An Index Theory For Quantum Dynamical Semigroups

W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

     

Representations of Hecke Algebras and Dilations of Semigroup Crossed Products

A family of Hecke C^*-algebras can be realised as crossed productsby semigroups of endomorphisms. It is shown by dilating representationsof the semigroup crossed product that the category of representationsof the Hecke algebra is equivalent to the category of continuousunitary representations of a totally disconnected locally compactgroup.     

Crossed products by a coalgebra

We introduce the notion of a crossed product of an al¬gebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.     

Actions of inverse semigroups arising from partial actions of groups

In this work we present a definition of crossed product for actions of inverse semigroups on C^∗-algebras, without resorting to covariant representations as done by Sieben in related work. We also show the existence of an isomorphism between the crossed product by a partial action of a group G and the crossed product by a related action of S(G), an inverse semigroup associated to G introduced by the first named author.     

I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

We introduce a notion of I-factorial quantum torsor, which consists of an integrable ergodic action of a locally compact quantum group on a type I-factor such that also the crossed product is a type I-factor. We show that any such I-factorial quantum torsor is at the same time a I-factorial quantum torsor for the dual locally compact quantum group, in such a way that the construction is involutive. As a motivating example, we show that quantized compact semisimple Lie groups, when amplified via a crossed product construction with the function algebra on the associated weight lattice, admit I-factorial quantum torsors, and give an explicit realization of the dual quantum torsor in terms of a deformed Heisenberg algebra for the Borel part of a quantized universal enveloping algebra.     

Crossed product of C*‐algebras by hypergroups

The crossed product of ‐algebras by groups, groupoids and semigroups are well studied. In this paper we introduce and study the crossed product of ‐algebras by (locally compact) hypergroups. We calculate the crossed products by finite hypergroups of orders 2 and 3.     

A quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups

Summary In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators. In this paper the case is considered of a finite-dimensional Hilbert space (n-level quantum system) and of instruments defined on a finite-dimensional Lie group. Then, the generator of a continuous semigroup of (quantum) probability operators is characterized. In this way a quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained.     

Compact quantum groups and group duality

In this paper, we consider the *-representations of compact quantum groups and group duality. The main results in the paper are: (1) there is a one-to-one correspondence between the *-representations of compact quantum groups and *-representations of the dual Banach *-algebra; (2) the category of commutative compact quantum groups (semigroups) is a dual category to the category of compact groups (semigroups); (3) the dual category of the category of locally compact groups (semigroups) is the category of commutative Hopf C^*-algebras with a particular property. Our group duality has the flavor of a Gelfand-Naimark type theorem for compact quantum groups, and for Hopf C^*-algebras.     

Crossed modules,quantum braided groups,and ribbon structures

Previous results about crossed modules over a braided Hopf algebra are applied to the study of quantum groups in braided categories. Cross products for braided Hopf algebras and quantum braided groups are constructed. Criteria for when a braided Hopf algebra or a quantum group is a cross product are obtained. A generalization of Majid's transmutation procedure for quantum braided groups is considered. A ribbon structure on a quantum braided group and its compatibility with cross product and transmutation are studied.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 368–387, June, 1995.     

An analogue of Hunt's representation theorem in quantum probability

In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of, instruments on groups and the associated semigroups of probability operators, which now are defined on spaces of functions with values in a von Neumann algebra. We consider a semigroup of probability operators with a continuity property weaker than uniform continuity, and we succeed in characterizing its infinitesimal generator under the additional hypothesis that twice differentiable functions belong to the domain of the generator. Such hypothesis can be proved in some particular cases. In this way a partial quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained. Our result provides also a closed characterization of generators of a new class of not norm continuous quantum dynamical semigroups.     

An elementary Green imprimitivity theorem for inverse semigroups

Semigroup Forum - A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let S be an...     

The set of quantum states and its averaged dynamic transformations

In this paper we consider the set of quantum states and passages to the limit for sequences of quantum dynamic semigroups in the mentioned set. We study the structure of the set of extreme points of the quantum state set and represent an arbitrary state as an integral over the set of one-dimensional orthogonal projectors; the obtained representation is similar to the spectral decomposition of a normal state. We apply the obtained results to the analysis of sequences of quantum dynamic semigroups which occur in the regularization of a degenerate Hamiltonian.     

Continuity and generators of dynamical semigroups for infinite Bose systems

For a class of quasifree quantum dynamical semigroups on the algebra of the canonical commutation relations (CCR) we give sufficient conditions for these semigroups to extend to ultraweakly continuous semigroups of normal operators on the von Neumann algebra associated with a representation of the CCR. Then the explicit form of the generators of the extended semigroups is calculated.     

A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups

We describe all [0-]simple semigroups that are nilpotent in the sense of Malcev. This generalizes the first Malcev theorem on nilpotent (in the sense of Malcev) semigroups. It is proved that if the extended standard wreath product of semigroups is nilpotent in the sense of Malcev and the passive semigroup is not nilpotent, then the active semigroup of the wreath product is a finite nilpotent group. In addition to that, the passive semigroup is uniform periodic. Necessary and sufficient conditions are found under which the extended standard wreath product of semigroups is nilpotent in the sense of Malcev in the case where each of the semigroups of the wreath product generates a variety of finite step.     

PERFECT rpp SEMIGROUPS

The aim of this paper is to study a class of rpp semigroups, namely the perfect rpp semigroups. We obtain some characterization theorems for such semigroups. In particular, the spined product structure of perfect rpp semigroups is established. As an application of spined product structure, we prove that a perfect rpp semigroup is a strong semilattice of left cancellative planks. By a left cancellative plank, we mean a product of a left cancellative monoid and a rectangular band. Thus, the work of J.B. Fountain on C-rpp semigroups is further developed.

     

On a Class of Dubreil-Jacotin Semigroups

The aim of this paper is to study a class of strong Dubreil-Jacotin semigroups, so-called split integral Dubreil-Jacotin semigroups. We establish the structure of split integral Dubreil-Jacotin semigroups in terms of the cartesian ordered semidirect product of an ordered monoid and an ordered group. Also, the lexicographic ordered semidirect product is considered. These results extend main results in [1].     

完备 wrpp 半群

研究一类 wrpp 半群,即完备 wrpp 半群,并给出完备 wrpp 半群的若干性质定理.特别地,得到完备 wrpp 半群的织积结构.作为织积的应用,我们证明完备 wrpp 半群是(R)-左消板的强半格.因此,唐向东关于 C-wrpp 半群的结果得到进一步发展.     

Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C ₀-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.     

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarason's characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.This research is supported by the Indian National Science Academy under Young Scientist Project.     

On idempotent-connected quasi-adequate semigroups

The aim of this paper is to study a class of F-abundant semigroups, that is, u-IC quasi-adequate semigroups. After giving some characterizations of u-IC quasiadequate semigroups, we establish the structure of such semigroups in terms of the semidirect product of bands and cancellative monoids.