arXiv subjects
Publications and source records attributed to Yoshimichi Ueda.
Orbital counterparts of the free pressure and its Legendre transform (or $η$-entropy) are introduced and studied in comparison with other entropy quantities in free probability theory and in relation to random multi-matrix models.
We show that the continuous core of any type III free product factor has no Cartan subalgebra. This is a complement to previous works due to Houdayer--Ricard and Boutonnet--Houdayer--Raum.
Let $M$ be a von Neumann algebra and let $M_\star$ be its (unique) predual. We study when for every $φ\in M_\star$ there exists $ψ\in M_\star$ solving the equation $\|φ\pm ψ\|=\|φ\|=\|ψ\|$. This is the case when $M$ does not contain type I nor type III$_1$ factors as direct summands and it is false at least for the unique hyperfinite type III$_1$ factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of $M_\star$ of length 4. An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.
We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors.
The present notes provide a proof of $i^*(\mathbb{C}P+\mathbb{C}(I-P)\,;\mathbb{C}Q+\mathbb{C}(I-Q)) = -χ_\mathrm{orb}(P,Q)$ for any pair of projections $P,Q$ with $τ(P)=τ(Q)=1/2$. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.
Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.
We give a complete answer to the questions of factoriality, type classification and fullness for arbitrary free product von Neumann algebras.
Type III_1 factors arising as (direct summands of) von Neumann algebraic free products are investigated. In particular we compute Connes' Sd- and tau- invariants for those type III_1 factors without any extra assumption.
The unique predual $M_\star/A_\perp$ of a non-commutative $H^\infty$-algebra $A = H^\infty(M,τ)$ is investigated. In particular, we will prove the liftability property of weakly relatively compact subsets in $M_\star/A_\perp$ to $M_\star$.
A non-commutative extension of Amar and Lederer's peak set result is given. As its simple applications it is shown that any non-commutative $H^\infty$-algebra $H^\infty(M,τ)$ has unique predual,and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality.
With a minor change made in the previous construction we observe that any reduced HNN extension is precisely a compressed algebra of a certain reduced amalgamated free product in both the von Neumann algebra and the $C^*$-algebra settings. It is also pointed out that the same fact holds true even for universal HNN extensions of C$^*$-algebras. We apply the observation to the questions of factoriality and type classification of HNN extensions of von Neumann algebras and also those of simplicity and $K$-theory of (reduced or universal) HNN extensions of $C^*$-algebras.
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 multi-variables"). 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 of $δ_{0,orb}$ with the original free entropy dimension $δ_0$ with applications to $δ_0$ itself.
We prove an inequality between the free entropy and the mutual free Fisher information for two projections, regarded as a free analog of the logarithmic Sobolev inequality. The proof is based on the random matrix approximation procedure via the Grassmannian random matrix model of two projections.
We study the microstate free entropy of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of their free entropy (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to the microstate free entropy of projections.
We provide an operator algebraic interpretation of discrete measurable groupoids in the course of re-proving (and slightly generalizing) a result on treeability due to Adams and Spatzier. Then, we reconstruct Gaboriau's beautiful work on costs of equivalence relations in operator algebra framework, avoiding any measure theoretic argument, and clarify what kind of his results can or cannot be generalized to the non-principal groupoid case.
We prove the free analogue of the transportation cost inequality for tracial distributions of non-commutative self-adjoint (also unitary) multi-variables based on random matrix approximation procedure.
Reduced HNN extensions of von Neumann algebras (as well as $C^*$-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.
Biane proved the free analog of the logarithmic Sobolev inequality for probability measures on the real line by means of random matrix approximation procedure. We show that the same method can be applied to reprove Biane and Voiculescu's free analog of Talagrand's transportation cost inequality for measures on the real line. Furthermore, we prove the free analogs of the logarithmic Sobolev inequality and the transportation cost inequality for measures on 1-dimensional torus as well by extending the method to special unitary random matrices.