Non-measurable automorphisms of Lie groups relative to the real- and non-archimedean-valued measures

关于原子阿基米德向量格上运算代数的正自同构

On positive automorphisms of algebras of operators on atomic Archimedean vector lattices

研究原子阿基米德向量格上运算代数的正自同构性质，证明其必为空间自同构，并利用Kakutani定理证明有限维子空间的序闭性。

Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms

In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.

A real-valued measure on non-Archimedean field extensions of $\mathbb{R}$

We introduce a real-valued measure ${m_L}$ on non-Archimedean ordered fields $(\mathbb{F},<)$ that extend the field of real numbers $(\mathbb{R},<)$. The definition of ${m_L}$ is inspired by the Loeb measures of hyperreal fields in the framework of Robinson's analysis with infinitesimals. The real-valued measure ${m_L}$ turns out to be general enough to obtain a canonical measurable representative in $\mathbb{F}$ for every Lebesgue measurable subset of $\mathbb{R}$, moreover, the measure of the two sets is equal. In addition, $m_L$ it is more expressive than a class of non-Archimedean uniform …

Uniform Algebras Over Complete Valued Fields

UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete valued field other than the complex numbers is used as the underlying field of the algebra. In the Archimedean setting, this generalisation is given by the theory of real function algebras introduced by S. H. Kulkarni and B. V. Limaye in the 1980s. This thesis establishes a broader theory accommodating any complete valued field as the underlying field by invo…

关于 Levi-Civita 域 $\mathcal{R}$ 上类 Lebesgue 积分的研究

On a Lebesgue-like integral over the Levi-Civita field $\mathcal{R}$

本文在 Levi-Civita 域 $\mathcal{R}$ 上发展了一种新积分理论，定义了新的可测函数族与积分，证明了其满足线性、可数可加性等类似于 Lebesgue 积分的重要性质，并推广了前人工作。

Infinite-dimensional transfer operators, endomorphisms, and measurable partitions

We develop a new duality between endomorphisms of measure spaces, on the one hand, and a certain family of positive operators, called transfer operators, acting in spaces of measurable functions on, on the other. A framework of standard Borel spaces is adopted; and this generality is wide enough to cover a host of applications. While the mathematical structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in a host of settings, both pure and applied, we propose here a unified study. This is the general setting of dynamics in Borel …