Kuiper's Theorem

The set of Kuiper is a mathematical theorem that in the transition area between the field of functional analysis and the field of topology is located and a work of the Dutch mathematician Nicolaas Hendrik Kuiper goes back to the year 1965th Kuiper treated here Homotopieeigenschaften the group of invertible bounded linear operators of an infinite - dimensional separable Hilbert space and confirmed his sentence of a Michael Atiyah , Albert Solomonovich Black and Richard Sheldon Palais established guess. ${\ displaystyle H}$

Formulation of the sentence

Kuiper's theorem can be formulated as follows:

Given is an infinitely-dimensional separable -Hilbert space , where , is the field of real numbers , or , the field of complex numbers , or , the skew field of quaternions . ${\ displaystyle \ mathbb {K}}$ ${\ displaystyle H}$ ${\ displaystyle \ mathbb {K} = \ mathbb {R}}$ ${\ displaystyle \ mathbb {K} = \ mathbb {C}}$ ${\ displaystyle \ mathbb {K} = \ mathbb {H}}$

Here , provided with the operator norm , let the topological ring of the constrained linear operators on and the topological group of invertible constrained linear operators contained therein and finally the subgroup of unitary operators also contained therein . ${\ displaystyle {\ mathfrak {L}} (H)}$ ${\ displaystyle H}$ ${\ displaystyle {\ mathfrak {LG}} (H) \ subset {\ mathfrak {L}} (H)}$ ${\ displaystyle H}$ ${\ displaystyle {\ mathfrak {U}} (H) \ subset {\ mathfrak {LG}} (H)}$ ${\ displaystyle H}$

Then:

(1) is as a topological subspace of a contractible space . ${\ displaystyle {\ mathfrak {LG}} (H)}$ ${\ displaystyle {\ mathfrak {L}} (H)}$

(2) is a retract of and therefore also contractible. ${\ displaystyle {\ mathfrak {U}} (H)}$ ${\ displaystyle {\ mathfrak {LG}} (H)}$

Remarks

We call the group sometimes called the Linear Group ( English linear group ) of . Kuiper calls them in his original work, the general linear group ( English general linear group ) of . ${\ displaystyle {\ mathfrak {LG}} (H)}$ ${\ displaystyle H}$ ${\ displaystyle H}$

${\ displaystyle {\ mathfrak {LG}} (H)}$ is within an open subset . ${\ displaystyle {\ mathfrak {L}} (H)}$

It is known from topology that a contractible space is always path-connected and that every retract of a contractible space is itself one.

If one lets certain classical Banach spaces take the place of the infinitely-dimensional, separable Hilbert space , then the above partial statement (1) on contractibility still applies , as Dietmar Arlt was able to show for. ${\ displaystyle H}$ ${\ displaystyle X}$ ${\ displaystyle {\ mathfrak {LG}} (X)}$ ${\ displaystyle X = c_ {0}}$

However, as Adrien Douady has shown, direct sums of two classical Banach spaces can be constructed, so that partial statement (1) is no longer valid for them.

According to Hirzebruch / Scharlau , the theorem is important for the relationships between algebraic topology and functional analysis .

literature

D. Arlt: Contractibility of the general linear group of the space c _{0 of} the zero sequences . In: Inventiones Mathematicae . tape 1 , 1966, p. 36-44 , doi : 10.1007 / BF01389697 ( MR0198259 ).
Bernhelm Booß : Topology and Analysis . Introduction to the Atiyah-Singer index formula (= university text ). Springer Verlag, Berlin / Heidelberg / New York 1977, ISBN 3-540-08451-7 ( MR0478242 ).
Adrien Douady: Un espace de Banach dont le groupe linéaire n'est pas connexe . In: Indagationes Mathematicae . tape 68 , 1965, pp. 787-789 ( MR0187056 ).
Nicolaas H. Kuiper: The homotopy type of the unitary group of Hilbert space . In: Topology . tape 3 , 1965, p. 19-30 ( MR0179792 ).
Friedrich Hirzebruch, Winfried Scharlau: Introduction to functional analysis (= BI university pocket books . Volume 296 ). Bibliographical Institute, Mannheim / Vienna / Zurich 1971, ISBN 3-411-00296-4 ( MR0463864 ).
Dieter Lutz : Topological Groups . BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1976, ISBN 3-411-01502-0 .
Albrecht Pietsch: History of Banach Spaces and Linear Operators . Birkhäuse, Boston / Basel / Berlin 2007, ISBN 0-8176-4367-2 ( MR2300779 ).
Stephen Willard : General Topology (= Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA (et al.) 1970 ( MR0264581 ).

References and comments

^ Nicolaas H. Kuiper: The homotopy type of the unitary group of Hilbert space. In: Topology , 3, pp. 19-30
↑ Friedrich Hirzebruch, Winfried Scharlau: Introduction to Functional Analysis. 1971, p. 150 ff
↑ ^a ^b ^c ^d ^e Albrecht Pietsch: History of Banach Spaces and Linear Operators. 2007, p. 538
↑ Kuiper, op.cit., P. 20
↑ ^a ^b Hirzebruch / Scharlau, op.cit., P. 151
↑ ^a ^b Kuiper, op.cit., P. 19
↑ However, some authors speak of a linear group only in relation to groups of linear automorphisms on finite-dimensional vector spaces ; see for example: Dieter Lutz: Topologische Gruppen , 1976, p. 61.
↑ Stephen Willard: General Topology. 1970, p. 226
↑ D. Arlt: Contractibility of the general linear group of the space c _{0 of}the zero sequences. in: Invent. Math. 1, pp. 36-44
^ Adrien Douady: Un espace de Banach dont le groupe linéaire n'est pas connexe. in: Indag. Math. 68, pp. 787-789

[NHK-1-1] Nicolaas H. Kuiper: The homotopy type of the unitary group of Hilbert space. In: Topology , 3, pp. 19-30

[FH-WS-1-2] Friedrich Hirzebruch, Winfried Scharlau: Introduction to Functional Analysis. 1971, p. 150 ff

[AP-1-3] Albrecht Pietsch: History of Banach Spaces and Linear Operators. 2007, p. 538

[NHK-2-4] Kuiper, op.cit., P. 20

[FH-WS-2-5] Hirzebruch / Scharlau, op.cit., P. 151

[NHK-3-6] Kuiper, op.cit., P. 19

[7] However, some authors speak of a linear group only in relation to groups of linear automorphisms on finite-dimensional vector spaces ; see for example: Dieter Lutz: Topologische Gruppen , 1976, p. 61.

[SW-1-8] Stephen Willard: General Topology. 1970, p. 226

[DA-1-9] D. Arlt: Contractibility of the general linear group of the space c _{0 of}the zero sequences. in: Invent. Math. 1, pp. 36-44

[AD-1-10] Adrien Douady: Un espace de Banach dont le groupe linéaire n'est pas connexe. in: Indag. Math. 68, pp. 787-789