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Difference between revisions of "Banach module"

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''(left) over a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151801.png" />''
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''(left) over a Banach algebra $A$''
  
A [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151802.png" /> together with a continuous bilinear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151803.png" /> defining on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151804.png" /> the structure of a left [[Module|module]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151805.png" /> in the algebraic sense. A right Banach module and a Banach bimodule over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151806.png" /> are defined in an analogous manner. A continuous homomorphism of two Banach modules is called a morphism. Examples of Banach modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151807.png" /> include a closed ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151808.png" /> and a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151809.png" />. A Banach module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b01518010.png" /> that can be represented as a direct factor of Banach modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b01518011.png" />, (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b01518012.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b01518013.png" /> with an added unit and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b01518014.png" /> is a Banach space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b01518015.png" />) is called projective. Cf. [[Topological tensor product|Topological tensor product]].
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A [[Banach space]] $X$ together with a continuous bilinear operator $m : A \times X \rightarrow X$ defining on $X$  the structure of a left [[module]] over $A$ in the algebraic sense. A right Banach module and a Banach bimodule over $A$ are defined in an analogous manner. A continuous homomorphism of two Banach modules is called a morphism. Examples of Banach modules over $A$ include a closed ideal in $A$ and a Banach algebra $B \supset A$. A Banach module over $A$ that can be represented as a direct factor of Banach modules $A_+ \hat\otimes E$, (where $A_+$ is $A$ with an added unit and $E$ is a Banach space and $m(a,b \otimes x) = ab \otimes x$) is called projective. Cf. [[Topological tensor product]].
  
 
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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Rieffel,  "Induced Banach representations of Banach algebras and locally compact groups"  ''J. Funct. Anal.'' , '''1'''  (1967)  pp. 443–491</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Rieffel,  "Induced Banach representations of Banach algebras and locally compact groups"  ''J. Funct. Anal.'' , '''1'''  (1967)  pp. 443–491</TD></TR>
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Latest revision as of 21:27, 17 December 2015

(left) over a Banach algebra $A$

A Banach space $X$ together with a continuous bilinear operator $m : A \times X \rightarrow X$ defining on $X$ the structure of a left module over $A$ in the algebraic sense. A right Banach module and a Banach bimodule over $A$ are defined in an analogous manner. A continuous homomorphism of two Banach modules is called a morphism. Examples of Banach modules over $A$ include a closed ideal in $A$ and a Banach algebra $B \supset A$. A Banach module over $A$ that can be represented as a direct factor of Banach modules $A_+ \hat\otimes E$, (where $A_+$ is $A$ with an added unit and $E$ is a Banach space and $m(a,b \otimes x) = ab \otimes x$) is called projective. Cf. Topological tensor product.

References

[1] M.A. Rieffel, "Induced Banach representations of Banach algebras and locally compact groups" J. Funct. Anal. , 1 (1967) pp. 443–491
How to Cite This Entry:
Banach module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_module&oldid=36964
This article was adapted from an original article by A.Ya. Khelemskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article