The analytic algebra associated to a quasi-analytic local algebra

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.

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Coordinates for analytic operator algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500007527 ◽

1989 ◽

Vol 31 (1) ◽

pp. 31-47

Author(s):

Baruch Solel

Keyword(s):

Abelian Group ◽

Von Neumann Algebra ◽

Operator Algebras ◽

Dual Group ◽

Positive Semigroup ◽

Analytic Operator ◽

Von Neumann ◽

Finite Von Neumann Algebra ◽

Analytic Algebra ◽

Image Position

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].

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A Remark on Structure of Projective Klingenberg Spaces over a Certain Local Algebra

Mathematics ◽

10.3390/math7080702 ◽

2019 ◽

Vol 7 (8) ◽

pp. 702 ◽

Cited By ~ 1

Author(s):

Marek Jukl

Keyword(s):

Local Ring ◽

Linear Algebra ◽

Nilpotent Element ◽

Local Algebra ◽

Geometric Description

This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using this, we present the geometric description of subsets of points of a projective Klingenberg space whose arithmetical representatives need not belong to a free submodule.

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