The Hahn-Banach Theorem and Symmetry Invariance II

Oct 112011

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on September 14.

* * * * *

Recall that we adopt the notation of Agnew and Morse [AM38] and write $[\mathscr{L},\rho,f,X]$ to denote the following:

$X$ is a real vector space;
$\rho$ is a sublinear functional on $X$ ;
$f$ is a linear functional defined on a subspace $Y$ of $X$ ;
$f$ is dominated by $\rho$ , viz., $f(x) \leq \rho(x)$ for all $x \in Y$ ;
$\Lambda$ is a collection of endomorphisms on $X$ ;
$\rho$ is invariant under each $\Lambda$ , viz., $\rho ( \Lambda x ) = \rho(x)$ for each $\Lambda \in \mathscr{L}$ ;
$f$ is invariant under each $\Lambda$ , viz., $f(\Lambda x) = f(x)$ for each $\Lambda \in \mathscr{L}$ ;
$Y$ is invariant under each $\Lambda$ , viz., $\Lambda(Y) \subseteq Y$ for each $\Lambda \in \mathscr{L}$ .

Furthermore, if $[\mathscr{L},\rho,f,X]$ is satisfied, we write $\{\mathscr{L},\rho,f,X\}$ to denote the collection of extensions $F$ of $f$ on $X$ such that $F$ is still dominated by $\rho$ , and that

$F ( \Lambda x ) = F(x)$

for all $\Lambda \in \mathscr{L}$ . In this notation, the classical Hahn-Banach theorem can be phrased as follows:

Theorem 1 (Hahn-Banach). Let $X$ be a real vector space, and $\mathscr{L}=\{I\}$ , where $I:X \to X$ is the identity mapping. If $[\mathscr{L},\rho,f,X]$ , then $\{\mathscr{L},\rho,f,X\}$ is nonempty.

Recall that a semigroup is a set with an associative binary operation. In the last talk, we have proved the following symmetry-invariant extension of Hahn-Banach:

Theorem 2 (Agnew-Morse). Let $\mathscr{L}$ be a semigroup of commuting endomorphisms on a real vector space $X$ . If $[\mathscr{L},\rho,f,X]$ , then $\{\mathscr{L},\rho,f,X]$ is nonempty.

For a proof, see pages 25-27 of Lax [Lax02].

We also recall the key example from the last talk. We write $X^*$ to denote the space of all real-valued sequences, and $X_b$ to denote the space $l^\infty(\mathbb{R})$ of all bounded real-valued sequences. The classical Banach limit extend the usual limit in the following manner:

Theorem 3. There exists an operator $L:X_b \to X_b$ , called a generalized limit or a Banach limit, such that

The generalized limit agrees with the usual limit if the usual limit exists;
$L ((x_n)_{n=1}^\infty + (y_n)_{n=1}^\infty) = L( (x_n)_{n=1}^\infty ) + L( (y_n)_{n=1}^\infty )$ ;
$L( (x_n)_{n=1}^\infty ) = L( (x_n)_{n=k}^\infty)$ for any $k \in \mathbb{N}$ ;
$\displaystyle \liminf_{n \to \infty} x_n \leq L( (x_n)_{n=1}^\infty ) \leq \limsup_{n \to \infty} x_n$ .

We now see that the above theorem is an easy consequence of the Agnew-Morse theorem. Indeed, the Agnew-Morse theorem furnishes an extension of the regular limit operator that is invariant under the translation operator. A slight modification of the Agnew-Morse theorem allows us to extend the limit operator even further. First, some definitions:

Definition 4. $X^*$ is the space of all real-valued sequences. Let $x = (x_n)_{n=1}^\infty \in X^*$ .

$T:X^* \to X^*$ is the translation operator
$T(x) = (x_2,x_3,\ldots)$ .
$H:X^* \to X^*$ is the Hölder mean operator
$H(x) = (x_1,[x_1+x_2]/2,[x_1+x_2+x_3]/3,\ldots)$ .
For each integer $r>0$ , the map $I_r:H^* \to H^*$ is the $r$ -fold iteration operator
$I_r(x) = (x_1,\ldots,x_1,x_2,\ldots,x_2,x_3\ldots,x_3,\ldots)$ ,

where each term is repeated $r$ times.

Definition 5. We define the following subspaces of $X^*$..

