Spaces of unbounded Fredholm operators: I. Homotopy equivalences
Marina Prokhorova
TL;DR
The paper analyzes spaces of unbounded regular operators on a separable infinite‑dimensional Hilbert space under the graph topology and related subspaces (compact resolvent, self‑adjoint variants), and relates them to classical bounded operator models representing $K^0$ and $K^1$ theory. It proves that natural maps between these unbounded operator spaces and the bounded operator models are homotopy equivalences, providing an accessible alternative to Joachim’s KK‑theory proofs. It also shows that the subspace of unbounded essentially positive Fredholm operators represents odd $K$‑theory and that invertible operators within these spaces are contractible. The results establish that these unbounded operator spaces form classifying spaces for $K^0$ and $K^1$, while the graph topology exhibits spectral‑flow phenomena not present in the Riesz topology, illustrating a rich unbounded‑operator panorama for operator $K$‑theory.
Abstract
This paper is devoted to the space of unbounded Fredholm operators equipped with the graph topology, the subspace of operators with compact resolvent, and their subspaces consisting of self-adjoint operators. Our main results are the following: (1) Natural maps between these four spaces and classical spaces of bounded operators representing K-theory are homotopy equivalences. This provides an alternative proof of a particular case of results of Joachim. (2) The subspace of unbounded essentially positive Fredholm operators represents odd K-theory. (3) The subspace of invertible operators in each of these spaces of unbounded operators is contractible.