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Notes on the Krupa-Zawisza ultrapower of self-adjoint operators

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Ando H. , Saigo H. , Ojima I.

tom Vol. 34, Fasc. 1

147--159

Let ω ∈ βN \ N be a free ultrafilter on N. It is known that there is a difficulty in constructing the ultrapower of unbounded operators. Krupa and Zawisza gave a rigorous definition of the ultrapower Aω of a self-adjoint operator A. In this note, we give an alternative description of Aω and the Hilbert space H(A) on which Aω is densely defined. This provides a criterion to determine a representing sequence (ξn)n of a given vector ξ ∈ dom(Aω) which has the property that Aωξ = (Aξn)ω holds. An explicit core for Aω is also described.

A note on Łoś theorem without the Axiom of Choice

70%

Usuba T.

2024

tom Vol. 72, no 1

17--44

We study some topics around Łoś’s theorem without assuming the Axiom of Choice. We prove that Łoś’s fundamental theorem on ultraproducts is equivalent to a weak form that every ultrapower is elementarily equivalent to its source structure. On the other hand, it is consistent that there is a structure M and an ultrafilter U such that the ultrapower of M by U is elementarily equivalent to M, but the fundamental theorem for the ultrapower of M by U fails. We also show that weak fragments of the Axiom of Choice, such as the Countable Choice, do not follow from Łoś’s theorem, even assuming the existence of non-principal ultrafilters.