# Quantum Structures

Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Technická 2, 166 27 Praha 6

## Who we are

**Martin Bohata**

He studies operator algebras and their applications in the axiomatics of
the quantum theory.

**Jan Hamhalter**

He studies operator algebras and quantum measure theory. He was awarded
by International Quantum Structures Association for research results in
2011. He wrote a basic monograph about the measure theory on von Neumann
algebras and its application in the axiomatics of the quantum theory.

**Mirko Navara**

He studies orthomodular structures (quantum logics), in particular
various algebraic and combinatorial constructions a techniques. He was
awarded by International Quantum Structures Association for research results
in 1996.

**Pavel Pták**

He studies orthomodular structures (quantum logics), in particular their
algebraical and topological properties. He is a coauthor of a basic
monograph about quantum logics and a founder of the seminar about quantum
logics.

**Josef Tkadlec**

He studies effect algebras and orthomodular posets, in particular
compatibility and states on these structures.

## Research description

We are solving actual problems of the theory of orthomodular algebraic structures or operator algebras and the measure theory built on these structures. These topics are motivated by the quantum theory. We are one of a few research teams in the world capable to combine methods of a functional analysis (continuos structures) and of an algebra and combinatorics (discrete structures).

## What is it good for?

The research results have an applications in the study of the axiomatics of the quantum theory, in quantum measure theory, in quantum measurement theory, in quantum information theory, in cybernetics.

## Research Areas

- Measure theory on von Neumann algebras (convergence theorems).
- The geometry of state spaces of C*-algebras and Jordan algebras.
- Concrete logics (Dynkin systems).
- General quantum logics, compatibility, state space, noncommutative probability.
- Constructions of quantum logics.
- Hilbert and pre-Hilbert spaces.
- The independence of operator algebras in the quantum field theory.

## Financial support

- Quantum logics as orthomodular structures, Czech Science Foundation 201/93/0953, 1993–1995.
- Mathematical formalism of quantum theories, Czech Science Foundation 201/96/0117, 1996–1998.
- Applied mathematics in technical sciences, Ministry of Education MSM 210000010, 1999–2004.
- Operator algebras, orthocomplemented structures noncommutative measure theory, Czech Science Foundation 201/00/0331, 2000–2002.
- Noncommutative measure theory, Czech Science Foundation 201/03/0455, 2003–2005.
- Applied mathematics in technical and physical sciences, Ministry of Education 6840770010, 2005–2011.
- Algebraical and measure-theoretical aspects of quantum structures, Czech Science Foundation 201/07/1051, 2007–2009.
- Topological and geometrical properties of Banach spaces and operator algebras, Czech Science Foundation P201/12/0290, 2012–2016.

## Scientific collaborations

- Charles University, Czech Republic
- Mathematical Institute of the Slovak Academy of Sciences, Slovakia
- New Mexico State University, USA
- Technishe Universität Wien, Austria
- University of Erlangen, Germany
- University of Lyon, France
- University of Malta, Malta
- University of Napoli, Italy
- University of Reading, United Kingdom
- University of Udine, Italy

## Selected recent results

- Conti, R., Hamhalter, J.: Independence of group algebras. Mathematische Nachrichten 238 (2010), 818–827.
- De Simone, A., Pták, P.: Measures on circle coarse-grained systems of sets. Positivity 14 (2010), 247–256.
- Hamhalter, J.: Absolute continuity and noncommutative measure theory. Internat. J. Theoret. Phys. 49 (2010), 3139–3145.
- Hamhalter, J., Bohata, M.: Bell's correlations and spin systems. Foundations Phys. 40 (2010), 1065–1075.
- Pták, P., Matou.ek, M.: On identities in orthocomplemented difference lattices. Mathematica Slovaca 60 (2010), 583–590.
- Tkadlec, J.: Common generalizations of orthocomplete and lattice effect algebras. Internat. J. Theoret. Phys. 49 (2010), 2279–2285.
- Bohata, M.: Star order on operator and function algebras. Publicationes Mathematicae. 79 (2011), 211–229.
- Caragheorgheopol, D., Tkadlec, J.: Atomic effect algebras with compression bases. J. Math. Phys. 52 (2011), 013512.
- Caragheorgheopol, D., Tkadlec, J.: Characterizations of spectral automorphisms and a Stone-type theorem in orthomodular Lattices. Internat. J. Theoret. Phys. 50 (2011), 3750–3760.
- Hamhalter, J.: Isomorphisms of ordered structures of abelian C*-subalgebras of C*-algebras. J. Math. Anal. Appl. 383 (2011), 391–399.
- Hamhalter, J., Turilova, E.: Subspace structures in inner product spaces and von Neumann algebras. Internat. J. Theoret. Phys. 50 (2011), 3812–3820.
- Matou.ek, M., Pták, P.: orthocomplemented difference lattices with few generators. Kybernetika 47 (2011), 60–73.
- Tkadlec, J.: Note on generalizations of orthocomplete and lattice effect algebras. Internat. J. Theoret. Phys. 50 (2011), 3915–3918.
- Tkadlec, J., Turunen, E.: Commutative bounded integral residuated orthomodular lattices are Boolean algebras, Soft Comput. 15 (2011), 635–636.
- Chetcutti, E., Hamhalter, J.: Completeness of *-symmetric Gelfand-Naimark-Segal inner product spaces. Quart. J. Math. 63 (2012), 367–373.
- Hamhalter, J.: Linear maps preserving maximal deviation and the Jordan structure of quantum systems. J. Math. Phys. 53 (2012).
- Matou.ek, M., Pták, P.: Orthocomplemented difference lattices in association with generalized rings. Math. Slovaca 62 (2012), 1063–1068.
- Bohata, M., Hamhalter, J.: Nonlinear maps on von Neumann algebras preserving the star order. Linear and Multilinear Algebra 61 (2013), 998–1009.
- Gabriëls, J., Navara, M.: Computer proof of monotonicity of operations on orthomodular lattices. Inform. Sci. 236 (2013), 205–217
- Hamhalter, J., Turilova, E.: Affilated subspaces and infiniteness of von Neumann algebras. Math. Nachr. 286 (2013), 976–985.
- Hamhalter, J., Turilova, E.: Affiliated subspaces and the structure of von Neumann algebras. J. Operator Theory 69 (2013), 101–115.
- Hamhalter, J., Turilova, E.: Structure of associative subalgebras of Jordan operator algebras. Quart. J. Math. 64 (2013), 397–408.
- Bohata, M, Hamhalter, J.: Star order on JBW algebras. J. Math. Anal. Appl. 2 (2014), 873–888.
- Hamhalter, J., Turilova, E.: Automorphisms of ordered structures of abelian parts of operator Algebras and their role in quantum theory. Internat. J. Theoret. Phys. 53 (2014), 3333–3345.
- Hamhalter, J., Turilova, E.: Classes of invariant subspaces for some operator algebras. Internat. J. Theoret. Phys. 53 (2014), 3397–3408.
- Simon, R., Mukunda, N., Chaturvedi, S., Srinivasan, V., Hamhalter, J.: Comment on: "Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics" [Phys. Lett. A 372 (2008) 6847]. Phys. Lett. A. 30–31 (2014), 2332–2335.