C ) equipped with thenorm k · k p , and for x ∈ M n ( C ) ⊗ M n ( C ) and 1 ≤ p, q ≤ ∞ , we let k x k p,q be the norm of the operator A x A as an operator S p ( n, tr) → S q ( n, tr) . So our discussion above shows that k x k , = k x k M n ( C ) ⊗ M n ( C ) . Because we are using the normalized trace, we have that k A k p ≤ k A k q for 1 ≤ p ≤ q ≤ ∞ and A ∈ M n ( C ) . So k x k p ,q ≤ k x k p ,q if p ≤ p , q ≤ q . We now state a conjecture weaker than our strong convergence conjecture in terms ofoperator norms M n ( C ) → S ( n, tr) . Conjecture 5. Fix an integer r ≥ . Then there is a constant C > with the following property. Let X ( k )1 , X ( k )2 , · · · , X ( k ) r , Y ( k )1 , Y ( k )2 , · · · , Y ( k ) r be random, self-adjoint k × k matrices which are independent andare each GUE distributed. Let s = ( s , · · · , s r ) be a free semicircular family each with mean zero and variance . Then for any P ∈ C h T , · · · , T r i we have that lim sup k →∞ k P ( X ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ Y ( k ) ) k ∞ , ≤ C k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ , where the norm on the right-hand side is taken in C ∗ ( s ) ⊗ min C ∗ ( s ) . Proposition 4.6. We have the following implications between the above conjectures and the Peterson-Thomconjecture. Conjecture 2 implies Conjecture 5, Conjecture 5 implies Conjecture 4, Conjecture 4 impliesConjecture 3, and Conjecture 3 implies the Peterson-Thom conjecture.Proof. Conjecture 2 implies Conjecture 5: Take C = 1 , and use that k x k ∞ , ≤ k x k , = k x k M n ( C ) ⊗ M n ( C ) for all n ∈ N , and all x ∈ M n ( C ) ⊗ M n ( C ) . Conjecture 5 implies Conjecture 4: It is well known (see [30, Proof of Lemma 3.3]) that we may find an R > k →∞ k log µ ( k ) r Y j =1 { A ∈ M k ( C ) : k A k ∞ ≤ R } c < . Suppose Q ≤ L ( F r ) is nonamenable, and apply [28, Lemma 2.2] to find a nonzero projection f ∈ Z ( Q ) and u , · · · , u r ∈ U ( Qf ) so D ′ = 1 r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 u j ⊗ u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < . By replacing (cid:16) r P rj =1 u j ⊗ u j (cid:17) with (cid:16) r P rj =1 u j ⊗ u j (cid:17) s for a suitably large s ∈ N , we may, and will, assumethat D ′ < τ ( f ) C . Let Ω be as in Theorem 4.3 (i). By Conjecture 5, we may choose a conull Ξ ⊆ Ξ × Ω sothat for all A = ( A ( k ) ) , B = ( B ( k ) ) ∈ Ξ , we havelim sup k →∞ k P ( A ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ B ( k ) ) k ∞ , ≤ C k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ . Suppose that the negation of Conjecture 4 holds. Then there is a positive measure Υ ⊆ Ξ and a freeultrafilter ω on N so that for all ( A, B ) ∈ Υ we have that Θ A,ω (cid:12)(cid:12) Q and Θ B,ω (cid:12)(cid:12) Q are unitarily conjugate. Fix( A, B ) ∈ Υ . Let v ∈ U ( Q k → ω M k ( C )) be such that v Θ B,ω ( x ) v ∗ = Θ A,ω ( x ) for all x ∈ Q ,and write v = ( V ( k ) ) k → ω with V ( k ) ∈ U ( k ) . Observe that for all ( X k ) k → ω ∈ Q k → ω M k ( C ) we have k ( X k ) k → ω k = lim k → ω k X k k . Indeed, this follows from the fact that | ( X k ) k → ω | = ( | X k | ) k → ω , which is in turn a consequence of the factthat continuous functional calculus commutes with operation of passing to the ultraproduct. Since C ∗ ( s ) isexact, and thus locally reflexive, as in the proof