Some operator inequalities for Hermitian Banach $*$-algebras

In this paper, we extend the Kubo-Ando theory from operator means on C$^{*}$-algebras to a Hermitian Banach $*$-algebra $\mathcal {A}$ with a continuous involution. For this purpose, we show that if $a$ and $b$ are self-adjoint elements in $\mathcal {A}$ with spectra in an interval $J$ such that $a \leq b$, then $f(a) \leq f(b)$ for every operator monotone function $f$ on $J$, where $f(a)$ and $f(b)$ are defined by the Riesz-Dunford integral. Moreover, we show that some convexity properties of the usual operator convex functions are preserved in the setting of Hermitian Banach $*$-algebras. In particular, Jensen's operator inequality is presented in these cases.

Integral representations and properties of some functions involving the logarithmic function

By using the Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such as being operator monotone function, being complete Bernstein function, and being Stieltjes function, for these functions, and verify a conjecture on complete monotonicity of a function involving the logarithmic function.

On (T,f)-connections of matrices and generalized inverses of linear operators

In this note, generalized connections σ_{T ,f} are investigated, where Aσ_{T ,f} B = T_Af(T_A)^−(B) for positive semidefinite matrix A and hermitian matrix B, and operator monotone function f : J → R on an interval J ⊂ R. Here the symbol (T_A)^− denotes a reflexive generalized inverse of a positive bounded linear operator T_A. The problem of estimating a given generalized connection by other ones is studied. The obtained results are specified for special cases of α-arithmetic, α-geometric and α-harmonic operator means.

AN INEQUALITY RELATED TO UNCERTAINTY PRINCIPLE IN VON NEUMANN ALGEBRAS

Recently Kosaki proved an inequality for matrices that can be seen as a kind of new uncertainty principle. Independently, the same result was proved by Yanagi et al. The new bound is given in terms of Wigner–Yanase–Dyson informations. Kosaki himself asked if this inequality can be proved in the setting of von Neumann algebras. In this paper we provide a positive answer to that question and moreover we show how the inequality can be generalized to an arbitrary operator monotone function.