## Final Report Summary - OPERADYNADUAL (Operator algebras and single operators via dynamical properties of dual objects)

The project lies at the intersection of three fields of pure mathematics: operator algebras (specifically C*-algebras), operator theory and dynamical systems, with applications to noncommutative structures, completely positive dynamics, spectral analysis of functional operators, and potentially to the theory of functional-differential equations, quantum physics and many other fields.

The theory of operator algebras originated in connection with quantum physics. Operator algebras can be viewed as far reaching generalization of number fields where in particular multiplication need not to be commutative. They provide a rigorous mathematical structure of quantum mechanics, as well as a link with classical physics. The general principle is that observables in a quantum systems are modelled by elements in genuinely noncommutative operator algebras, and if the algebra is commutative then it is in fact an algebra of functions (classical observables) on some classical space:

noncommutativity <---> quantum

An important very general class of operator algebras, which are also of fundamental importance in mathematics, in particular in harmonic analysis and representation theory are called C*-algebras.

The theory of dynamical system studies properties of actions (evolutions) of groups or semigroups. Group is a mathematical object that is used to describe symmetries of the system or reversible dynamics. Semigroups model evolution systems which are irreversible. The theory of operators studies single linear maps between infinite dimensional spaces. In particular, spectral properties of the so called functional operators model asymptotic and ergodic properties of evolutions of systems modeling complicated physical processes containing both dynamical contributory factors and interaction with outer media, e.powerful. the process of motion and transformation of particles.

The most significant achievements of the project can be divided into 5 items:

1. CONSTRUCTION OF ALGEBRAS FROM DYNAMICAL DATA. In the “noncommutative world” dynamics is usually irreversible. Moreover, it usually is given by maps which preserve only positivity, do not preserve multiplication – completely positive dynamics. In fact, it might be given by even more general structures – C*-correspondences. These are very abstract and general notions which are hard to analyze. So far it is not known how to construct operator algebras incorporating these kind of dynamics in general. The project provides a huge contribution to this subject.

A new mathematical language of right tensor C*-precategories was developed. This led to general constructions which unify and extend the most successful constructions of this sort. This includes the so called Cuntz-Pimsner algebras and Nica-Toeplitz algebras associated with semigroups of C*-correspondences. This also concerns crossed products by completely positive maps. Moreover, the structure of these algebras is deeply analyzed.

2. NONCOMMUTATIVE GRAPHS. As we already mentioned noncommutative dynamics involve more sophisticated objects than maps. In fact it is known that C*-correspondences with commutative coefficient algebras correspond to graphs –combinatorial objects. This allows a detailed study of the corresponding C*-algebra, and the theory of graph C*-algebra is nowadays a well-established fundamental part of the theory of operator algebras. One of the main result of the project is development of the theory of graphs dual to C*-correspondences with noncommutative algebras of coefficients. This provides combinatorial technics to study these genuinely noncommutative objects.

3. UNIQUENESS THEOREMS. Among the first important examples of operator algebras modeling quantum systems are CAR and CCR algebras. They are generated respectively by quantum anti-commutation and canonical commutation relations (they model observables in Bose-Einstein and Fermi-Dirac quantum statistics, respectively). The essential property of relations of CAR and CCR type is that whenever you have family of operators satisfying the prescribed relations the C*-algebras they generate are the same. This uniqueness property has a fundamental physical meaning, as if we had no such uniqueness, different representations would yield different physics! Similar results of this type for graph C*-algebras and more general structures are called "Uniqueness Theorems". They are analysis tools to study the structure of the corresponding C*-algebras.

Using the machinery described in item 2, general Uniqueness Theorems have been established. They unified and extended a number of previous results of these type. In particular, they were applied to study the structure algebras described in item 1. Also some new phenomena were discovered which led to results in item 4.

4. PARADOXICALITY AND PURE INFINITENESS. The famous Banach–Tarski paradox is a theorem saying that given a solid ball (e.g. an orange) in 3 dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together using only physical motions to yield two identical copies of the original ball (two oranges).

One of the results of the project is a discovery of conditions for the noncommutative (quantum) version of Banach-Tarski paradox to hold. Also its relationship to pure infiniteness of the corresponding algebras has been clarified, see fig1.png. This [*]analyzis[/*] is phrased in terms of Fell bundles over groups which correspond to actions of dual quantum groups on genuinely noncommutative spaces. These significant results give strong tools to construct purely infinite C*-algebras, which are of crucial importance in the theory of classification of C*-algebras.

5. APPLICATIONS TO FUNCTIONAL OPERATORS. The fundamental spectral information of functional operators is encoded in the algebras they generate. These have a structure of generalized crossed products. As an application of the previously mentioned results of the project, spectral properties where deeply analyzed.

One of fundamental tools that has been developed is a canonical dilation (extension) of an irreversible system to a reversible one. In the simplest example of a map that wraps a unit circle twice, the corresponding reversible object is a solenoid – it can be imaged as an infinitely thin solid torus wrapped infinitely many times around itself, see fig2.png. Calculation of spectra of the corresponding operator involves integrating over such a complicated object, or its noncommutative counterpart.

The powerful, innovative constructions and tools described above are expected to give impulse for new lines of research. The outputs and developed innovative methods of analysis brings to the ERA a unique expertise of a great impact and scientific value, with a wide range of potential interdisciplinary applications. Thanks to the training received during the project the fellow integrated with an excellent Scandinavian scientific network of world leaders in crossing the boundaries between operator algebras and other fields. This strengthens this network and provides a new strong outside link with Poland.

