Complete ccc Boolean algebras, the order sequential topology, and a problem of von Neumann

Topological density of ccc Boolean algebras - every cardinality occurs.

For every uncountable cardinal mu there is a ccc Boolean algebra whose topological density is mu .

The sequential topology on complete Boolean algebras

We investigate the sequential topology $τ_s$ on a complete Boolean algebra $B$ determined by algebraically convergent sequences in $B$. We show the role of weak distributivity of $B$ in separation axioms for the sequential topology. The main result is that a necessary and sufficient condition for $B$ to carry a strictly positive Maharam submeasure is that $B$ is ccc and that the space $(B,τ_s)$ is Hausdorff. We also characterize sequential cardinals.

一个可数的、满足赖兴巴赫共同原因完备性的布尔代数

A countable Boolean algebra that is Reichenbach's common cause complete

本文构造了一个可数的布尔代数，使其满足赖兴巴赫的共同原因完备性（CCC），回答了是否存在此类小布尔代数的问题。

Abundance of independent sequences in compact spaces and Boolean algebras

It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal $\mathcal{C}^\perp$ as the class of all compact spaces for which every continuous image in $\mathcal{C}$ is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and $ccc$ compacta), and also classes where independence is scarce (most notably weakly Radon-Nikodým compacta) with classes where i…

通过冯·诺依曼正则$\u003cspan style="font-style: italic;"\u003eC\u003c/span\u003e^∞$-环的布尔代数分类

Classification of Boolean Algebras through von Neumann regular $\mathcal{C}^{\infty}-$Rings

本研究引入冯·诺依曼正则C^∞-环概念，通过函子实现布尔空间的有效表示，并证明布尔代数同态可由这类环的C^∞-环同态表示，为布尔代数分类提供新方法。

On regular $^*$-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras

We study (von Neumann) regular $^*$-subalgebras of $B(H)$, which we call R$^*$-algebras. The class of R$^*$-algebras coincides with that of "E$^*$-algebras that are pre-C$^*$-algebras" in the sense of Z. Szűcs and B. Takács. We give examples, properties and questions of R$^*$-algebras. We observe that the class of unital commutative R$^*$-algebras has a canonical one-to-one correspondence with the class of Boolean algebras. This motivates the study of R$^*$-algebras as that of noncommutative Boolean algebras. We explain that seemingly unrelated topics of functional analysis, like AF C$^*$-alge…