Suppose that is a Banach algebra and is a Banach -bimodule. For , 1, 2, ..., let be the Banach space of all bounded -linear mappings from into together with multilinear operator norm
(1)
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and . The elements of are called -dimensional cochains. Consider the sequence
(2)
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where
(3)
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and for , 1, 2, ...,
(4)
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where , , and .
The sequence is a complex, i.e., for each , . This complex is called the standard cohomology complex or Hochschild-Kamowitz complex for and . The th cohomology group of is said to be -dimensional (ordinary or Hochschild) cohomology group of with coefficients in and denoted by . The spaces and are denoted and , and their elements are called -dimensional cocycles and -dimensional coboundaries, respectively. Hence (Helemskii 1989, 1993, 1997).