Polymeric complexes of cornstarch and waxy cornstarch phosphates with milk casein and their performance as biodegradable materials

Complexes of phosphated cornstarch and waxy cornstarch with casein were prepared and characterised. They were prepared from casein in defatted milk and corn and waxy corn starches phosphated to degree of substitution values (DS) of 0.0637 and 0.0968, respectively. The components were blended in starch to casein ratios of 2:1, 1:1, and 1:2, then precipitated with hydrochloric acid. Aqueous solubility, water binding capacity, IR spectra, and thermal analysis (thermogravimetry, TG, and differential thermogravimetry, DTG) of the precipitates revealed that they were not simple physical mixtures of the components. The components interact with one another electrostatically with involvement of the starch phosphate groups and the peptide bonds of casein as documented by the IR spectra. Because of their insolubility in 7 M aqueous urea solution they might also be considered as complexes in which the components were chemically bound. Enzymatic studies showed that they are biodegradable materials.     

Multiplicatively and non-symmetric multiplicatively norm-preserving maps

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖ _X and ‖.‖ _Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖ _Y = ‖fg‖ _X , for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖ _X = ‖Tf Tg + α‖ _Y , f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A,

Isometries between completely regular vector-valued function spaces

Periodica Mathematica Hungarica - In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces A and B of $$C_0(X,E)$$ and $$C_0(Y,F)$$ where X...     

Maps between Banach function algebras satisfying certain norm conditions

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\bar A$ and $\bar B$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ?\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)?α‖Y=‖ρ(p)τ(p′) ? α‖ _x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M _B and a homeomorphism ?: M _B → M _A between the maximal ideal spaces of B and A such that for all f ∈ A, where $\hat \cdot$ denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from $\bar A$ onto $\bar B$ inducing a homeomorphism between $M_{\bar B}$ and $M_{\bar A}$ . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.     

Nonlinear Diameter Preserving Maps Between Certain Function Spaces

Let X, Y be compact Hausdorff spaces and A, B be subspaces of C(X) and C(Y), respectively, containing the constant functions such that B is point separating and the evaluation functionals are linearly independent on B. In this paper, we give the general form of a surjective, not assumed to be linear, diameter preserving map \({T:A \longrightarrow B}\) for the case where A is dense in C(X). Fixing a point \({x_1\in X}\), we show that there exist a subset \({Y_0}\) of Y, a scalar \({\beta\in \mathbb{T}}\), a bijective continuous map \({\Psi: Y_0 \longrightarrow X}\) and a constant function \({\alpha: Y_0 \longrightarrow \{-1,1\}}\) such that

$$\begin{aligned}T_{1} f(y) - T_{1} f(y_{1}) = & \beta ({\rm Re} (f(\Psi(y)) - f(\Psi(y_{1}))) \\ & + \alpha(y) i {\rm Im} (f(\Psi(y)) - f(\Psi(y_{1}))))\end{aligned}$$

for all \({f\in A}\) and \({y\in Y_0}\), where \({T_1=T-T0}\) and \({\Psi(y_1)=x_1}\). In particular, either

$$T_1(f)(y)=\beta f(\Psi(y))+L(f) \qquad (f\in A,y\in Y_0),$$

or

$$T_1(f)(y)=\beta \overline{f(\Psi(y))}+L(f) \qquad (f\in A, y\in Y_0),$$

holds for some functional L on A, which is linear (resp. real-linear) whenever T is so.

     

Complete order amenability of the Fourier algebra

We define complete order amenability and first complete order cohomology groups for quantized Banach ordered algebras and show that the vanishing of the latter is equivalent to the operator amenability for the Fourier algebra.     

Generalized norm preserving maps between subsets of continuous functions

Positivity - Let X and Y be locally compact Hausdorff spaces. In this paper we study surjections $$T: A \longrightarrow B$$ between certain subsets A and B of $$C_0(X)$$ and $$C_0(Y)$$ ,...