Symmetric linear functionals on the Banach space generated by pseudometrics
Abstract
In this work we consider the notion of $B$-equivalence of pseudometrics.
Two pseudometrics $d_1$ and $d_2$ on a set $X$ are called $B$-equivalent, where $B$ is a subgroup of the group of all bijections on $X,$ if there exists an element $b$ of $B$ such that $d_1(x,y) = d_2(b(x),b(y))$ for every $x,y\in X,$ that is, $d_1$ can be obtained from $d_2$ by permutating elements of $X$ with the aid of the bijection $b.$
The group $B$ generates the group $\widehat B$ of transformations of the set of all pseudometrics
on $X,$ elements of which act as $d(\cdot, \cdot)\mapsto d(b(\cdot),b(\cdot)),$ where $d$ is a pseudometrics on $X$ and $b\in B.$ A function $f$ on the set of all pseudometrics on $X$
is called $\widehat B$-symmetric if $f$ is invariant under the action on its argument of elements of the group $\widehat B.$
If two pseudometrics $d_1$ and $d_2$ are $B$-equivalent, then $f(d_1)=f(d_2)$ for every $\widehat B$-symmetric function $f.$
In general, the technique of symmetric functions is well-developed for the case of symmetric continuous polynomials and, in particular, for the case of symmetric continuous linear functionals on Banach spaces. To use this technique for the construction of $\widehat B$-symmetric
functions on sets of pseudometrics, we map these sets to some appropriate Banach space $V$, which is isometrically isomorphic to the Banach space $\ell_1$
of all absolutely summing real sequences.
We
investigate symmetric (with respect to an arbitrary group of symmetry, elements of which
map the standard Schauder basis of $\ell_1$ into itself) linear continuous functionals
on $\ell_1.$
We obtain the complete description of the structure of these functionals.
Also we establish analogical results for symmetric linear continuous functionals on the space $V.$ These results are used for the construction of $\widehat B$-symmetric functionals on the set of all pseudometrics on an arbitrary set $X$ for the following case:
the group
$B$ of bijections on $X,$ that generates the group $\widehat B,$ is such that the set of all $x\in X,$ for which there exists $b\in B$ such that $b(x)\neq x,$
is finite.
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