Notes on Microstate Free Entropy of Projections

Hiai, Fumio; Ueda, Yoshimichi

doi:10.2977/prims/1207921076

Cited by 6 publications

(13 citation statements)

References 29 publications

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“…The next definition is a natural generalization of the projection free entropy χ proj introduced and studied in [11] following Voiculescu's proposal in [26, 14.2]. Definition 2.3.…”

Section: Lemma 13 (Voiculescu [25])mentioning

confidence: 99%

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Orbital Approach to Microstate Free Entropy

2009

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Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy χ orb for non-commutative self-adjoint random variables (also for "hyperfinite random multivariables"). Besides its basic properties, the relation of χ orb with the usual free entropy χ is shown. Moreover, the dimension counterpart δ 0,orb of χ orb is discussed, and we obtain the relation between δ 0,orb and the original free entropy dimension δ 0 together with applications to δ 0 itself. Throughout this section, let (M, τ) be a tracial W * -probability space and (X 1 , . . . , X n ) be an n-tuple of self-adjoint random variables in (M, τ). We will use the standard notations such as the microstate set Γ R (X 1 , . . . , X n ; N, m, δ) appearing in the course of defining the microstate free entropy χ(X 1 , . . . , X n ) (see [22]). We define a free entropy-like quantity as follows.Definition 2.1. For each δ > 0, m, N ∈ N, R > 0 and 1 ≤ i ≤ n, we denote byand by Γ orb,R (X 1 , . . . , X n ; N, m, δ) the set of all n-tuples (U 1 , . . . , U n ) of N × N unitary matrices such that there exists an n-tuple (D 1 , . . . , D n ) in n i=1 ∆ R (X i ; N, m, δ) satisfying

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“…The next definition is a natural generalization of the projection free entropy χ proj introduced and studied in [11] following Voiculescu's proposal in [26, 14.2]. Definition 2.3.…”

Section: Lemma 13 (Voiculescu [25])mentioning

confidence: 99%

“…To prove the theorem, we will provide a certain transportation cost inequality similarly to the projection case in [11,Sec. 5].…”

Section: Characterization Of Freeness By χ Orb =mentioning

confidence: 99%

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Orbital Approach to Microstate Free Entropy

2009

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“…This concept was used to define several important regularized versions of measures associated to free entropy and free information theory, and to this days plays an important role in free probability theory. The special case that A, B are algebras generated by two projections has been extensively studied [15,16,17,18,19,25,26,27], as the best special case where one can hope to compute all quantities fairly explicitly.…”

Section: Strong Convergence Of the Processmentioning

confidence: 99%

Strong Convergence of Unitary Brownian Motion

Collins,

Dahlqvist,

Kemp

2015

Preprint

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The Brownian motion (U N t ) t≥0 on the unitary group converges, as a process, to the free unitary Brownian motion (u t ) t≥0 as N → ∞. In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time t > 0, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.

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“…, i r ) ∈ [n] r . Following Voiculescu's remarks in [31,Sect 14], in [17,18] Hiai and Petz defined the projection free entropy as follows: γ G(N,k i (N )) (Γ proj (p 1 , . .…”

Section: Free Entropy For Projectionsmentioning

confidence: 99%

Liberation of projections

Collins

Kemp

2014

Journal of Functional Analysis

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We study the liberation process for projections: (p, q) → (p t , q) = (u t pu * t , q) where u t is a free unitary Brownian motion freely independent from {p, q}. Its action on the operator-valued angle qp t q between the projections induces a flow on the corresponding spectral measures µ t ; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure µ t possesses a piecewise analytic density for any t > 0 and any initial projections of trace 1 2 . We us this to prove the Unification Conjecture for free entropy and information in this trace 1 2 setting.Let V and W be subspaces of a the finite-finite dimensional complex space C d . From elementary linear algebra, it follows that dim V ∩ W ≥ max{dim V + dim W − d, 0}. In fact, this inequality is almost surely equality:The almost surely can be interpreted in a number of ways: for example, it can be taken with respect to any reasonable probability measure on the (product) Grassmannian manifold of subspaces. We will discuss a more probabilistic interpretation shortly. When two subspaces satisfy the equality of Equation 1.1, they are said to be in general position. For convenience, we may rewrite this relation as follows. Let P, Q be the orthogonal projections onto V and W respectively, and let P ∧Q denote the orthogonal projection onto V ∩W . The dimension of a subspace is the trace of the projection onto it, so the subspaces are in general position if and only if Tr (P ∧ Q) = max{Tr P + Tr Q − d, 0}. Normalizing, letting tr = 1 d Tr , two projections (i.e. subspaces) are in general position if and only ifIn this language, a good way to express the genericity of general position is as follows: let P, Q be as above, and let U be a random unitary matrix. IfP = U P U * (i.e. the projection onto the rotation of the image of P by U ), thenP and Q are in general position almost surely. This statement is valid for all reasonable notions of random unitary matrix; for example, it holds true if U is sampled from some measure that has a continuous, strictly positive density with respect to the Haar measure. (Indeed, the subset of the Grassmannian product where general position fails to hold is a subvariety of lower dimension, and so it is easy to see that any reasonable measure will assign it probability 0.) Since any neighborhood of the identity has positive measure, it follows that rotations arbitrarily close to the identity produce projections in general position, agreeing with our intuition. Note: the same result applies as well even if P, Q are random projections, provided that the random unitary U is independent from {P, Q}.An important flow of random unitaries is given by the unitary Brownian motion. The group U d of unitary d×d matrices is a compact Lie group, and so its left-invariant Riemannian metric (given by the Hilbert-Schmidt norm on the Lie algebra u d ) gives rise to a heat kernel, which generates a Markov process: Brownian m...

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Notes on Microstate Free Entropy of Projections

Cited by 6 publications

References 29 publications

Orbital Approach to Microstate Free Entropy

Orbital Approach to Microstate Free Entropy

Strong Convergence of Unitary Brownian Motion

Liberation of projections

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