Stable isomorphism of dual operator spaces

We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations ? and ψ of X and Y, respectively, and ternary rings of operators M₁, M₂ such that and . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras.     

Quantum Tori, Mirror Symmetry and Deformation Theory

We suggest to compactify the universal covering of the moduli space of complex structures by noncommutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of Abelian varieties, this approach gives quantum tori as a noncommutative boundary of the moduli space. Relations to mirror symmetry, modular forms and deformation theory are discussed.     

Functoriality of the Bimodule Associated to a Hilsum–Skandalis Map

We extend the functoriality of the Connes convolution algebra to the category of Hilsum–Skandalis maps between separated smooth étale groupoids. Thereby we establish that Morita equivalent separated smooth étale groupoids have Morita equivalent convolution algebras, analogously to the results of Muhly, Renault and Williams, and others, on the C^*algebras of Morita equivalent groupoids.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C (M) and ( (E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C ^*-algebras. We also discuss the semi-classical geometry arising from these deformations.     

Enantioselective Allylic Amination of Morita‐Baylis‐Hillman Acetates Catalyzed by Chiral Thiourea‐Phosphine

The enantioselective allylic amination of Morita‐Baylis‐Hillman acetates catalyzed by chiral cyclohexane‐based thiourea‐phosphine catalysts was investigated. In the presence of 20 mol% rosin‐derived thiourea‐phosphine 3j , the chiral amines were obtained in up to 88% yield and up to 85% ee.     

Characterization of Morita Equivalence Pairs of Quantales

We characterize the pairs of sup-lattices which occur as pairs of Morita equivalence bimodules between quantales in terms of the mutual relation between the sup-lattices.     

Obstructions to the extension of partial maps

One of the most important problems in topology is the minimization (in some sense) of obstructions to extending a partial map , i.e., of a subsetF ⊂ Z/A such thatf can be globally extended to its complement. It is shown that ifZ is a fixed metric space with dimZ ≤ n andp, q ≥−1 are fixed numbers, then obstructions to extending all partial maps can be concentrated in preselected fairly thin subsets ofZ. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 803–812, December 1997 Translated by O. V. Sipacheva     

Morita equivalence for crossed products by Hilbert -bimodules

We introduce the notion of the crossed product of a -algebra by a Hilbert -bimodule . It is shown that given a -algebra which carries a semi-saturated action of the circle group (in the sense that is generated by the spectral subspaces and ), then is isomorphic to the crossed product . We then present our main result, in which we show that the crossed products and are strongly Morita equivalent to each other, provided that and are strongly Morita equivalent under an imprimitivity bimodule satisfying as Hilbert -bimodules. We also present a six-term exact sequence for -groups of crossed products by Hilbert -bimodules.

     

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.