en>David Eppstein: /* Triangles */ one more main template

2014-01-11T02:05:56Z

Triangles: one more main template

← Older revision		Revision as of 04:05, 11 January 2014
Line 1:		Line 1:
	Hi there~~! :~~) ~~My name~~ is ~~Maria~~, ~~I'm~~ a ~~student studying American Politics from Bollingen~~, ~~Switzerland~~.<br><br>~~Feel free~~ to ~~visit my website~~ :: ~~[http:~~//~~somossamarios.com~~/~~wiki~~/index.~~php?title=No_More_Mistakes_With_Http:_casinobloggen~~.~~blogg~~.~~se_2014_february_den~~-~~basta~~-~~casinoon~~-i-~~karibien~~-~~aruba~~.~~html http://casinobloggen~~.~~blogg~~.se/~~2014/february/den~~-~~basta~~-~~casinoon~~-i-~~karibien~~-~~aruba.html~~]		In [[mathematics]], '''operator K-theory''' is a variant of [[K-theory]] on the [[Category (mathematics)\|category]] of [[Banach algebras]] (In most applications, these Banach algebras are [[C*-algebras]]).

			Its basic feature that distinguishes it from [[algebraic K-theory]] is that it has a [[Bott periodicity]]. So there are only two K-groups, namely <math>K_0</math>, equal to algebraic <math>K_0</math>, and <math>K_1</math>. As a consequence of the periodicity theorem, it satisfies [[excision theorem\|excision]]. This means that it associates to an [[Algebraic extension\|extension]] of [[C*-algebra]]s to a [[long exact sequence]], which, by Bott periodicity, reduces to an exact cyclic 6-term-sequence.

			Operator K-theory is a generalization of [[topological K-theory]], defined by means of [[vector bundle]]s on [[locally compact]] [[Hausdorff space]]s. Here, an n-dimensional vector bundle over a topological space X is associated to a projection in the C* algebra of matrix-valued—that is, <math>M_n(\mathbb{C})</math>-valued—continuous functions over X. Also, it is known that isomorphism of vector bundles translates to Murray-von Neumann equivalence of the associated projection in <math>K \otimes C(X)</math>, where <math>K</math> is the compact operators on a separable Hilbert space.

			Hence, the <math>K_0</math> group of a (not necessarily commutative) C* algebra A is defined as [[Grothendieck group]] generated by the Murray-von Neumann equivalence classes of projections in <math>K \otimes C(X)</math>. <math>K_0</math> is a functor from the category of C* Algebras and -homomorphisms, to the category of abelian groups and group homomorphisms. The higher K-functors are defined via a C-version of the suspension:

			<math>K_n(A) = K_0(S^n(A))</math> where

			:<math>SA = C_0(0,1) \otimes A.</math>

			However, by Bott periodicity, it turns out that <math>K_{n+2}(A)</math> and <math>K_n(A)</math> are isomorphic for each ''n'', and thus the only groups produced by this construction are <math>K_0</math> and <math>K_1</math>.

			The key reason for the introduction of K-theoretic methods into the study of C-algebras was the [[Fredholm index]]: Given a bounded linear operator on a Hilbert space that has finite dimensional kernel and co-kernel, one can associate to it an integer, which, as it turns out, reflects the 'defect' on the operator - i.e. the extent to which it is not invertible. The Freholm index map appears in the 6-term exact sequence given by the [[Calkin algebra]]. In analysis on manifolds, this index and its generalizations played a crucial role in the [[Atiyah-Singer index theorem\|index theory]] of Atiyah and Singer, where the topological index of the manifold can be expressed via the index of elliptic operators on it. Later on, [[BDF theory\|Brown, Douglas and Fillmore]] observed that the Fredholm index was the missing ingredient in classifying [[essentially normal operator]]s up to certain natural equivalence. These ideas, together with [[George A. Elliott\|Elliott]]'s classification of [[AF C-algebra]]s via K-theory led to a great deal of interest in adapting methods such as K-theory from algebraic topology into the study of operator algebras.

			This, in turn, led to [[K-homology]], [[Gennadi Kasparov\|Kasparov]]'s bivariant [[KK-Theory]], and, more recently, [[Alain Connes\|Connes]] and [[Nigel Higson\|Higson]]'s E-theory.

			==References==
			* {{Citation \| last1=Rordam \| first1=M. \| last2=Larsen \| first2=Finn \| last3=Laustsen \| first3=N. \| title=An introduction to ''K''-theory for ''C<sup>∗</sup>-algebras \| publisher=[[Cambridge University Press]] \| series=London Mathematical Society Student Texts \| isbn=978-0-521-78334-7; 978-0-521-78944-8 \| year=2000 \| volume=49}}

			[[Category:K-theory]]

en>David Eppstein: unstub, unreferenced

2012-08-25T14:52:47Z

unstub, unreferenced

New page

Hi there! :) My name is Maria, I'm a student studying American Politics from Bollingen, Switzerland.<br><br>Feel free to visit my website :: [http://somossamarios.com/wiki/index.php?title=No_More_Mistakes_With_Http:_casinobloggen.blogg.se_2014_february_den-basta-casinoon-i-karibien-aruba.html http://casinobloggen.blogg.se/2014/february/den-basta-casinoon-i-karibien-aruba.html]

Bicentric polygon - Revision history

en>David Eppstein: /* Triangles */ one more main template

en>David Eppstein: unstub, unreferenced