C*-algebras and the Uncountable: A Systematic Study of the Combinatorics of the Uncountable in the Noncommutative Framework

Abstract

Next, we turn our attention to the parallel product. In joint work with Dzhafarov, Hirschfeldt, Patey, and Pauly, we investigate the infinite pigeonhole principle for different numbers of colors and how these problems behave under Weihrauch reducibility with respect to parallel products. Finally, we leave the setting of computable reducibilities for the setting of reverse mathematics. First, we define a Σ 1 1 axiom of finite choice and investigate its relationships with other theorems of hyperarithmetic analysis. For one, we show that it follows from Arithmetic Bolzano-Weierstrass. On the other hand, using an elaboration of Steel's forcing with tagged trees, we show that it does not follow from Δ 1 1 comprehension. Second, in joint work with James Barnes and Richard A. Shore, we analyze a theorem of Halin about disjoint rays in graphs. Our main result shows that Halin's theorem is a theorem of hyperarithmetic analysis, making it only the second "natural" (i.e., not formulated using concepts from logic) theorem with this property.

Trivial Endomorphisms of the Calkin Algebra

“…The second consequence of OCA we prove in this paper is related to the recent works [FHV19], [FKV19] and [Vac19,Chapter 2], where methods from set theory are employed in the study of nonseparable subalgebras of Q(H). Let E be the class of all C * -algebras that embed into the Q(H).…”