Amenable Banach algebra

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.

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).
Volker Runde, "Amenable Banach Algebras. A Panorama", Springer Verlag (2020).

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

Operators

Algebras

Open problems

Applications

Advanced topics

Category

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	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure 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 Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
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 Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem 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 Wiener–Khinchin theorem