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 d1 and d2 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 d1(x,y)=d2(b(x),b(y)) for every x,y∈X, that is, d1 can be obtained from d2 by permutating elements of X with the aid of the bijection b.
The group B generates the group ˆB of transformations of the set of all pseudometrics
on X, elements of which act as d(⋅,⋅)↦d(b(⋅),b(⋅)), where d is a pseudometrics on X and b∈B. A function f on the set of all pseudometrics on X
is called ˆB-symmetric if f is invariant under the action on its argument of elements of the group ˆB.
If two pseudometrics d1 and d2 are B-equivalent, then f(d1)=f(d2) for every ˆB-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 ˆB-symmetric
functions on sets of pseudometrics, we map these sets to some appropriate Banach space V, which is isometrically isomorphic to the Banach space ℓ1
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 ℓ1 into itself) linear continuous functionals
on ℓ1.
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 ˆB-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 ˆB, is such that the set of all x∈X, for which there exists b∈B such that b(x)≠x,
is finite.
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