Amenable Banach algebra

Last updated November 03, 2020 • 1 min readFrom Wikipedia, The Free Encyclopedia

In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form $a\mapsto a.x-x.a$ for some $x$ in the dual module).

An equivalent characterization is that A is amenable if and only if it has a virtual diagonal.

Examples

If A is a group algebra $L^{1}(G)$ for some locally compact group G then A is amenable if and only if G is amenable.
If A is a C*-algebra then A is amenable if and only if it is nuclear.
If A is a uniform algebra on a compact Hausdorff space then A is amenable if and only if it is trivial (i.e. the algebra C(X) of all continuous complex functions on X).
If A is amenable and there is a continuous algebra homomorphism $\theta$ from A to another Banach algebra, then the closure of ${\overline {\theta (A)}}$ is amenable.

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References

F.F. Bonsall, J. Duncan, "Complete normed algebras", Springer-Verlag (1973).
H.G. Dales, "Banach algebras and automatic continuity", Oxford University Press (2001).
B.E. Johnson, "Cohomology in Banach algebras", Memoirs of the AMS 127 (1972).
J.-P. Pier, "Amenable Banach algebras", Longman Scientific and Technical (1988).

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v t e Functional analysis (topics – glossary)
Spaces	Hilbert space Banach space Fréchet space topological vector space
Theorems	Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
Operators	bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
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	Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics	locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory

v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition of a spectrum	(Continuous Point Residual) Approximate point Compression Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Uniform algebra
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Limiting absorption principle Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law