Martijn Caspers

Intuitionistic quantum logic of an n-level system

AbstractA decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra Mn(ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σn and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ℘(Mn(ℂ)) of projections in Mn(ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice

The Haagerup Approximation Property for von Neumann Algebras via Quantum Markov Semigroups and Dirichlet Forms

\mathcal{O}(\Sigma_{n})

Weak Amenability of Locally Compact Quantum Groups and Approximation Properties of Extended Quantum SU(1, 1)

of functions from the poset

Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

\mathcal{C}(M_{n}(\mathbb{C}))

SCHUR AND FOURIER MULTIPLIERS OF AN AMENABLE GROUP ACTING ON NON-COMMUTATIVE L p -SPACES

of all unital commutative C*-subalgebras C of Mn(ℂ) to ℘(Mn(ℂ)). The lattice

Locally compact quantum groups

\mathcal{O}(\Sigma_{n})

Connes embeddability of graph products

is essentially the (pointfree) topology of the quantum phase space Σn, and as such defines a Heyting algebra. Each element of

The lp-fourier transform on locally compact quantum groups

\mathcal{O}(\Sigma_{n})

Graph products of operator algebras

corresponds to a “Bohrified” proposition, in the sense that to each classical context

The Haagerup Property for Arbitrary von Neumann Algebras

C\in\mathcal{C}(M_{n}(\mathbb{C}))