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Bounded Hochschild Cohomology

Bounded Hochschild cohomology of a Banach algebra A with coefficients in a normed Banach bimodule X is the cohomology of the Hochschild-Kamowitz complex of bounded multilinear maps. If d^k:C_b^k(A,X)->C_b^(k+1)(A,X) is its coboundary map, then for k>=1

H_b^k(A,X)=Kerd^k/Imd^(k-1).

The denominator is the actual image, without taking its norm closure. For X=A, the bimodule actions are multiplication in A. The adjective bounded refers to ordinary multilinear operator norms; it does not impose complete boundedness. The Kadison-Ringrose cohomology conjecture concerns self-coefficients for von Neumann algebras.

See also

Banach Algebra, Hochschild-Kamowitz Complex, Kadison-Ringrose Cohomology Conjecture, Normed Banach Bimodule

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References

Sinclair, A. M. and Smith, R. R. "The Hochschild Cohomology Problem for von Neumann Algebras." Proc. Natl. Acad. Sci. USA 95, 3376-3379, 1998. https://doi.org/10.1073/pnas.95.7.3376.

Cite this as:

Weisstein, Eric W. "Bounded Hochschild Cohomology." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BoundedHochschildCohomology.html

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