Banach algebra cohomology

In mathematics, Banach algebra cohomology of a Banach algebra with coefficients in a bimodule is a cohomology theory defined in a similar way to Hochschild cohomology of an abstract algebra, except that one takes the topology into account so that all cochains and so on are continuous.[1][2][3]

## References

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1. ^ Johnson, Barry Edward (1972), Cohomology in Banach algebras, Memoirs of the American Mathematical Society, vol. 127, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1827-5, MR 0374934
2. ^ Helemskii, Alexander Ya. (1989) [1986], The homology of Banach and topological algebras, Mathematics and its Applications (Soviet Series), vol. 41, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-0217-9, MR 1093462
3. ^ Khelemskii, A.Ya. (2001) [1994], "Cohomology of Banach algebras", Encyclopedia of Mathematics, EMS Press

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Functional analysis (topics – glossary)

Spaces|

* Banach
* Besov
* Fréchet
* Hilbert
* Hölder
* Nuclear
* Orlicz
* Schwartz
* Sobolev
* Topological vector

Properties|

* Barrelled
* Complete
* Dual (Algebraic / Topological)
* Locally convex
* Reflexive
* Separable

Theorems|

* Hahn–Banach
* Riesz representation
* Closed graph
* Uniform boundedness principle
* Kakutani fixed-point
* Krein–Milman
* Min–max
* Gelfand–Naimark
* Banach–Alaoglu

Operators|

* Adjoint
* Bounded
* Compact
* Hilbert–Schmidt
* Normal
* Nuclear
* Trace class
* Transpose
* Unbounded
* Unitary

Algebras|

* Banach algebra
* C*-algebra
* Spectrum of a C*-algebra
* Operator algebra
* Group algebra of a locally compact group
* Von Neumann algebra

Open problems|

* Invariant subspace problem
* Mahler's conjecture

Applications|

* Hardy space
* Spectral theory of ordinary differential equations
* Heat kernel
* Index theorem
* Calculus of variations
* Functional calculus
* Integral linear operator
* Jones polynomial
* Topological quantum field theory
* Noncommutative geometry
* Riemann hypothesis
* Distribution (or Generalized functions)

Advanced topics|

* Approximation property
* Balanced set
* Choquet theory
* Weak topology
* Banach–Mazur distance
* Tomita–Takesaki theory

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