Unitary Actions Research Articles

Characterizations of Morita equivalent inverse semigroups

We prove that four different notions of Morita equivalence for inverse semigroups motivated by C ∗ -algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.

Unitary braid matrices: bridge between topological and quantum entanglements

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body en-tanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0; 1). Thus, braiding operators correspond-ing to over- and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entan-glements. For higher dimensions, starting with di erent initial triplets one can entangle by turns, each state with all the rest. A speci c coupling of three angular momenta is brie y discussed to throw more light on three body entanglements.

Unitary braid matrices: bridge between topological and quantum entanglements

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body en-tanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0; 1). Thus, braiding operators correspond-ing to over- and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entan-glements. For higher dimensions, starting with di erent initial triplets one can entangle by turns, each state with all the rest. A speci c coupling of three angular momenta is brie y discussed to throw more light on three body entanglements.

The basic bundle gerbe on unitary groups

We consider the construction of the basic bundle gerbe on S U ( n ) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U ( n ) and U p ( H ) , the Banach Lie group of unitaries differing from the identity by an element of a Schatten ideal. In all these cases we give an explicit connection and curving on the basic bundle gerbe and calculate the real Dixmier–Douady class. Extensive use is made of the holomorphic functional calculus for operators on a Hilbert space.

Unitary representations and modular actions

We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense “orthogonal” to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity and strong mixing. Bibliography:s 25 titles.

Dynamical decoupling in quantum control: A system theoretic approach

Suppressing decoherence is one of the most challenging problems in the control of quantum dynamical systems. Dynamical decoupling is an open-loop decoherence control technique based on high-frequency and high-amplitude periodic controls. Here, we reformulate the effects of the basic strategy in terms of linear, symmetric matrix equations. Such a reformulation proves to be useful both in the analysis and in the synthesis of the needed unitary control actions. A general framework is provided, and simple, but significant, particular cases are studied in detail.

The geometry of sets of parameters of wave packet frames

We study wave packet systems WP ( ψ , M ) ; that is, countable collections of dilations, translations, and modulations of a single function ψ ∈ L 2 ( R ) . The parameters of these unitary actions form a discrete subset M ⊂ R + × R × R . We introduce analogues of the notion of Beurling density, adapted to the geometry of discrete subsets of R + × R × R , and notions of lower and upper dimensions associated with these densities. Our goal is to describe completeness properties of wave packet systems via geometric properties of the sets of their parameters. In particular, we show necessary conditions for WP ( ψ , M ) to be a Bessel system, and we construct multiple examples of non-standard wave packet frames with prescribed dimensions.

Weakly wandering vectors for Abelian unitary actions

A result of Krengel on the density of the set of vectors with infinitely many pairwise orthogonal iterates under powers of a unitary operator with continuous spectrum is generalized to actions of countable Abelian groups.

Regulatory mechanisms underlying novelty-induced grooming in the laboratory rat

Bouts of pelage cleaning can be readily evoked in laboratory rodents under conditions of exposure to novelty. Such novelty-induced grooming is described as stereotyped and rostro-caudal in its progression. The patterned structure of novelty-induced grooming makes it particularly attractive for research on the organizational and motivational underpinnings of co-ordinated behaviour. Micro-characteristics of stereotyped novelty-induced grooming bouts were studied in 27 female Wistar rats that were exposed to a novel arena with shelters for a period of 15 min. The order of grooming acts within the initial bouts was rostro-caudal, but subsequent bouts became progressively disorganized in their sequencing. The observed pattern of progressive bout disorganization may be attributed to the gradual dearousal from stress. Differences between consecutive bouts in their micro-characteristics suggest that at least some grooming actions emitted within the context of those bouts operate as relatively independent units of behaviour. Those unitary component actions appear to be integrated into protracted pelage cleaning sequences by a separate mechanism. Similar endogenous mechanisms have been proposed for other co-ordinated motor actions, which suggests that the organizational principles identified in the context of novelty-induced grooming may represent general principles that govern co-ordinated behaviour.

Vectors with orthogonal iterates along IP-sets in unitary group actions

We prove the following result in commutative harmonic analysis.THEOREM 1. Suppose\muis a probability measure on the dual\hat Gof a countable Abelian groupG, Eis a symmetric subset ofGand\sum_{g \in E-E, g \neq 0} |\widehat{\mu} (g)| 0there are anx'\buildrel\epsilon\over\sim xand an IP-set\Gammagenerated by a subsequence of\{g_i\}such that the vectors\{U_gx':g\in\Gamma\}are pairwise orthogonal. We also show that even in the case of a \mathbb{Z}-action (i.e. a single unitary operator U) the hypothesis \{g_i\} mixing alone is not sufficient for the conclusion of Theorem 2. Finally we give conditions on G and on the spectrum of the action under which that assumption is sufficient.

The structure of unitary actions of finitely generated nilpotent groups

Let $G$ be a finitely generated nilpotent group of unitary operators on a Hilbert space ${\cal H}$. We prove that ${\cal H}$ is decomposable into a direct sum ${\cal H}=\bigoplus_{\alpha\in A}{\cal L}_{\alpha}$ of pairwise orthogonal closed subspaces so that elements of $G$ permute the subspaces ${\cal L}_{\alpha}$, and if $T({\cal L}_{\alpha})={\cal L}_{\alpha}$, then the action of $T$ on ${\cal L}_{\alpha}$ is either scalar or has continuous spectrum. We also provide examples showing that analogous results do not hold for solvable non-nilpotent groups.