$X_0$ is the space of all convergent sequences with limit 0.
$X_c$ is the space of all convergent sequences.
$X_b= l^\infty(\mathbb{R})$ is the space of all bounded sequences.
For each integer $k>0$ , $X_k$ is the space of all $x \in X^*$ such that $H^k x \in X_b$ .
$X_{o(n)}$ is the set of all sequences $(x_1,\ldots,x_n,\ldots)$ in $X^*$ such that $|x_n| = o(n)$ , that is, $|x_n|$ is dominated by $n$ asymptotically.
$X = X_{o(n)} \cap [\bigcup_{k=1}^\infty X_k]$ .

The extended result is the following, which is Theorem 4 in [MWW69]:

Theorem 6. On $\bigcup_{k=1}^\infty X_k$ , there exists a positive linear functional $f_1$ such that $f_1(x)$ is the limit of the sequence $H^k x$ whenever $H^k x \in X_c$ . Furthermore, $f_1$ on $\bigcup_{k=1}^\infty$ is invariant under all elements in the semigroup generated by $H$ and $T$ , and $f_1$ on $X$ is invariant under all elements in the semigroup generated by $H$ , $T$ , and $I_r$ , where $r \in \mathbb{N}$ .

We omit the proof of this result. Interested readers can find a proof in [MWW69]. We merely point out the variant of the Agnew-Morse theorem required to establish the above theorem. To state the variant, we need some definitions.

Definition 7. A set $G$ of endomorphisms on $X$ is said to act on a subset $X_1$ of $X$ commutatively to within left $H$ -factors if each $x \in X_1$ and every $g_1,g_2 \in G$ furnish $h_1,h_2 \in H$ such that

$h_1g_1g_2(x) = h_2g_2g_1(x)$ .

Definition 8. Let $G$ be s set of endomorphisms on $X$ and $X_1$ a subset of $X$ . We say that $G$ is left-solvable over $X_1$ if there exists a sequence

$G = G_n \supseteq G_{n-1} \supseteq \cdots \supseteq G_0$

of endomorphisms such that, for each $0 \leq k leq n-1$ , the set $G_{k+1}$ acts on $X_1$ commutatively to within left $G_k$ -factors, and if $G_0$ is commutative.

Here is the required variant, which is Theorem 2 in [MWW69]; interested readers can find a proof in [MWW69]:

Theorem 9. Let $X$ be a partially ordered linear space and $G$ a set of order-preserving linear transformations of $X$ into itself. Let

$X_0 \subseteq X_1 \subseteq \cdots$

be a set of $\bar{n}$ $G$ -invariant subspaces ( $\bar{n}$ may be infinity) such that the union is $X$ . We assume that, for each $1 < n < \bar{n}$ , either

$G$ is left-solvable over $X_n$ , and for each $x \in X$ , there is an $s \in X_{n-1}$ such that $s \geq x$ ; or
for each $x \in X_n$ , there is a $g \in G$ such that $g(x) \in X_{n-1}$ ; and for each $x \in X_n$ and $g_1,g_2,g_3 \in G$ such that $g_1(x)$ and $g_2g_3(x)$ are in $X_{n-1}$ , there are $h_1,h_2 \in G$ such that
$h_1g_1g_2g_3(x) = h_2g_2g_3g_1(x)$ .

Then every $G$ -invariant positive linear functional on $X_0$ can be extended to a $G$ -invariant positive linear functional on $X$ .

As another application of the Agnew-Morse theorem, we consider generalizations of the Lebesgue measure:

Theorem 10. There exists a nonnegative finitely additive set function $m: \mathcal{P}(\mathbb{S}^1) \to [0,\infty)$ that is invariant under rotation.

A proof was presented in the talk, which is taken directly from [Lax02]. Interested readers can find a proof in pages 33-34 of [Lax02]. Powered by Theorem 9, we can generalize the above theorem to the following:

Theorem 11. Let $X$ be a compact Hausdorff space and $G$ a semigroup of continuous endomorphisms on $X$ containing the identity such that $G$ is right-solvable over itself. Then there is a Baire measure on $X$ which is $G$ -invariant and ergodic.

This is Theorem 6 in [MWW69]. No proof of the above result was presented in the talk. The paper contains a partial proof.

References

[AM38] A. G. Agnew and A. P. Morse, “Extensions of linear functionals, with applications to limits, integrals, measures, and densities,” Annals of Mathematics 39 (1938), no. 1, 20-30.
[Lax02] Peter D. Lax, Functional Analysis, John Wiley & Sons. 2002.
[MWW69] E. J. McShane, R. B. Warfield, and V. M. Warfield. “Invariant extensions of linear functionals, with applications to measures and stochastic processes,” Pacific Journal of Mathematics 28 (1969), no. 1, 121-142.

Mark H. Kim

The Hahn-Banach Theorem and Symmetry Invariance II

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