Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a linear bijective map ψA →A satisfying ψ（A）ψ（B） - ψ（B）ψ（A）＊ = AB - BA＊ for A, B ∈ A and a real functional h on A with h（0） -= 0 such that Ф（A） = ψ（A） ＋ h（A）I for every A ∈ A. In particular, if .4 is a type I factor, then, Ф（A） = cA ＋ h（A）I for every A ∈ .4, where c = ±1.
The author establishes operator-valued Fourier multiplier theorems on multi-dimensional Hardy spaces Hp(Td;X),where 1 ≤ p < ∞,d ∈ N,and X is an AUMD Banach space having the property (α).The suffcient condition on the multiplier is a Marcinkiewicz type condition of order 2 using Rademacher boundedness of sets of bounded linear operators.It is also shown that the assumption that X has the property (α) is necessary when d ≥ 2 even for scalar-valued multipliers.When the underlying Banach space does not have the property (α),a suffcient condition on the multiplier of Marcinkiewicz type of order 2 using a notion of d-Rademacher boundedness is also given.
We study the well-posedness of the equations with fractional derivative D^αu（t）= Au（t） ＋ f（t） （0 ≤ t ≤ 2π）, where A is a closed operator in a Banach space X, 0 〈 α 〈 1 and D^αis the fractional derivative in the sense of Weyl. Although this problem is not always well-posed in L^P（0, 2π; X） or periodic continuous function spaces Cper（[0, 2π]; X）, we show by using the method of sum that it is well-posed in some subspaces of L^P（0, 2π; X） or Cper （[0, 2π]; X）.
We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.
