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Localizations for construction of quantum coset spaces

Zoran Škoda (2003)

Banach Center Publications

Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum...

Metrics in the sphere of a C*-module

Esteban Andruchow, Alejandro Varela (2007)

Open Mathematics

Given a unital C*-algebra 𝒜 and a right C*-module 𝒳 over 𝒜 , we consider the problem of finding short smooth curves in the sphere 𝒮 𝒳 = x ∈ 𝒳 : 〈x, x〉 = 1. Curves in 𝒮 𝒳 are measured considering the Finsler metric which consists of the norm of 𝒳 at each tangent space of 𝒮 𝒳 . The initial value problem is solved, for the case when 𝒜 is a von Neumann algebra and 𝒳 is selfdual: for any element x 0 ∈ 𝒮 𝒳 and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ 𝒜 ( 𝒳 ) , Z* = −Z and ∥Z∥ ≤ π, such...

Non compact boundaries of complex analytic varieties in Hilbert spaces

Samuele Mongodi, Alberto Saracco (2014)

Complex Manifolds

We treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H.We deal with the problem by cutting with a family of complex hyperplanes and applying the already known result for the compact case.

Noncommutative Borsuk-Ulam-type conjectures

Paul F. Baum, Ludwik Dąbrowski, Piotr M. Hajac (2015)

Banach Center Publications

Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if δ : A A m i n H is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra A δ H . For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated Borsuk-Ulam...

Non-commutative Hodge structures

Claude Sabbah (2011)

Annales de l’institut Fourier

This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.

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