The Space of Measurement Outcomes as a Spectral Invariant for Non-Commutative Algebras

Abstract

The recently developed technique of Bohrification associates to a (unital) C*-algebra A 1. the Kripke model, a presheaf topos, of its classical contexts; 2. in this Kripke model a commutative C*-algebra, called the Bohrification of A; 3. the spectrum of the Bohrification as a locale internal in the Kripke model.We propose this locale, the 'state space', as a (n intuitionistic) logic of the physical system whose observable algebra is A.We compute a site which externally captures this locale and find that externally its points may be identified with partial measurement outcomes. This prompts us to compare Scott-continuity on the poset of contexts and continuity with respect to the C*-algebra as two ways to mathematically identify measurement outcomes with the same physical interpretation. Finally, we consider the not-not-sheafification of the Kripke model on classical contexts and obtain a space of measurement outcomes which for commutative C*-algebras coincides with the spectrum. The construction is functorial on the category of C*-algebras with commutativity reflecting maps.

Domains of Commutative C*-Subalgebras

“…This is claimed in [24,Prop. 15] without complete proof of sufficiency; note that this direction does not always hold in the setting of von Neumann algebras (see Section IX) since the adaptation of Lemma II.22 to the relevant topology of von Neumann algebras fails.…”

“…This article fits into a wider programme: its associated partial order determines a C*-algebra to a great extent [21], [22], and has therefore become popular as a substitute [23], [24], [25]. In the case of deterministic computing it can be axiomatized [26].…”

The Many Classical Faces of Quantum Structures

“…Domain theory is mostly concerned with partial orders where every element can be approximated by finite ones, as those are the ones we can measure in practice, leading to the following definitions. 37,118]). If A is a C*-algebra, then C(A) is a directed complete partially ordered set, in which i C i is the norm-closure of i C i .…”

“…Connecting back to Theorem 3.5 and Corollary 3.6, let us notice that C can also be regarded as a domain using the interval topology: smaller intervals approximate an ideal complex number better than larger ones. Moreover, (piecewise) states A → C respect such approximations: the induced functions from C(A) to the interval domain on C are Scott continuous [37,118].…”

“…[73,118,20]). There is a bijective correspondence between piecewise states on an AW*-algebra A, and states of the canonical contextual set C → C inside the topos of contextual sets over C(A).…”