\(p\)-Banach spaces and \(p\)-totally convex spaces. II
Let \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\) and let \(p\) be denote a real number with \(0<p\leq 1\) in the following. A \(p\)-absolutely convex set \(C\) (with respect to \(\mathbb{K})\) is a non-empty subset of a \(\mathbb{K}\) vector space such that, for any \(x_i\in C\), \(1\leq i\leq n\), and any \(\alpha_i\in \mathbb{K}\), \(1\leq i\leq n\), with \(\sum^n_{i=1} |\alpha_i |^p\leq 1\), \(\sum^n_{i=1} \alpha_ix_i\in C\) follows. It should be noted that the author here uses a notation different from the one used in the literature: His ``\(p\)-absolutely convex'' is usually called ``absolutely \(p\)-convex''. A \(p\)-absolutely convex space \(C\) is defined as a natural generalization of a \(p\)-absolutely convex set. It is a non-empty set admitting all \(p\)-absolutely convex combinations \[ (\alpha_i\mid 1\leq i\leq n)\in \Omega_{p,fin}: =\left\{(\alpha_\nu \mid 1\leq \nu\leq m)\mid \alpha_\nu \in\mathbb{K}, \sum^m_{\nu=1} |\alpha_\nu |^p\leq 1,\;m\in\mathbb{N} \right\} \] and satisfying the usual equations, but it is not, in general, a subset of a vector space with the induced operations. These structures were introduced and investigated in the first part of the paper by the author [\textit{R. Kemper}, ``\(p\)-Banach spaces and \(p\)-totally convex spaces'', Appl. Categ. Struct. 7, No. 3, 279-295 (1999; Zbl 0949.18002)]. The \(p\)-absolutely convex spaces are equationally defined algebras with \(\Omega_{p,fin}\) as operator domain and, together with the operation preserving mappings, form a category \({\mathcal A}{\mathcal C}_p\). By taking as operator domain \[ \Omega_p: =\left\{(\alpha_i\mid i\in\mathbb{N}) \mid\alpha_i \in\mathbb{K}, \sum^\infty_{i=1} |\alpha_i |^p\leq 1\right\} \] one gets in an analogous way the category \({\mathcal T}{\mathcal C}_p\) of \(p\)-totally convex spaces (with respect to \(\mathbb{K})\) [loc. cit.]. A paradigmatic example of a \(p\)-totally convex space is the closed unit ball \(O_p(E)\) of a \(p\)-Banach space \(E\). The present paper is a continuation of [loc.cit.]. One of the fundamental theorems of this theory is 4.11 in [loc. cit.] concerning the set \[ S:= \Bigl\{\beta \in[0,1] \mid\beta^{1 \over p}x+ (1-\beta)^{1\over p}z\sim \beta^{1 \over p}y+ (1-\beta)^{1\over p}z\Bigr\}, \] if \(\sim\) is a congruence relation on a \(p\)-absolutely convex space \(C\) and \(x,y,z\in C\), 4.11 states that (i) If \(\|z\|<1\leq\|x\|^{p\over 1-p} +\|y\|^{p \over 1-p}\), then, with \(\Delta:= (\|x\|^{p\over 1-p}+ \|y\|^{p \over 1-p})^{1-p}-\|z\|^p\), \([0,(1-\|z\|^p)\Delta^{-1} [\subset S\) holds, where \(\|\square\|\) denotes the so-called \(p\)-seminorm of \(C\). (ii) \(\|x\|^{p\over 1-p}+\|y\|^{p\over 1-p}<1\) implies \(S=[0,1]\). In \(\S 1\) of this paper it is proved that this result is sharp, i.e., cannot be improved (cf. theorem 1.5) and that \(S\) may be any subset of \([0,1[\) which contains 0. Moreover, several important results for absolutely and totally convex spaces [cf. \textit{D. Pumplün} and \textit{H. Röhrl}, ``Banach spaces and totally convex spaces. I'', Commun. Algebr. 12, 953-1019 (1984; Zbl 0552.46042)] and ``Banach spaces and totally convex spaces. II'', Commun. Algebra 13, 1047-1113 (1985; Zbl 0574.46054)] are proved for \(p\)-absolutely and \(p\)-totally convex spaces and partially even generalized (e.g. 1.13, 1.15, 1.17). \(\S 2\) contains explicit constructions of the so-called ``comparison functors'' \(S_p:{\mathcal T}{\mathcal C}_p\to {\mathcal B}an_p\), \(S_{p,fin}: {\mathcal A}{\mathcal C}_p\to {\mathcal V}ec_p\), where \({\mathcal B}an_p\) is the category of \(p\)-Banach spaces and \({\mathcal V}ec_p\) the category of \(p\)-normed vector spaces (over \(\mathbb{K})\). These functors show the close relation between the algebraic theories represented by \({\mathcal A}{\mathcal C}_p\) and \({\mathcal T}{\mathcal C}_p\) on the one side and the functional analytic theories of \(p\)-Banach spaces and \(p\)-normed vector spaces, respectively.
scientific article; zbMATH DE number 1553290
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