More information on the project can be found the project Website:

http://math.uwb.edu.pl/~zaf/kwasniewski/operadynadual

Additional information can be provided by e-mail: bartoszk@math.uwb.edu.pl

The theory of operator algebras originated in connection with quantum physics. Operator algebras can be viewed as far reaching generalization of number fields where in particular multiplication need not to be commutative. They provide a rigorous mathematical structure of quantum mechanics, as well as a link with classical physics. The general principle is that observables in a quantum systems are modelled by elements in genuinely noncommutative operator algebras, and if the algebra is commutative then it is in fact an algebra of functions (classical observables) on some classical space:

noncommutativity <---> quantum

An important very general class of operator algebras, which are also of fundamental importance in mathematics, in particular in harmonic analysis and representation theory are called C*-algebras.

The theory of dynamical system studies properties of actions (evolutions) of groups or semigroups. Group is a mathematical object that is used to describe symmetries of the system or reversible dynamics. Semigroups model evolution systems which are irreversible. The theory of operators studies single linear maps between infinite dimensional spaces. In particular, spectral properties of the so called functional operators model asymptotic and ergodic properties of evolutions of systems modeling complicated physical processes containing both dynamical contributory factors and interaction with outer media, e.powerful. the process of motion and transformation of particles.

The most significant achievements of the project can be divided into 5 items:

1. CONSTRUCTION OF ALGEBRAS FROM DYNAMICAL DATA. In the “noncommutative world” dynamics is usually irreversible. Moreover, it usually is given by maps which preserve only positivity, do not preserve multiplication – completely positive dynamics. In fact, it might be given by even more general structures – C*-correspondences. These are very abstract and general notions which are hard to analyze. So far it is not known how to construct operator algebras incorporating these kind of dynamics in general. The project provides a huge contribution to this subject.

A new mathematical language of right tensor C*-precategories was developed. This led to general constructions which unify and extend the most successful constructions of this sort. This includes the so called Cuntz-Pimsner algebras and Nica-Toeplitz algebras associated with semigroups of C*-correspondences. This also concerns crossed products by completely positive maps. Moreover, the structure of these algebras is deeply analyzed.

2. NONCOMMUTATIVE GRAPHS. As we already mentioned noncommutative dynamics involve more sophisticated objects than maps. In fact it is known that C*-correspondences with commutative coefficient algebras correspond to graphs –combinatorial objects. This allows a detailed study of the corresponding C*-algebra, and the theory of graph C*-algebra is nowadays a well-established fundamental part of the theory of operator algebras. One of the main result of the project is development of the theory of graphs dual to C*-correspondences with noncommutative algebras of coefficients. This provides combinatorial technics to study these genuinely noncommutative objects.

3. UNIQUENESS THEOREMS. Among the first important examples of operator algebras modeling quantum systems are CAR and CCR algebras. They are generated respectively by quantum anti-commutation and canonical commutation relations (they model observables in Bose-Einstein and Fermi-Dirac quantum statistics, respectively). The essential property of relations of CAR and CCR type is that whenever you have family of operators satisfying the prescribed relations the C*-algebras they generate are the same. This uniqueness property has a fundamental physical meaning, as if we had no such uniqueness, different representations would yield different physics! Similar results of this type for graph C*-algebras and more general structures are called "Uniqueness Theorems". They are analysis tools to study the structure of the corresponding C*-algebras.

Using the machinery described in item 2, general Uniqueness Theorems have been established. They unified and extended a number of previous results of these type. In particular, they were applied to study the structure algebras described in item 1. Also some new phenomena were discovered which led to results in item 4.

4. PARADOXICALITY AND PURE INFINITENESS. The famous Banach–Tarski paradox is a theorem saying that given a solid ball (e.g. an orange) in 3 dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together using only physical motions to yield two identical copies of the original ball (two oranges).

One of the results of the project is a discovery of conditions for the noncommutative (quantum) version of Banach-Tarski paradox to hold. Also its relationship to pure infiniteness of the corresponding algebras has been clarified, see fig1.png. This [*]analyzis[/*] is phrased in terms of Fell bundles over groups which correspond to actions of dual quantum groups on genuinely noncommutative spaces. These significant results give strong tools to construct purely infinite C*-algebras, which are of crucial importance in the theory of classification of C*-algebras.

5. APPLICATIONS TO FUNCTIONAL OPERATORS. The fundamental spectral information of functional operators is encoded in the algebras they generate. These have a structure of generalized crossed products. As an application of the previously mentioned results of the project, spectral properties where deeply analyzed.

One of fundamental tools that has been developed is a canonical dilation (extension) of an irreversible system to a reversible one. In the simplest example of a map that wraps a unit circle twice, the corresponding reversible object is a solenoid – it can be imaged as an infinitely thin solid torus wrapped infinitely many times around itself, see fig2.png. Calculation of spectra of the corresponding operator involves integrating over such a complicated object, or its noncommutative counterpart.

The powerful, innovative constructions and tools described above are expected to give impulse for new lines of research. The outputs and developed innovative methods of analysis brings to the ERA a unique expertise of a great impact and scientific value, with a wide range of potential interdisciplinary applications. Thanks to the training received during the project the fellow integrated with an excellent Scandinavian scientific network of world leaders in crossing the boundaries between operator algebras and other fields. This strengthens this network and provides a new strong outside link with Poland.

More information on the project can be found the project Website:

http://math.uwb.edu.pl/~zaf/kwasniewski/operadynadual

Additional information can be provided by e-mail: bartoszk@math.uwb.edu.pl