Hideki Kosaki

A remark on the minimal index of subfactors

Let M ⫆ N be a factor-subfactor pair with a faithful normal conditional expectation E: M → N satisfying Ind E < + ∞. Let N ⫅ M ⫅ M1 ⫅ M2 ··· be the Jones tower of basic extensions, and Ek: Mk → Mk − 1 be the dual expectation constructed canonically from the given E. We prove that, if E: M → N is minimal, then so is the composition E ° E1 ° ··· ° Ek: Mk → N. Some consequences of this result are also presented.

IRREDUCIBLE BIMODULES ASSOCIATED WITH CROSSED PRODUCT ALGEBRAS

Crossed-product factors F⋊Γ⊃F⋊H, F⋊K and bimodules naturally arising from them are considered. Basic properties of these bimodules (necessary to study inclusion relations of factors) are established together with some applications.

Matrix Trace Inequalities Related to Uncertainty Principle

In their recent article, Luo and Zhang conjectured the matrix trace inequality mentioned in the introduction below, which is motivated by uncertainty principle. We present a proof for the conjectured inequality.

Automorphisms in the Irreducible Decomposion of Sectors

Automorphisms for an inclusion of factors with finite index are considered. For such automorphisms, we introduce the notion of strong outerness and obtain a characterization of the strong outerness based on the sector technique.

Index Theory for Type III Factors

We describe the structure of (finite-index) inclusion of type III factors based on analysis of involved flows of weights. Roughly speaking, a type HI index theory splits into a “purely type III” index theory and an (essentially) type II index theory. The factor flows constructed in [1] serve as the complete invariant for the former in the AFD case while the latter can be analyzed by paragroups or quantized groups (as announced in [7]). Therefore, classification of subfactors in an AFD type III factor reduces to classification of factor flows and an “equivariant” paragroup theory.

On equality condition for trace Jensen inequality in semi-finite von Neumann algebras

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

A certain generalization of the Heinz inequality for positive operators

Norm inequalities of the form 1 2 HαXK1−α + H1−αXKα ≤ 1 2n ∑m=0n nmHmnXKn−m n with n = 1, 2,… and α ∈ [0, 1] are studied. Here, H,K,X are operators with H,K ≥ 0 and ⋅ is an arbitrary unitarily invariant norm. We show that the inequality holds true if and only if 1 2(1 − 1 n) ≤ α ≤ 1 2(1 + 1 n).

TRACE JENSEN INEQUALITY FOR SELF-ADJOINT OPERATORS IN SEMI-FINITE VON NEUMANN ALGEBRAS

Let be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace τ, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality in our previous work states τ(f(a*xa)) ≤ τ(a*f(x)a) (as long as the both sides are well-defined) for a contraction and a semi-bounded τ-measurable operator x. Validity of this inequality for (not necessarily semi-bounded) self-adjoint τ-measurable operators is investigated.

8 Certain alternating sums of operators

8.1 Preliminaries 8.2 Uniform bounds for norms 8.3 Monotonicity of norms 8.4 Notes and references

7 Binomial means B α

7.1 Majorization Bα⪯M∞ 7.2 Equivalence of |||B α (H,K)X||| for α >0 7.3 Norm continuity in parameter 7.4 Notes and references