Orbital Approach to Microstate Free Entropy

Hiai, Fumio; Miyamoto, Takuho; Ueda, Yoshimichi

doi:10.1142/s0129167x09005261

Cited by 19 publications

(56 citation statements)

References 25 publications

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“…Arguing as in Lemma 4.2 of [20], we may find a m ′ ∈ N, γ ′ > 0 and a finite G 0 ⊆ Y so that Γ RX ∨RY (F : G 0 ; m ′ , γ ′′ , k) is contained in the unitary conjugation orbit of Ξ (A k ),RX ∨RY (F : G; m, γ, k).…”

Section: Appendix a Properties Of 1-bounded Entropymentioning

confidence: 99%

1-Bounded entropy and regularity problems in von Neumann algebras

Hayes

2016

Int Math Res Notices

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Abstract. We define and investigate the singular subspace Hs(N ⊆ M ) of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N -N subbimodule of L 2 (M ) containing the quasinormalizer introduced in [38], the one-sided quasinormalizer introduced in [11], and the wq-normalizer introduced in [14] (following upon work in [26] and [40]). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of N ⊆ M generates M. By abstracting Voiculescu's original proof of absence of Cartan subalgebras in [53] we show that there can be no diffuse, hyperfinite subalgebra of L(Fn) which is spectrally regular. Our techniques are robust enough to repeat this process by transfinite induction and rule out chains of spectrally regular inclusions of algebras starting from a diffuse, hyperfinite subalgebra and ending in L(Fn). We use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II 1 -factors (see [14]). Our results may be regarded as a consistency check for the possibility of existence of a "good" cohomology theory of II 1 -factors. We can also use our techniques to show that if Ut is a one-parameter orthogonal group on a real Hilbert space H and the spectral measure of its generator is singular with respect to the Lebesgue measure, then the continuous core of the free Araki-Woods factor Γ(H, Ut) ′′ is not isomorphic to L(Ft⊗B(ℓ 2 (N)) for any t ∈ (1, ∞]. In particular, Γ(H, Ut) ′′ ∼ = Γ(L 2 (R, m), λt) ′′ where m is Lebesgue measure and λ is the left regular representation. This was previously only know when the spectral measure of the generator of Ut had all of its convolution powers singular with respect to Lebesgue measure. We give similar applications to crossed products by free Bogoliubov actions in the spirit of [22].

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Section: Appendix a Properties Of 1-bounded Entropymentioning

confidence: 99%

1-Bounded entropy and regularity problems in von Neumann algebras

Hayes

2016

Int Math Res Notices

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“…However, we will give more 'exact' proofs to them later for the sake of completeness. In fact, (8) and (10) imply that lim N →∞…”

Section: Convergence Of Conditional Expectationmentioning

confidence: 99%

Matrix Liberation Process I: Large Deviation Upper Bound and Almost Sure Convergence

Ueda

2018

J Theor Probab

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We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution and several properties on its rate function. As a simple consequence we obtain the almost sure convergence of the empirical distribution of the matrix liberation process to that of the corresponding liberation process as continuous processes in the large N limit.

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“…Thus, t 0 s∂ s [H(s, φ s (z))]ds is a continuous function of z on V t as consequence of the theorem of continuity under integral sign. It follows from (4.2) that H(t, φ t (z)) extends continuously 6 to V t ∩ T. Now, the identity K(t, φ t (z)) = K(0, z), implies that H(t, φ t (z)) rewrites as…”

Section: Proof Of the Main Resultsmentioning

confidence: 98%