of Theorem 1.2 (ii)) we may choose a D ∈ ( D ′ , τ ( f ) C ) and asequence P j,m ∈ C h ( T j ) j ∈ J i so that • k P j,m k R, ∞ ≤ , • k P j,m ( s ) − u j k → m →∞ , • r (cid:13)(cid:13)(cid:13)P rj =1 P j,m ( s ) ⊗ P j,m ( s ) (cid:13)(cid:13)(cid:13) ∞ ≤ D for all m. Note that as a consequence of the second item k P j,m ( s ) − u j k ≤ k P j,m ( s ) − u j k → m →∞ . RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 39 Then for every m ∈ N ,τ ( f ) = k vf k = 1 r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 Θ A,ω ( u j ) v Θ B,ω ( f )Θ B,ω ( u j ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ r r X j =1 k P j,m ( s ) − u j k + 1 r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 P j,m (( A ( k ) ) k → ω ) v Θ B,ω ( f ) P j,m (( B ( k ) ) k → ω ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , where in the last step we use that k P j,m ( s ) k ∞ ≤ A,ω ,Θ B,ω are k · k − k · k , k · k ∞ − k · k ∞ isometries. Write Θ B,ω ( f ) = ( F ( k ) ) k → ω where F ( k ) are projections in M n ( k ) ( C ). We can estimate the secondterm above as follows:1 r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 P j,m (( A ( k ) ) k → ω ) v Θ B,ω ( f ) P j,m (( B ( k ) ) k → ω ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim k → ω r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 P j,m ( A ( k ) ) V ( k ) F ( k ) P j,m ( B ( k ) ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ lim sup k →∞ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 P j,m ( A ( k ) ) ⊗ P j,m ( B ( k ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ , ≤ Cr (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X j =1 P j,m ( s ) ⊗ P j,m ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ CD. So we have shown that for every m ∈ N we have τ ( f ) − CD ≤ r r X j =1 k P j,m ( s ) − u j k . Since D < τ ( f ) C , we obtain a contradiction by letting m → ∞ . Conjecture 4 implies Conjecture 3: This follows from Theorem 4.3 (ii).Conjecture 3 implies the Peterson-Thom conjecture: This is the content of Proposition 2.7. (cid:3) We remark that it is likely helpful to consider operator spaces and operator space tensor products to tackleConjecture 5. For instance, one can imagine that instead of working with M n ( C ) ⊗ M n ( C ) one considers S p ( n, tr) ⊗ α S q ( n, tr) for some p, q ∈ [1 , ∞ ] and some operator space tensor product ⊗ α . One would wantto choose α so the map S p ( n, tr) ⊗ α S q ( n, tr) → CB ( M n ( C ) , S ( n, tr)) given by A ⊗ B ( C ACB t ) iscompletely bounded. It is natural to choose p, q with p + q = 1 so that k ACB t k ≤ k A k p k B k ∞ k C k q . Thus it would make sense to consider an operator space tensor norm on S ( n, tr) ⊗ M n ( C ) or on OS ( n, tr) ⊗ OS ( n, tr) where OS ( n, tr) is Pisier’s operator space structure on S ( n, tr) (or potentially other naturaloperator space structures on S ( n, tr)). We close this section by mentioning that the full strength of Conjecture 2 is not needed to deduceConjecture 4. In fact, we only need that for all P , · · · , P l ∈ C h T , · · · , T r i we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l X j =1 P l ( X ( k ) ) ⊗ P j ( Y ( k ) ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l X j =1 P j ( s ) ⊗ P j ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ . And so we can allow a certain