Robert Sheckley\s Answerer for two orthogonal projections

TL;DR

This paper explores the deep connections between Halmos' two projections theorem and supersymmetry in operator algebras, showing they are essentially equivalent frameworks for understanding questions about orthogonal projections.

Contribution

It demonstrates that Halmos' theorem and the supersymmetry approach are fundamentally equivalent, providing a unified perspective on questions about two orthogonal projections.

Findings

Halmos' two projections theorem acts as an Answerer for questions about orthogonal projections.

Supersymmetry equality is shown to be equivalent to Halmos' approach.

The paper offers an alternative perspective linking operator algebra theorems and supersymmetry.

Abstract

This paper is the written version of our talk (presented by the second author) at the IWOTA in Chemnitz in August 2017. The meta theorem of the paper is that Halmos' two projections theorem is something like Robert Sheckley's Answerer: no question about the W*- and C*-algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supersymmetry equality introduced by Avron, Seiler, and Simon. A noteworthy insight of the paper reveals that the supersymmetric approach is nothing but Halmos in different language and hence an equivalent Answerer.