arXiv subjects

David Kruml

Publications and source records attributed to David Kruml.

7 recordsLinked to original sources

arXiv2026

Quantales carrying ortholattice structure

This paper investigates the intersection of residuated structures from many-valued logic and orthomodular lattices from quantum logic. We explore whether non-Boolean structures can simultaneously satisfy residuation principles and orthocomplementation requirements. Our main contribution is a study of Girard posets with inversions, providing a characterization theorem where a unital residuated poset is Girard if and only if it admits an inversion satisfying specific adjointness conditions. We prove that any complemented lattice admitting an integral residuated structure must be Boolean, which motivates our search for orthomodular examples in the non-integral case. We answer this by demonstrating that the lattice $C(\mathbb{R}^n)$ of closed subspaces of $n$-dimensional real coordinate space carries both an orthomodular and a commutative Girard quantale structure. This construction provides a concrete non-Boolean framework unifying quantum-logical and many-valued logical reasoning.

math.LO↗

arXiv2021

Linear orthogonality spaces as a new approach to quantum logic

The notion of an orthogonality space was recently rediscovered as an effective means to characterise the essential properties of quantum logic. The approach can be considered as minimalistic; solely the aspect of mutual exclusiveness is taken into account. In fact, an orthogonality space is simply a set endowed with a symmetric and irreflexive binary relation. If the rank is at least $4$ and if a certain combinatorial condition holds, these relational structures can be shown to give rise in a unique way to Hermitian spaces. In this paper, we focus on the finite case. In particular, we investigate orthogonality spaces of rank at most $3$.

math.LO↗

arXiv2008

Girard couples of quantales

We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice $S$, a Girard quantale whose right-sided part is isomorphic to $S$.

math.CT↗

arXiv2008

Quantum triads: an algebraic approach

A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frames, simple and Galois quantales, operator algebras, or orthomodular lattices.

math.CT↗

arXiv2004

On quantales that classify C*-algebras

The functor Max of Mulvey assigns to each unital C*-algebra A the unital involutive quantale Max A of closed linear subspaces of A, and it has been remarked that it classifies unital C*-algebras up to *-isomorphism. In this paper we provide a proof of this and of the stronger fact that for every isomorphism u : Max A -> Max B of unital involutive quantales there is a *-isomorphism u' : A -> B such that Max u' coincides with u when restricted to the left-sided elements of Max A. But we also show that isomorphisms u : Max A -> Max B may exist for which no isomorphism v : A -> B is such that Max v = u.

math.OA↗

arXiv2002

On quantales and spectra of C*-algebras

We study properties of the quantale spectrum Max A of an arbitrary unital C*-algebra A. In particular we show that the spatialization of Max A with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-)preservation of limits and colimits. Although Max is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C*-algebras.

math.OA↗

arXiv2002

On simple and semisimple quantales

In a recent paper, J. W. Pelletier and J. Rosicky published a characterization of *-simple *-quantales. Their results were adapted for the case of simple quantales by J. Paseka. In this paper we present similar characterizations which do not use a notion of discrete quantale. We also show a completely new characterization based on separating and cyclic sets. Further we explain a link to simple quantale modules. To apply these characterizations, we study (*-)semisimple (*-)quantales and discuss some other perspectives. Our approach has connections with several earlier works on the subject.

math.OA↗