symmetry in the elements of C h T , · · · , T r , S , · · · , S r i we are testing strongconvergence on. Similar remarks apply to the other conjectures in this section. Lastly, in Conjectures 2, 4,5we may replace the GUE ensemble with Haar unitaries, or any other ensemble provided it has exponentialconcentration, and converges in law to the law of a generator x of a free group factor with the propertythat C ∗ ( x ) is locally reflexive. The details as to why these alternate conjectures imply the Peterson-Thomconjecture are the same as in Proposition 4.6.5. Closing Remarks We close with some comments around Theorem 1.1. First is that in Theorem 1.1 (iii) it is crucial thatwe are taking X ( k ) , Y ( k ) independent of each other. In fact, tensoring tends to behave rather poorly in thestrong topology, as we now show. Definition 5.1. Let ( M, τ ) be a tracial von Neumann algebra, and J a countable index set. We say that x ∈ M J is a nonamenability tuple if • sup j k x j k ∞ < ∞ , • there is a µ ∈ Prob( J ) so that P j ∈ J µ j | x j ⊗ − ⊗ x opj | ∈ M ⊗ M op is invertible.The sum in question in the second item converges in k · k ∞ -norm. By [17] (see also [1, Theorem 10.2.9]),every nonamenable von Neumann algebra admits a finite nonamenability tuple. Proposition 5.2. Let ( M, τ ) be a tracial von Neumann algebra, J a countable index set and x ∈ M J . Suppose either that x is a nonamenability tuple, or that W ∗ ( x ) is nonamenable and that C ∗ ( x ) is locallyreflexive. Fix an R > with sup j k x j k ∞ < ∞ . Given any sequence n ( k ) ∈ N , and x k ∈ M k ( C ) J with sup k,j k x k,j k ∞ ≤ R and ℓ x k → ℓ x strongly we have that ℓ x k ⊗ , ⊗ x tk does not converge strongly to ℓ x ⊗ , ⊗ x op . Proof. The case that C ∗ ( x ) is locally reflexive and that W ∗ ( x ) is nonamenable follows from Proposition 4.5,so we assume that x is a nonamenability set. Let µ ∈ Prob( J ) be so that X j ∈ J µ j | x j ⊗ − ⊗ x opj | is invertible. Since invertible elements in a Banach algebra are open and the sum above converges in k · k ∞ , it follows that we may choose a finite F ⊆ J so that X j ∈ F µ j | x j ⊗ − ⊗ x opj | is invertible. RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 41 By strong convergence and Lemma 2.9, the spectrum of P j ∈ F µ j | x k,j ⊗ − ⊗ x tk,j | Hausdorff convergesto the spectrum of P j ∈ F µ j | x j ⊗ − ⊗ x opj | . Since 0 is not in the spectrum of P j ∈ F µ j | x j ⊗ − ⊗ x opj | ,it follows that 0 is not in the spectrum of P j ∈ F µ j | x k,j ⊗ − ⊗ x tk,j | for all sufficiently large k. But since X j ∈ F µ j | x k,j ⊗ − ⊗ x tk,j | , we have that 0 is in the spectrum of P j ∈ J µ j | x k,j ⊗ − ⊗ x tk,j | for all k. So we have a contradiction, andthis completes the proof. (cid:3) More positively, we remark that many previous proofs of strong convergence (e.g. for a mixture ofdeterministic and random matrices see [49, 16]) involve replacing some coordinates of the tuple with theirstrong limits. A similar approach holds here. Proposition 5.3. Let r ≥ be an integer and s = ( s , · · · , s r ) a free semicircular family each with meanzero and variance one. Let X ( k ) be as in Theorem 1.1 (iii). In order to prove Theorem 1.1, it is enough toshow that for any P ∈ C h ( T j ) rj =1 , ( S j ) rj =1 i we have (cid:12)(cid:12)(cid:12) k P ( X ( k ) ⊗ M k ( C ) , M k ( C ) ⊗ Y ( k ) ) k ∞ − k P ( X ( k ) ⊗ C ∗ ( s ) , M k ( C ) ⊗ s ) k ∞ (cid:12)(cid:12)(cid:12) → in probability.Proof. It suffices to show that k P ( X ( k ) ⊗ C ∗ ( s ) , M k ( C ) ⊗ s ) k ∞ → k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ in probability. Let µ ( k ) ∈ Prob( M k ( C ) rs.a ) be the distribution of ( X ( k ) ) . By Haagerup-Thorjbørnsen [29,Theorem A], we may find a sequence Ω k ⊆ M k ( C ) rs.a. so that • µ ( k ) (Ω k ) → , • for all ( A ( k ) ) k ∈ Q k Ω k we have that ℓ A ( k ) → ℓ s strongly.Let B = { ( a k ) k ∈ Y k M k ( C ) : sup k k a k k ∞ < ∞} , J = { ( a k ) k ∈ Y k M k ( C ) : k a k k ∞ → k →∞ } , and set A = B/J . Then A is a C ∗ -algebra under the norm k ( a k ) k + J k = lim sup k →∞ k a k k ∞ and we have an exact sequence of C ∗ -algebras(9) 0 −−−−→ J −−−−→ B −−−−→ A −−−−→ . For ( A ( k ) ) k ∈ Q k Ω k , strong convergence guarantees that we have a ∗ -homomorphism π : C ∗ ( s ) → B satisfying π ( P ( s )) = ( P ( A ( k ) )) k + J for all P ∈ C h T , T , · · · , T r i . We thus have a natural ∗ -homomorphism(10) π ⊗ id : C ∗ ( s ) ⊗ min C ∗ ( s ) → B ⊗ min C ∗ ( s ) . Since C ∗ ( s ) is exact, the exact sequence (9) produces an exact sequence(11) 0 −−−−→ J ⊗ min C ∗ ( s ) −−−−→ B ⊗ min C ∗ ( s ) −−−−→ A ⊗ min C ∗ ( s ) −−−−→ . We have a natural identification J ⊗ min C ∗ ( s ) ∼ = ( ( a k ) k ∈ Y k ( M k ( C ) ⊗ min C ∗ ( s )) : k a k k ∞ → ) , and a natural isometric embedding B ⊗ min C ∗ ( s ) ֒ → ( ( a k ) k ∈ Y k ( M k ( C ) ⊗ min C ∗ ( s )) : sup k k a k k ∞ < ∞ ) . Combining this with (10) , (11) , we have produced a ∗ -homomorphism C ∗ ( s ) ⊗ C ∗ ( s ) → { ( a k ) k ∈ Q k ( M k ( C ) ⊗ min C ∗ ( s )) : sup k k a k k ∞ < ∞}{ ( a k ) k ∈ Q k M k ( C ) ⊗ min C ∗ ( s ) : k a k k ∞ → } satisfying P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) ( P ( A ( k ) ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s )) k + ( ( a k ) k ∈ Y k ( M k ( C ) ⊗ min C ∗ ( s )) : k a k k ∞ → ) for all P ∈ C h T , · · · , T r , S , · · · , S r i . Since ∗ -homomorphisms between C ∗ -algebras are contractive, thisimplies that k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ ≤ lim sup k →∞ k P ( A ( k ) ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ . The inequality k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ ≥ lim inf k →∞ k P ( A ( k ) ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ is a consequence of the weak ∗ -convergence of the law of ( A ( k ) ⊗ C ∗ ( s ) , M k ( C ) ⊗ s ) to the law of ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ). So we have shown k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ = lim k →∞ k P ( A ( k ) ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ for all ( A ( k ) ) k ∈ Q k Ω k and all P ∈ C h T , · · · , T r , S , · · · , S r i . Since µ ( k ) (Ω k ) → , it follow that k P ( X ( k ) ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ → k P ( s ⊗ C ∗ ( s ) , C ∗ ( s ) ⊗ s ) k ∞ in probability. (cid:3) References [1] C. Anantharaman and S. Popa. An introduction to II factors. book in progress , 2016.[2] M. F. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. pages 43–72. Ast´erisque, No. 32–33, 1976.[3] S. Atkinson and S. K. Elayavalli. On ultraproduct embeddings and amenability for tracial von neumann algebras, 2019.