Dirichlet's unit theorem: Difference between revisions

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In [[functional analysis]], a branch of mathematics, '''nest algebras''' are a class of [[operator algebra]]s that generalise the [[upper-triangular matrix]] algebras to a [[Hilbert space]] context. They were introduced by {{harvs|txt|authorlink=John Ringrose|last=Ringrose|year=1965}} and have many interesting properties. They are non-[[selfadjoint]] algebras, are [[closure (mathematics)|closed]] in the [[weak operator topology]] and are [[reflexive operator algebra|reflexive]].
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Nest algebras are among the simplest examples of [[commutative subspace lattice algebra]]s. Indeed, they are formally defined as the algebra of [[bounded operator]]s leaving [[Invariant (mathematics)|invariant]] each [[Linear subspace|subspace]] contained in a [[subspace nest]], that is, a set of subspaces which is [[total order|totally ordered]] by [[subset|inclusion]] and is also a [[complete lattice]]. Since the [[orthogonal projection]]s corresponding to the subspaces in a nest [[commutative operation|commute]], nests are commutative subspace lattices.
 
By way of an example, let us apply this definition to recover the finite-dimensional upper-triangular matrices. Let us work in the <math>n</math>-[[dimension]]al [[complex number|complex]] [[vector space]] <math>\mathbb{C}^n</math>, and let <math>e_1,e_2,\dots,e_n</math> be the [[standard basis]]. For <math>j=0,1,2,\dots,n</math>, let <math>S_j</math> be the <math>j</math>-dimensional subspace of <math>\mathbb{C}^n</math> [[linear span|span]]ned by the first <math>j</math> basis vectors <math>e_1,\dots,e_j</math>. Let
 
:<math>N=\{ (0)=S_0, S_1, S_2, \dots, S_{n-1}, S_n=\mathbb{C}^n \};</math>
 
then ''N'' is a subspace nest, and the corresponding nest algebra of ''n''&nbsp;&times;&nbsp;''n'' complex matrices ''M'' leaving each subspace in ''N'' invariant &nbsp; that is, satisfying <math>MS\subseteq S</math> for each ''S'' in ''N'' &ndash; is precisely the set of upper-triangular matrices.
 
If we omit one or more of the subspaces ''S<sub>j</sub>'' from ''N'' then the corresponding nest algebra consists of block upper-triangular matrices.
 
==Properties==
* Nest algebras are [[hyperreflexive]] with distance constant 1.
 
==See also==
*[[flag manifold]]
 
==References==
 
*{{Citation | last1=Ringrose | first1=John R. | title=On some algebras of operators | doi=10.1112/plms/s3-15.1.61  | id={{MR|0171174}} | year=1965 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=15 | pages=61–83}}
 
 
{{DEFAULTSORT:Nest Algebra}}
[[Category:Operator theory]]
[[Category:Operator algebras]]

Latest revision as of 11:57, 5 May 2014

Mix a tablespoon of lemon juice with equal amounts of honey and milk powder. Application to the affected area a couple times a day can be very beneficial during the healing process. The cream effectively removes tanning to give you a lighter skin. Yet, most of the time, it is great to use more than one ingredient to experience better outcomes. A great skin lightening cream is the solution you need for your skin.

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These and other similar properties will help skin restore quicker after the whitening treatment and will avoid the appearance of any adverse reactions. While it is true that physical beauty is just skin deep, it is still no reason why people, particularly women, should disregard basic skin care. Licorice inhibit tyrosinase activity of melanocytes without any cytotoxicity, it also showed that UV-B–induced pigmentation and erythema can be inhibited by topical application of 0. The use of sunscreen products helps protects the skin from the ultra violet rays of the sun, keeping melanin production at a minimum. White needles in today has become a beauty new fashion, many stars are playing the white needle to maintain their own white skin.

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