## Vector-valued invariant means and projections from the bidual space

Invariant means on amenable groups are an important tool in many parts of mathematics, especially in harmonic analysis. Invariant means and their generalizations for vector-valued functions play also an important role in the stability of functional equations and selections of set-valued functions.

Some generalization of the invariant mean for vector-valued functions was investigated in  and the existence of such means is connected with reflexive spaces.

Some generalized definition of an invariant mean has been used by many mathematicians as folklore (e.g. by A. Pełczyński . The explicit form of this definition we can find e.g. in the work of R. Ger .

Definition:

Let $$(S,+)$$ be a left [right] amenable semigroup, $$X$$ be a real Banach space. A linear map $$M$$ from the space of all bounded functions from a set $$S$$ into a Banach space $$X$$ into $$X$$ is called left [right] $$X$$–valued invariant mean if
1) norm of $$M$$ is 1;
2) $$M$$ on a constant function is equal this constant,
3) $$M$$ is invariant under a left [right] shifts of the argument of the function.

If $$M$$ is a left and right invariant mean, then $$M$$ is called $$X$$-valued invariant mean.

If in the above definition norm of map $$M$$ is equal at most $$\lambda \geq 1$$, then $$M$$ is called $$X$$-valued invariant $$\lambda$$-mean.

The existence of such invariant means for a fixed Banach space and for all amenable semigroups has been studied by H.B. Domecq  (his paper has a gap in the proof which was corrected by T. Kania ).

In this talk, we will show a connection between the existence of $X$ -valued invariant $$\lambda$$-means on a Banach $$X$$ and projections from subspaces of bidual space $$X^{**}$$ (with large enough density) onto $$X$$ .

References:
 H. Bustos Domecq, Vector-valued invariant means revisited, J. Math. Anal. Appl. 275(2) (2002) 512–520.
 Z. Gajda, Invariant means and representations of semigroups in the theory of functional equations, Prace Naukowe Uniwersytetu Śląskiego, Katowice (1992).
 R. Ger, The singular case in the stability behavior of linear mappings, Grazer Math. Ber. {316} (1992), 59–70.
 T. Kania, Vector-valued invariant means revisited once again, J. Math. Anal. Appl. 445 (2017), 797–802.
 A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Rozprawy Mat. 58 (1968), 92 pp.