Geometrical constants of Day-James spaces (The generalization of function spaces and its enviroment)

We describe some recent results on the von Neumann-Jordan (NJ-) constant CNJ(X) and the related geometrical constants of concrete Banach spaces X. In particular, we calculate the constants for X being a class of Day-James spaces lp-lq by using the Banach-Mazur distance d(X, H) between X and H, where H is a two-dimensional inner product space

Schatten p-norm inequalities related to an extended operator parallelogram law

Let $\mathcal{C}_p$ be the Schatten $p$-class for $p>0$. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: If $\mathbf{A}=\{A_1,A_2,...,A_n\}$ and $\mathbf{B}=\{B_1,B_2,...,B_n\}$ are two sets of operators in $\mathcal{C}_2$, then \sum_{i,j=1}^n\|A_i-A_j\|_2^2 + \sum_{i,j=1}^n\|B_i-B_j\|_2^2 = 2\sum_{i,j=1}^n\|A_i-B_j\|_2^2 - 2\Norm{\sum_{i=1}^n(A_i-B_i)}_2^2. In this paper, we give generalizations of this as pairs of inequalities for Schatten $p$-norms, which hold for certain values of $p$ and reduce to the equality above for $p=2$. Moreover, we present some related inequalities for three sets of operators.Comment: 8 page

On the calculation of the James constant of Lorentz sequence spaces

In [M. Kato and L. Maligranda, On James and Jordan-von Neumannconstants of Lorentz sequence spaces, J. Math. Anal. Appl., 258(2001), 457–465], theJames constant of the 2-dimensional Lorentz sequence space d(2)(!; q) is computed inthe case where 2 · q < 1. It is an open problem to compute it in the case where1 · q < 2. In this paper, we completely determine the James constant of d(2)(!; q) inthe case where 1 · q < 2

A note on maximality of analytic crossed products

AbstractLet G be a compact abelian group with the totally ordered dual group Gˆ which admits the positive semigroup Gˆ+. Let N be a von Neumann algebra and α={αgˆ}gˆ∈Gˆ be an automorphism group of Gˆ on N. We denote N⋊αGˆ+ to the analytic crossed product determined by N and α. We show that if N⋊αGˆ+ is a maximal σ-weakly closed subalgebra of N⋊αGˆ, then Gˆ+ induces an archimedean order in Gˆ