[4] C. Bordenave and B. Collins. Eigenvalues of random lifts and polynomial of random permutations matrices. arXiv:1801.00876 .[5] J. Bosa, N. Brown, Y. Sato, A. Tikuisis, S. White, and W. Winter. Covering dimension of C ∗ -algebras and -colouredclassification , volume 257 of Mem. Amer. Math. Soc. Amer. Math. Soc., 2019.[6] R. Boutonnet and A. Carderi. Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. Geometric and Functional Analysis , 25(6):1688–1705, Dec 2015. RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 43 [7] R. Boutonnet and C. Houdayer. Amenable absorption in amalgamated free product von Neumann algebras. Kyoto J.Math. , 58(3):583–593, 07 2018.[8] A. Brothier. The cup subalgebra of a II factor given by a subfactor planar algebra is maximal amenable. Pacific J. Math. ,269(1):19–29, 2014.[9] A. Brothier and C. Wen. The cup subalgebra has the absorbing amenability property. Internat. J. Math. , 27(2):1650013,6, 2016.[10] N. P. Brown and N. Ozawa. C ∗ -algebras and finite-dimensional approximations , volume 88 of Graduate Studies in Math-ematics . American Mathematical Society, Providence, RI, 2008.[11] J. Cameron, J. Fang, M. Ravichandran, and S. White. The radial masa in a free group factor is maximal injective. J. Lond.Math. Soc. (2) , 82(3):787–809, 2010.[12] J. Castillejos, S. Evington, A. Tikuisis, and S. White. Classifying maps into uniform tracial sequence algebras, 2020.[13] J. Castillejos, S. Evington, A. Tikuisis, S. White, and W. Winter. Nuclear dimension of simple c*-algebras, 2019.[14] I. Chifan and T. Sinclair. On the structural theory of II factors of negatively curved groups. Ann. Sci. ´Ec. Norm. Sup´er.(4) , 46(1):1–33 (2013), 2013.[15] B. Collins, A. Guionnet, and F. Parraud. On the operator norm of non-commutative polynomials in deterministic matricesand iid gue matrices. arXiv:1912.04588 , 2019.[16] B. Collins and C. Male. The strong asymptotic freeness of Haar and deterministic matrices. Ann. Sci. ´Ec. Norm. Sup´er.(4) , 47(1):147–163, 2014.[17] A. Connes. Classification of injective factors. Cases II , II ∞ , III λ , λ = 1. Ann. of Math. (2) , 104(1):73–115, 1976.[18] K. J. Dykema. Two applications of free entropy. Math. Ann. , 308(3):547–558, 1997.[19] K. J. Dykema. Exactness of reduced amalgamated free product C ∗ -algebras. Forum Math. , 16(2):161–180, 2004.[20] K. J. Dykema and D. Shlyakhtenko. Exactness of Cuntz-Pimsner C ∗ -algebras. Proc. Edinb. Math. Soc. (2) , 44(2):425–444,2001.[21] E. G. Effros and Z.-J. Ruan. Operator spaces , volume 23 of London Mathematical Society Monographs. New Series . TheClarendon Press, Oxford University Press, New York, 2000.[22] J. Fang. On maximal injective subalgebras of tensor products of von Neumann algebras. J. Funct. Anal. , 244(1):277–288,2007.[23] D. Gaboriau. Invariants l de relations d’´equivalence et de groupes. Publ. Math. Inst. Hautes ´Etudes Sci. , (95):93–150,2002.[24] M. Gao. On maximal injective subalgebras. Proc. Amer. Math. Soc. , 138(6):2065–2070, 2010.[25] L. Ge. On maximal injective subalgebras of factors. Adv. Math. , 118(1):34–70, 1996.[26] L. Ge. Applications of free entropy to finite von Neumann algebras. II. Ann. of Math. (2) , 147(1):143–157, 1998.[27] L. Ge and S. Popa. On some decomposition properties for factors of type II . Duke Math. J. , 94(1):79–101, 1998.