View source for Banach module

''(left) over a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015180/b0151801.png" />''

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]].

====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>