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Hochschild-Kamowitz Complex

Suppose that A is a Banach algebra and X is a Banach A-bimodule. For n=0, 1, 2, ..., let C^n(A,X) be the Banach space of all bounded n-linear mappings from A×...×A into X together with multilinear operator norm

||f||=sup{||f(a_1,...,a_n)||: a_i in A,||a_i||<=1,1<=i<=n}
(1)

and C^0(A,X)=X. The elements of C^n(A,X) are called n-dimensional cochains. Consider the sequence

0->C^0(A,X)->^(delta^0)C^1(A,X)->^(delta^1)...(C^~(A,X)),
(2)

where

delta^0x(a)=ax-xa,
(3)

and for n=0, 1, 2, ...,

delta^nf(a_1,...,a_(n+1))=a_1f(a_2,...,a_(n+1))+sum_(k=1)^n(-1)^kf(a_1,...,a_(k-1),a_ka_(k+1),...,a_(n+1))+(-1)^(n+1)f(a_1,...,a_n)a_(n+1),
(4)

where x in X, a,a_1,...,a_(n+1) in A, and f in C^n(A,X).

The sequence C^~(A,X) is a complex, i.e., for each n, delta^(n+1) degreesdelta^n=0. This complex is called the standard cohomology complex or Hochschild-Kamowitz complex for A and X. The nth cohomology group of C^~(A,X) is said to be n-dimensional (ordinary or Hochschild) cohomology group of A with coefficients in X and denoted by H^n(A,X). The spaces Ker(delta^n) and Im(delta^(n-1)) are denoted Z^n(A,X) and B^n(A,X), and their elements are called n-dimensional cocycles and n-dimensional coboundaries, respectively. Hence H^n(A,X)=Z^n(A,X)/B^n(A,X) (Helemskii 1989, 1993, 1997).

This entry contributed by Mohammad Sal Moslehian

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References

Helemskii, A. Ya. The Homology of Banach and Topological Algebras. Dordrecht, Netherlands: Kluwer, 1989.Helemskii, A. Ya. Banach and Locally Convex Algebras. Oxford, England: Oxford University Press, 1993.Helemskii, A. Ya. "The Homology in Algebra of Analysis." In Handbook of Algebra, Vol. 2. Amsterdam, Netherlands: Elsevier, 1997.

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Hochschild-Kamowitz Complex

Cite this as:

Moslehian, Mohammad Sal. "Hochschild-Kamowitz Complex." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Hochschild-KamowitzComplex.html

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