This article can be linked with articles of mathematics on Amenability, Amenable Locally Compact Groups, Invariant Mean, Operator Algebras, Functional Analysis, Abstract Harmonic Analysis, Topological Group, Locally Compact Group, Banach Algebra, C*-Algebra, von Neumann Algebra.

Amenability is a fundamental concept in the study of locally compact groups with far-reaching implications in functional analysis and operator algebras. As group operator algebras began to flourish, the theory of amenability saw significant developments. In 1975, Edward G. Effros introduced the notion of inner amenability for countable discrete groups. This idea was later generalized in 1986 by Chuan-Kuan Yuan (also known as CK Yuan) to arbitrary locally compact groups. Inner amenability, a generalization of classical amenability, plays a pivotal role in the study of group von Neumann algebras and C*-algebras.

The concept of inner amenability was introduced by Effros in the study of property ${\displaystyle \Gamma }$ for type II₁ von Neumann algebras.

Note: Property ${\displaystyle \Gamma }$ indicates the existence of non-trivial central sequences — bounded sequences in the algebra that asymptotically commute with all elements and are not approximately scalar.

Let ${\displaystyle G}$ be a countable discrete group. ${\displaystyle G}$ is said to be inner amenable if there exists a finitely additive function ${\displaystyle m}$ on ${\displaystyle \ell ^{\infty }(G)}$ such that:

• Positivity and Normalization:

${\displaystyle m(f)\geq 0}$ for all ${\displaystyle f\in \ell ^{\infty }(G)}$, and ${\displaystyle m(1)=1}$.

• Inner Invariance:

For all ${\displaystyle x,y\in G}$, define ${\displaystyle xfx(y)=f(x^{-1}yx)}$. Then ${\displaystyle m(f)=m(xfx)}$.

• Nontriviality:

${\displaystyle m\neq \delta _{e}}$, the Dirac measure concentrated on identity e of G, i.e. ${\displaystyle \delta _{e}(f)=f(e)}$.

In 1986, Yuan extended inner amenability to locally compact groups by defining a mean on ${\displaystyle L^{\infty }(G)}$ and introducing inner invariant means.

Let ${\displaystyle X\subseteq L^{\infty }(G)}$ include the constant function ${\displaystyle 1}$ and be closed under complex conjugation.

• Definition 1: A mean ${\displaystyle m}$ on ${\displaystyle X}$ is a positive linear functional on ${\displaystyle X}$ with ${\displaystyle m(1)=1}$.

• Definition 2: A mean ${\displaystyle m}$ is inner invariant if ${\displaystyle m(xfx)=m(f)}$ for all ${\displaystyle x\in G}$ and ${\displaystyle f\in X}$.

• Definition 3: A locally compact group is called an [IA] group if there exists an inner invariant mean m on ${\displaystyle L^{\infty }(G)}$.

• Every abelian group.
• Every compact group (via normalized left Haar measure on G).
• Every discrete group (via the Dirac measure where ${\displaystyle \delta _{e}(f)=f(e)}$).
• Every amenable group.

• ${\displaystyle \mathrm {GL} (n,\mathbb {C} )}$ for ${\displaystyle n\geq 2}$.
• ${\displaystyle \mathrm {SL} (n,\mathbb {C} )}$ for ${\displaystyle n\geq 2}$.

Recall that the positive linear functionals of norm 1 on a C*-algebra A are called states on A and S(A), the set of all states on A, is convex and it is weakly*compact if and only if A contains identity.

Let ${\displaystyle J=\{\varphi \in L^{1}(G):\int \varphi \,d\lambda =0\}}$. The set ${\displaystyle J}$ is a closed two-sided ideal in ${\displaystyle L^{1}(G)}$.

Let ${\displaystyle \beta (J)}$ be the operator norm closure of ${\displaystyle \{\beta (\varphi ):\varphi \in J\}}$.

A number of equivalent conditions for a locally compact group ${\displaystyle G}$ to be [IA] have been established. For instance, two of these conditions are as follows:

• There exists a state ${\displaystyle \omega }$ on ${\displaystyle B(L^{2}(G))}$ such that ${\displaystyle \omega (\beta (\varphi ))=\int \varphi \,d\lambda }$ for all ${\displaystyle \varphi \in L^{1}(G)}$.

• There exists a state ${\displaystyle \omega }$ on ${\displaystyle B(L^{2}(G))}$ such that ${\displaystyle \omega (\beta (x))=1}$ for all ${\displaystyle x\in G}$.

With these preparations, we are now ready to define inner amenability for locally compact groups. Let ${\displaystyle \mathrm {CB} (G)\subseteq L^{\infty }(G)}$ denote all complex-valued, continuous, bounded functions on ${\displaystyle G}$. Note that ${\displaystyle \mathrm {CB} (G)}$ is a closed subspace of ${\displaystyle L^{\infty }(G)}$.