[28] U. Haagerup. Injectivity and decomposition of completely bounded maps. In Operator algebras and their connections withtopology and ergodic theory (Bu¸steni, 1983) , volume 1132 of Lecture Notes in Math. , pages 170–222. Springer, Berlin, 1985.[29] U. Haagerup and S. Thorbjø rnsen. A new application of random matrices: Ext( C ∗ red ( F )) is not a group. Ann. of Math.(2) , 162(2):711–775, 2005.[30] U. Haagerup and S. Thorbjørnsen. Random matrices with complex Gaussian entries. Expo. Math. , 21(4):293–337, 2003.[31] B. Hayes. Polish models and sofic entropy. J. Inst. Math. Jussieu to appear .[32] B. Hayes. 1-bounded entropy and regularity problems in von Neumann algebras. Int. Math. Res. Not. IMRN , (1):57–137,2018.[33] B. Hayes, D. Jekel, B. Nelson, and T. Sinclair. A random matrix approach to absorption in free products. arXiv:1912.11569 ,2019.[34] C. Houdayer. A class of II factors with an exotic abelian maximal amenable subalgebra. Trans. Amer. Math. Soc. ,366(7):3693–3707, 2014. [35] C. Houdayer. Structure of II factors arising from free Bogoljubov actions of arbitrary groups. Adv. Math. , 260:414–457,2014.[36] C. Houdayer. Gamma stability in free product von Neumann algebras. Comm. Math. Phys. , 336(2):831–851, 2015.[37] C. Houdayer and D. Shlyakhtenko. Strongly solid I I factors with an exotic MASA. Int. Math. Res. Not. IMRN , (6):1352–1380, 2011.[38] A. Ioana. Classification and rigidity for von Neumann algebras. In European Congress of Mathematics , pages 601–625.Eur. Math. Soc., Z¨urich, 2014.[39] A. Ioana. Rigidity for von Neumann algebras. In Proceedings of the International Congress of Mathematicians—Rio deJaneiro 2018. Vol. III. Invited lectures , pages 1639–1672. World Sci. Publ., Hackensack, NJ, 2018.[40] D. Jekel. An elementary approach to free entropy theory for convex potentials. arXiv:1805.08814 , 2018. To appear inAnalysis and PDE.[41] D. Jekel. Conditional expectation, entropy, and transport for convex gibbs laws in free probability. arXiv:1906.10051 , 2019.[42] D. Jekel. Evolution equations in non-commutative probability . PhD thesis, University of California, Los Angeles, 2020.[43] K. Jung. Amenability, tubularity, and embeddings into R ω . Math. Ann. , 338(1):241–248, 2007.[44] K. Jung. Strongly 1-bounded von Neumann algebras. Geom. Funct. Anal. , 17(4):1180–1200, 2007.[45] E. Kirchberg. Commutants of unitaries in UHF algebras and functorial properties of exactness. J. Reine Angew. Math. ,452:39–77, 1994.[46] E. Kirchberg. On subalgebras of the CAR-algebra. J. Funct. Anal. , 129(1):35–63, 1995.[47] B. Leary. Maximal Amenability with Asymptotic Orthogonality in Amalgamated Free Products. arXiv e-prints , pagearXiv:1912.06061, Dec 2019.[48] M. Ledoux. The concentration of measure phenomenon , volume 89 of Mathematical Surveys and Monographs . AmericanMathematical Society, Providence, RI, 2001.[49] C. Male. The norm of polynomials in large random and deterministic matrices. Probab. Theory Related Fields , 154(3-4):477–532, 2012. With an appendix by Dimitri Shlyakhtenko.[50] E. S. Meckes and M. W. Meckes. Spectral powers of random matrices. Electron. Comm. Probab. , 18(78), 2013.[51] N. Ozawa. Dixmier approximation and symmetric amenability for C ∗ -algebras. J. Math. Sci. Univ. Tokyo , 20:349–374,2013.