The Crossed Product of Finite Hopf C*-Algebra and C*-Algebra

Let H be a finite Hopf C*-algebra and A a C*-algebra of finite dimension. In this paper, we focus on the crossed product A⋊H arising from the action of H on A, which is a ∗-algebra. In terms of the faithful positive Haar measure on a finite Hopf C*-algebra, one can construct a linear functional on the ∗-algebra A⋊H, which is further a faithful positive linear functional. Here, the complete positivity of a positive linear functional plays a vital role in the argument. At last, we conclude that the crossed product A⋊H is a C*-algebra of finite dimension according to a faithful ∗- representation.

Integration Theory

The Lebesgue measure on ?n (presented in section 8.4) is a pivotal component of this chapter. The approach in the chapter is to extend the positive linear functional provided by the Riemann integral on the space of continuous, compactly supported functions on ?n (presented in section 8.1). An excursion on Radon measures is included at the end of section 8.4. The rest of the sections are largely independent of sections 8.1 and 8.4 and constitute a deep introduction to general measure and integration theories. Topics include measurable spaces and measurable functions, Carathéodory’s theorem, abstract integration and convergence theorems, complex measures and the Radon-Nikodym theorem, Lp spaces, product measures and Fubini’s theorem, and a good collection of approximation theorems. The closing section of the book provides a glimpse of Fourier analysis and gives a nice conclusion to the discussion of Fourier series and orthogonal polynomials started in section 4.10.

Oscillation results for second order matrix differential equations with damping

By using the positive linear functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order matrix differential equations (r(t)P(t)?(X(t))K(X'(t)))' + p(t)R(t)?(X(t))K(X'(t)) + Q(t)F(X'(t))G(X(t)) = 0,t ? t0 > 0. The results improve and generalize those given in some previous papers.

Polymer-Fourier quantization of the scalar field revisited

The polymer quantization of the Fourier modes of the real scalar field is studied within algebraic scheme. We replace the positive linear functional of the standard Poincaré invariant quantization by a singular one. This singular positive linear functional is constructed as mimicking the singular limit of the complex structure of the Poincaré invariant Fock quantization. The resulting symmetry group of such polymer quantization is the subgroup [Formula: see text] which is a subgroup of [Formula: see text] formed by spatial volume preserving diffeomorphisms. In consequence, this yields an entirely different irreducible representation of the canonical commutation relations, nonunitary equivalent to the standard Fock representation. We also compared the Poincaré invariant Fock vacuum with the polymer Fourier vacuum.