Definition 4: A locally compact group is inner amenable if there exists an inner invariant mean ${\displaystyle m}$ on ${\displaystyle L^{\infty }(G)}$ such that ${\displaystyle m|_{\mathrm {CB} (G)}\neq \delta _{e}}$.

Yuan also investigated whether the restriction of an inner invariant mean on ${\displaystyle L^{\infty }(G)}$ to ${\displaystyle \mathrm {CB} (G)}$ is just the Dirac measure ${\displaystyle \delta _{e}}$. This corresponds to Effros’s original definition for discrete ${\displaystyle G}$ that is not inner amenable, where ${\displaystyle \delta _{e}}$ is the only inner invariant mean on ${\displaystyle \mathrm {CB} (G)}$.

Thus, Definition 4 extends the inner invariance property to the setting of locally compact groups. Further literature provides sufficient conditions for the existence of inner invariant means on ${\displaystyle L^{\infty }(G)}$, and explores connections to topological group properties and operator algebras.

The notion of inner amenability is strictly weaker than classical amenability. All amenable groups are inner amenable, but the converse does not hold. Similarly, inner amenability implies [IA], but the reverse is false.

Examples of [IA] Groups:

• Free group ${\displaystyle F_{2}}$: Not inner amenable.
• ICC groups with property ${\displaystyle \Gamma }$: Inner amenable, but not amenable.
• Thompson’s group ${\displaystyle F}$: Inner amenable. Its amenability remains open.
• Baumslag–Solitar group ${\displaystyle \mathrm {BS} (1,2)=\langle a,b\mid bab^{-1}=a^{2}\rangle }$: Inner amenable but not amenable.
• Tarski monster groups: Not amenable and typically not inner amenable.

The relationships between different classes of groups can be summarized as:

${\displaystyle \{{\text{Amenable Groups}}\}\subset \{{\text{Inner Amenable Groups}}\}\subset \{{\text{[IA] Groups}}\}}$

• Akemann, C. A., & Walter, M. E. (1981). "Unbounded negative definite functions." Canadian Journal of Mathematics, 33(4), 862–871. DOI

• Day, M. M. (1957). "Amenable semigroups." Illinois Journal of Mathematics, 1(4), 509–544. DOI

• Dixmier, J. (1977). C*-algebras. North-Holland Publishing Company. ISBN: 978-0720407624

• Effros, E. G. (1975). "Property Γ and inner amenability." Proc. Amer. Math. Soc., 47(2), 483–486. DOI

• Greenleaf, F. P. (1969). Invariant Means on Topological Groups and Their Applications. Van Nostrand Reinhold. ISBN: 978-0442028572

• Lau, A. T.-M., & Paterson, A. L. T. (1988). "Operator theoretic characterizations of [IN]-groups and inner amenability." Proc. Amer. Math. Soc., 102(4), 893–897. DOI

• Lau, A. T.-M., & Paterson, A. L. T. (1991). "Inner amenable locally compact groups." Trans. Amer. Math. Soc., 325(1), 155–169. DOI

• Losert, V., & Rindler, H. (1987). "Conjugate invariant means." Colloquium Mathematicum, 51(1), 221–225. PDF

• Paschke, W. L. (1978). "Inner Amenability and Conjugation Operators." Proc. Amer. Math. Soc., 71(1), 117–118. DOI

• Pedersen, G. K. (1979). C*-Algebras and Their Automorphism Groups. Academic Press. ISBN: 978-0125494502

• Pier, J.-P. (1982). "Quasi-Invariance Intérieure sur les Groupes Localement Compacts." In: Actualités Mathématiques, Gauthier-Villars. DOI

• Pier, J.-P. (1984). Amenable Locally Compact Groups. Wiley-Interscience. ISBN: 978-0471893905

• Pier, J.-P. (1988). Amenable Banach Algebras. Pitman Research Notes in Mathematics Series, No. 172. ISBN: 978-0582014808

• Yeadon, F. J. (1974). "Fixed Points and Amenability: A Counterexample." J. Math. Anal. Appl., 45(3), 718–720. DOI

• Yuan, C. K. (1988). "The Existence of Inner Invariant Means on L∞(G)." J. Math. Anal. Appl., 130(2), 514–524. DOI

• Yuan, C. K. (1988). "Inner invariant means and the regular conjugation representation of L1(G)." In Eymard, P., & Pier, J.-P. (Eds.), Harmonic Analysis, Lecture Notes in Mathematics, Vol. 1359, pp. 283–287. Springer. DOI

• Yuan, C. K. (1991). "Conjugate Convolutions and Inner Invariant Means." J. Math. Anal. Appl., 157(1), 166–178. DOI