Unitary Actions on Nests and the Weyl Relations

Unitary Actions on Nests and the Weyl Relations

Weakly Wandering Vectors for Unitary Actions of Commutative Groups

We extend the main result of [V. Bergelson, I. Kornfeld, and B. Mityagin, Proc. Amer. Math. Soc. 119 (1993), 1127-1134.] to more general commutative group actions satisfying different mixing conditions. The more general setup required introduction of new techniques based on spectral theory as well as refinement of those used in that paper.

Unitary Actions of Levy Flows of Diffeomorphisms

A stochastic integral representation is obtained for unitary operators induced by a class of flows of diffeomorphisms of a smooth manifold which are driven by stochastic processes with stationary and independent increments.

Dixmier-Douady Classes of Dynamical Systems and Crossed Products

AbstractContinuous-trace C*-algebras A with spectrum T can be characterized as those algebras which are locally Monta equivalent to C0(T). The Dixmier-Douady class δ(A) is an element of the Čech cohomology group Ȟ3(T, ℤ) and is the obstruction to building a global equivalence from the local equivalences. Here we shall be concerned with systems (A, G, α) which are locally Monta equivalent to their spectral system (C0(T),G, τ), in which G acts on the spectrum T of A via the action induced by α. Such systems include locally unitary actions as well as N-principal systems. Our new Dixmier-Douady class δ (A, G, α) will be the obstruction to piecing the local equivalences together to form a Monta equivalence of (A, G, α) with its spectral system. Our first main theorem is that two systems (A, G, α) and (B, G, β) are Monta equivalent if and only if δ (A, G, α) = δ (B, G, β). In our second main theorem, we give a detailed formula for δ (A ⋊α G) when (A, G, α) is N-principal.

Complex Bordism and Free Unitary Actions of Finite Solvable Groups with Periodic Cohomology on Spheres.

Complex Bordism and Free Unitary Actions of Finite Solvable Groups with Periodic Cohomology on Spheres.

Homotopy groups of some homogeneous spaces and some remarks on their sections over spheres

Stable homotopy groups and a few of the unstable homotopy groups of the homogeneous spacesSO(2n)/U(s) andU(2n)/Sp(s) fors<n, and those ofSO(4n)/Sp(s) fors≤n are computed. The enumeration of homotopy classes of normalized almost complex (n−1) and (n−2) substructures with oriented framed complements are given. Some results about the existence of the equivariant cross sections of these homogeneous spaces over the spheres with respect to certain unitary group actions are proved.

Pointwise unitary automorphism groups

Let ( A , G , α ) be a separable C ∗ -dynamical system in which G is abelian and A has continuous trace. The action α is said to pointwise unitary if for each π in A ̂ there is a covariant representation ( π , U ) of the system ( A , G , α ). It is shown that for pointwise unitary actions the spectrum ( A × α G ) ∧ of the crossed product is a locally compact Hausdorff space and the restriction map Res, which takes π × U in ( A × G ) ∧ to π, maps ( A × G ) ∧ onto A ̂ . Furthermore, Res induces a homeomorphism of ( (A × α G) ∧ G ̂ onto A ̂ . These results lead to a characterization of crossed products A × α G with continuous trace: when A has continuous trace and G acts freely on A ̂ , this happens precisely when the action of G on A ̂ is also proper. This extends Green's characterization of free and proper transformation groups. Finally, it is shown that the results have applications to the problem of determining the dual topology on a locally compact group.

GENERATORS OFS1-BORDISM

In this paper generators are found for the rings US1* (the unitary S1-bordism ring) and U*(S1,{ZS}) (the unitary bordism ring with actions of the group S1 without fixed points). The generators found are S1-manifolds of the form (S3)k × CPn/(S1)k. By an obvious construction the ring US1* allows one to establish a relation between numerical invariants of manifolds with unitary actions of S1 and the set of fixed points, without using a theorem of the type of an integrality theorem. In particular, we obtain a new proof of the Atiyah-Hirzebruch formula for the generalized Todd genus of S1-manifolds. Bibliography: 9 titles.

Decreased resistance to extinction after haloperidol: Implications for the role of dopamine in reinforcement

Previous experiments have noted that the reduction in operant behavior following treatment with neuroleptic drugs resembles an extinction curve. From this it has been argued that neuroleptic drugs disrupt operant responding by blocking the hedonic properties of primary reinforcers. The present series of experiments challenge this interpretation on several grounds. Rats were trained to bar-press either for food or for brain-stimulation reward on a variable interval (VI-60 sec) schedule. They were subsequently put into a condition of non-reward (i.e. extinction) and the effects of haloperidol (0.1 mg/kg) on the rate of responding during extinction were examined. According to the anhedonia hypothesis haloperidol should not further reduce responding during extinction. Contrary to the prediction it was found that the rate of responding during haloperidol and extinction was greatly reduced compared to that measured during extinction alone. Furthermore, the anhedonia hypothesis has maintained that following neuroleptic treatment, response patterns change only after the animal has been reinforced on several occasions. However, in the third experiment of the present study which employed a VI-4 min schedule of food reinforcement, the response rate often was attenuated prior to the first reinforcement. These data indicate that the effects of neuroleptics on operant behavior cannot be accounted for in terms of unitary actions such as specific motor impairments or blockade of primary reinforcement. Rather these drugs appear to have multiple behavioral effects.