[52] N. Ozawa. A remark on amenable von Neumann subalgebras in a tracial free product. Proc. Japan Acad. Ser. A Math.Sci. , 91(7):104, 2015.[53] N. Ozawa and S. Popa. On a class of II factors with at most one Cartan subalgebra. Ann. of Math. (2) , 172(1):713–749,2010.[54] N. Ozawa and S. Popa. On a class of II factors with at most one Cartan subalgebra, II. Amer. J. Math. , 132(3):841–866,2010.[55] S. Parekh, K. Shimada, and C. Wen. Maximal amenability of the generator subalgebra in q -Gaussian von Neumannalgebras. J. Operator Theory , 80(1):125–152, 2018.[56] J. Peterson. L -rigidity in von Neumann algebras. Invent. Math. , 175(2):417–433, 2009.[57] J. Peterson and A. Thom. Group cocycles and the ring of affiliated operators. Invent. Math. , 185(3):561–592, 2011.[58] G. Pisier. Random matrices and subexponential operator spaces. Israel J. Math. , 203(1):223–273, 2014.[59] S. Popa. Maximal injective subalgebras in factors associated with free groups. Adv. in Math. , 50(1):27–48, 1983.[60] S. Popa. On a class of type II factors with Betti numbers invariants. Ann. of Math. (2) , 163(3):809–899, 2006.[61] S. Popa. Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. , 230(2):273–328, 2006.[62] S. Popa. Strong rigidity of II factors arising from malleable actions of w -rigid groups. I. Invent. Math. , 165(2):369–408,2006.[63] S. Popa. Strong rigidity of II factors arising from malleable actions of w -rigid groups. II. Invent. Math. , 165(2):409–451,2006. RANDOM MATRIX APPROACH TO THE PETERSON-THOM CONJECTURE 45 [64] S. Popa. Deformation and rigidity for group actions and von Neumann algebras. In International Congress of Mathemati-cians. Vol. I , pages 445–477. Eur. Math. Soc., Z¨urich, 2007.[65] J. Shen. Maximal injective subalgebras of tensor products of free group factors. J. Funct. Anal. , 240(2):334–348, 2006.[66] M. Takesaki. Theory of operator algebras. I , volume 124 of Encyclopaedia of Mathematical Sciences . Springer-Verlag,Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5.[67] S. Vaes. Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa). Ast´erisque , (311):Exp.No. 961, viii, 237–294, 2007. S´eminaire Bourbaki. Vol. 2005/2006.[68] S. Vaes. Rigidity for von Neumann algebras and their invariants. In Proceedings of the International Congress of Mathe-maticians. Volume III , pages 1624–1650. Hindustan Book Agency, New Delhi, 2010.[69] D. Voiculescu. Limit laws for random matrices and free products. Invent. Math. , 104(1):201–220, 1991.[70] D. Voiculescu. The analogues of entropy and of Fisher’s information measure in free probability theory. II. Invent. Math. ,118(3):411–440, 1994.[71] D. Voiculescu. The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absenceof Cartan subalgebras. Geom. Funct. Anal. , 6(1):172–199, 1996.[72] D. Voiculescu. A strengthened asymptotic freeness result for random matrices with applications to free entropy. Internat.Math. Res. Notices , (1):41–63, 1998.[73] D. V. Voiculescu, K. J. Dykema, and A. Nica. Free random variables , volume 1 of CRM Monograph Series . AmericanMathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applicationsto random matrices, operator algebras and harmonic analysis on free groups.[74] C. Wen. Maximal amenability and disjointness for the radial masa. J. Funct. Anal. , 270(2):787–801, 2016. University of Virginia, Charlottesville, VA 22904 E-mail address ::