Non-commutative Gelfand-Naimark theorem

Janusz Migda

Displaying similar documents to “Non-commutative Gelfand-Naimark theorem”

The Group of Invertible Elements of the Algebra of Quaternions

Irina A. Kuzmina, Marie Chodorová (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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We have, that all two-dimensional subspaces of the algebra of quaternions, containing a unit, are 2-dimensional subalgebras isomorphic to the algebra $ℂ$ of complex numbers. It was proved in the papers of N. E. Belova. In the present article we consider a 2-dimensional subalgebra $(i)$ of complex numbers with basis $1, i$ and we construct the principal locally trivial bundle which is isomorphic to the Hopf fibration.

The natural operators lifting projectable vector fields to some fiber product preserving bundles

W. M. Mikulski (2003)

Annales Polonici Mathematici

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Admissible fiber product preserving bundle functors F on $ℱ ℳ_{m}$ are defined. For every admissible fiber product preserving bundle functor F on $ℱ ℳ_{m}$ all natural operators $B : T_{p r o j | ℱ ℳ_{m, n}} \to T F$ lifting projectable vector fields to F are classified.

Lifting to the r-frame bundle by means of connections

J. Kurek, W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let m and r be natural numbers and let $P^{r} : ℳ f_{m} \to ℱ ℳ$ be the rth order frame bundle functor. Let $F : ℳ f_{m} \to ℱ ℳ$ and $G : ℳ f_{k} \to ℱ ℳ$ be natural bundles, where $k = d i m (P^{r} ℝ^{m})$ . We describe all $ℳ f_{m}$ -natural operators A transforming sections σ of $F M \to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G (P^{r} M) \to P^{r} M$ . We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic...

Equivalence bimodule between non-commutative tori

Sei-Qwon Oh, Chun-Gil Park (2003)

Czechoslovak Mathematical Journal

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The non-commutative torus $C^{*} (ℤ^{n}, ω)$ is realized as the $C^{*}$ -algebra of sections of a locally trivial $C^{*}$ -algebra bundle over $\hat{S_{ω}}$ with fibres isomorphic to $C^{*} (ℤ^{n} / S_{ω}, ω_{1})$ for a totally skew multiplier $ω_{1}$ on $ℤ^{n} / S_{ω}$ . D. Poguntke [9] proved that $A_{ω}$ is stably isomorphic to $C (\hat{S_{ω}}) \otimes C^{*} (ℤ^{n} / S_{ω}, ω_{1}) ≅ C (\hat{S_{ω}}) \otimes A_{ϕ} \otimes M_{k l} (ℂ)$ for a simple non-commutative torus $A_{ϕ}$ and an integer $k l$ . It is well-known that a stable isomorphism of two separable $C^{*}$ -algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_{ω}$ - $C (\hat{S_{ω}}) \otimes A_{ϕ}$ -equivalence bimodule.

On natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$

A. Ntyam, A. Mba (2009)

Annales Polonici Mathematici

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Some properties and applications of natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$ are presented.

Linear liftings of affinors to Weil bundles

Jacek Dębecki (2003)

Colloquium Mathematicae

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We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A} M$ , where $T^{A}$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

Constructions on second order connections

J. Kurek, W. M. Mikulski (2007)

Annales Polonici Mathematici

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We classify all $ℱ ℳ_{m, n}$ -natural operators $: J ² ⟿ J ² V^{A}$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $(Γ) : V^{A} Y \to J ² V^{A} Y$ on the vertical Weil bundle $V^{A} Y \to M$ corresponding to a Weil algebra A.

The natural operators lifting 1-forms to some vector bundle functors

J. Kurek, W. M. Mikulski (2002)

Colloquium Mathematicae

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Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{| ℳ f ₙ} ⇝ T^{(0, 0)} (F_{| ℳ f ₙ}) *$ transforming vector fields to functions on the dual bundle functor $(F_{| ℳ f ₙ}) *$ . Next, we study the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ lifting 1-forms to $(F_{| ℳ f ₙ}) *$ . As an application we classify the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ for some well known vector bundle functors F.

Liftings of 1-forms to $(J^{r} T ) $

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

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Let $J^{r} T * M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r} T * M) *$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r} T * M) *$ is given.

Natural operators lifting linear vector fields from a vector bundle to its r-jet prolongations

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

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All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r} E$ of E are given. Similar results are deduced for the r-jet prolongations $J_{v}^{r} E$ and $J^{[r]} E$ in place of $J^{r} E$ .

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

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We prove that the problem of finding all $ℳ f_{m}$ -natural operators $B : Q ⟿ Q T^{A}$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_{M} (\nabla)$ on the Weil bundle $T^{A} M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳ f_{m}$ -natural operators $C : Q ⟿ (T ¹_{p - 1}, T * \otimes T * \otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_{M} (\nabla) : T ¹_{p - 1} M = ⨁_{M}^{p - 1} T M \to T * M \otimes T * M \otimes T M$ .

The kh-socle of a commutative semisimple Banach algebra

Youness Hadder (2020)

Mathematica Bohemica

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Let $𝒜$ be a commutative complex semisimple Banach algebra. Denote by $kh (soc (𝒜))$ the kernel of the hull of the socle of $𝒜$ . In this work we give some new characterizations of this ideal in terms of minimal idempotents in $𝒜$ . This allows us to show that a “result” from Riesz theory in commutative Banach algebras is not true.