arXiv subjects
Publications and source records attributed to Debashish Goswami.
Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in \cite{Ma}. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
Given an action of a discrete quantum group (in the sense of Van Daele, Kustermans and Effros-Ruan) ${\cal A}$ on a $C^*$-algebra ${\cal C}$, satisfying some regularity assumptions resembling the proper $Γ$-compact action for a classical discrete group $Γ$ on some space, we are able to construct canonical maps $μ^i_r$ (\$μ_i$ respectively) ($i=0,1$) from the ${\cal A}$-equivariant K-homology groups $KK_i^{\cal A}({\cal C}, C |)$ to the K-theory groups $K_i(\hat{{\cal A}_r})$ ($K_i(\hat{\cal A})$ respectively), where $\hat{{\cal A}_r}$ and $\hat{\cal A}$ stand for the quantum analogues of the reduced and full group $C^*$-algebras (c.f. [11],[6]). We follow the steps of the construction of the classical Baum-Connes map, although in the context of quantum group the nontrivial modular property of the invariant weights (and the related fact that the square of the antipode is not identity) has to be taken into serious consideration, making it somewhat tricky to guess and prove the correct definitions of relevant Hilbert module structures.
We study various noncommutative geometric aspects of the compact quantum group SU_q(2) for positive q (not equal to 1), following the suggestion of Connes and his coauthors [CL, CD] for considering the so-called true Dirac operator. However, it turns out that the method of the above references do not extend to the case of positive (not equal to 1) values of q in the sense that the true Dirac operator does not have bounded commutators with "smooth" algebra elements in this case, in contrast to what happens for complex q of modulus 1. Nevertheless, we show how to obtain the canonical volume form, i.e. the Haar state, using the true Dirac operator.
We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, H(phi)egh-Krohn and Olsen [Alb] for the special case of the finite dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.
We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [KMT]. With very similar definitions and techniques as those used in [jlo], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator-theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry ([Con]) and an additional positive operator having specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant under the action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not be invariant under the above action.
This is a continuation of the study of the theory of quantum stochastic dilation of completely positive semigroups on a von Neumann or $C^*$ algebra, here with unbounded generators. The additional assumption of symmetry with respect to a semifinite trace allows the use of the Hilbert space techniques, while the covariance gives rise to better handle on domains. An Evans-Hudson flow is obtained, dilating the given semigroup.
In a noncommutative torus, effect of perturbation by inner derivation on the associated quantum stochastic process and geometric parameters like volume and scalar curvature have been studied. Cohomological calculations show that the above perturbation produces new spectral triples. Also for the Weyl C^*-algebra, the Laplacian associated with a natural stochastic process is obtained and associated volume form is calculated. Keywords: noncommutative torus, Laplacian, Dixmier trace, quantum stochastic process.
Explicit sufficient and necessary conditions for separability of higher dimensional quantum systems with rank two density matrices are given. A nonseparability inequality is also presented, for the case where one of the eigenvectors corresponding to nonzero eigenvalues is a maximally entangled state.