Reports of the Academy of Sciences of the USSR
1965, Volume 163, No. 5

MATHEMATICS

V. L. LEVIN

TENSOR PRODUCTS AND FUNCTORS IN CATEGORIES OF BANACH SPACES DEFINED BY \(KB\)-LINEALS

(Presented by Academician P. S. Novikov on 23 IV 1965)

In the author’s note \((^{1})\), for each \(KB\)-lineal \(E\) a functor \(\Phi_E\) was constructed in the category of Banach spaces, which is a broad and natural generalization of the functors defined by minimal normed ideals of sequences and measurable functions \((^{2})\). The functor \(\Phi_E\) assigns to a Banach space \(X\) the completion of the algebraic tensor product \(E \otimes X\) with respect to the crossnorm \(n_E\), defined by the formula

\[ n_E\left(\sum_{k=1}^{n} e_k \otimes x^k\right)= \]

\[ = \inf \left\{\|u\|_E:\ u \geqslant \left|\sum_{k=1}^{n} e_k \langle x_k, x' \rangle\right| \text{ for all } x' \in X',\ \|x'\|\leqslant 1\right\}; \]

this completion will henceforth be denoted by \(E \widetilde{\otimes} X\). In \((^{1})\) the properties of the functor \(\Phi_E\) were studied; in particular, the space of mappings \(\{\Phi_E \to \Phi_F\}\) of one such functor into another and the dual functor \(D\Phi_E\) were described.

In the present note, which is a continuation of \((^{1})\), further properties of the tensor product \(E \widetilde{\otimes} X\) and of the functor \(\Phi_E\) are reported. For definitions relating to functors in categories of Banach spaces, see \((^{2})\).

An \(L\)-space is called, following S. Kakutani \((^{3})\), a \(KB\)-lineal \(E\) satisfying the condition \(\|e_1+e_2\|=\|e_1\|+\|e_2\|\) for any positive \(e_1,e_2\in E\); in \((^{3})\) it is proved that every \(L\)-space is linearly isometric and structurally isomorphic to the \(KB\)-lineal \(L_M^1\), where \(M=(S,\Sigma,\mu)\) is some measure space.

An \(M\)-space is called, following S. Kakutani \((^{4})\), a \(KB\)-lineal \(E\) satisfying the condition \(\|\sup(e_1,e_2)\|=\max(\|e_1\|,\|e_2\|)\) for any positive \(e_1,e_2\in E\); in \((^{4})\) it is proved that every \(M\)-space is linearly isometric and structurally isomorphic to a sublineal of a special type of the \(KB\)-lineal \(C_Q\), where \(Q\) is some bicompactum.

Theorem 1. Let \(E\) be a \(KB\)-lineal.

1) In order that, for every \(X\), the crossnorm \(n_E\) coincide on \(E\otimes X\) with the maximal crossnorm \(\pi\), it is necessary and sufficient that \(E\) be an \(L\)-space.

2) In order that, for every \(X\), the crossnorm \(n_E\) coincide on \(E\otimes X\) with the minimal crossnorm \(\varepsilon\), it is necessary and sufficient that \(E\) be an \(M\)-space.

We shall need some definitions from note \((^{1})\); let us recall them. We say that for a \(KB\)-lineal \(E\) condition \((*)\) is fulfilled if, for every monotonically increasing sequence bounded above of elements \(e_n \geqslant 0\) of \(E\), \(\|e_n\|\to\infty\) as \(n\to\infty\); condition \((**)\), if for every monotonically increasing sequence of elements \(e_n\geqslant 0\) of \(E\) having a supremum \(e=\sup_n e_n\), \(\|e_n\|\to\|e\|\) as \(n\to\infty\); condition \((***)\), if for every Banach space \(X\) and every

for \(z\in E\otimes X\)

\[ n_E(z)=\inf \left\|\sum_{k=1}^{n}|e_k|\,\|x_k\|\right\|_E, \]

where the lower bound is taken over all possible representations

\[ z=\sum_{k=1}^{n} e_k\otimes x_k . \]

Condition \((*)\) is satisfied by the majority of known \(KB\)-lineals, in particular by all \(KB\)-spaces \((^{5,6})\); it is not satisfied by the \(KB\)-lineal \(C_0\). Conditions \((**)\) and \((***)\) are fulfilled in all known cases.

We call a mapping \(\alpha:E\to Y\) of a \(KB\)-lineal \(E\) into a Banach space \(Y\) summing if it carries every series \(\sum_{k=1}^{\infty} e_k\) from \(E\) for which the series \(\sum_{k=1}^{\infty} |e_k|\) converges into an absolutely convergent series \(\sum_{k=1}^{\infty} \alpha e_k\) in \(Y\).

We call a mapping \(\alpha:X\to E\) of a Banach space \(X\) into a \(KB\)-lineal \(E\) proper if it carries the unit ball of \(X\) into a subset of \(E\) that is bounded in the sense of the ordering.

Theorem 2. Let \(E\) be a \(KB\)-lineal satisfying condition \((***)\), and let \(X\) be a Banach space. Then the space \((E\widetilde{\otimes}X)'\), conjugate to \(E\oplus X\), is isometric to the space of summing mappings of \(E\) into \(X'\), endowed with the norm

\[ \nu_E(\alpha)=\sup \left[\sum_{k=1}^{n}\|\alpha e_k\|_{X'} \,/\, \left\|\sum_{k=1}^{n}|e_k|\right\|_E\right], \]

where the least upper bound is taken over all possible finite collections of elements \(e_1,\ldots,e_n\in E\).

The space \(E'\), conjugate to the \(KB\)-lineal \(E\), is a conditionally complete \(KB\)-lineal with respect to the natural order (see \((^{5})\) or \((^{6})\)).

Proposition 1. Let \(E\) be a \(KB\)-lineal, \(X\) a Banach space, and let \(\alpha:X\to E\) be a proper mapping. Then its conjugate \(\alpha^*:E'\to X'\) is a summing mapping and

\[ \nu_{E'}(\alpha^*)\le n(\alpha)=\inf \{\|u\|_E:\ u\ge |\alpha x|\ \text{for all }x\in X,\ \|x\|\le 1\}. \]

Proposition 2. Let \(E\) be a \(KB\)-lineal, \(Y\) a Banach space, and \(X\) its closed subspace. Then \(E\widetilde{\otimes}X\) is a closed subspace of \(E\widetilde{\otimes}Y\).

Proposition 3. Let \(E\) be a \(KB\)-lineal satisfying condition \((***)\), \(X\) a Banach space, \(X_0\) its closed subspace, and \(Y=X/X_0\). Then \(E\widetilde{\otimes}Y\) is isometric to the quotient space

\[ E\widetilde{\otimes}X\,/\,E\widetilde{\otimes}X_0 . \]

Let \(E\) be a \(KB\)-lineal and \(X\) a Banach space. The identity mapping of \(E\widetilde{\otimes}X\) onto itself extends to a continuous linear mapping

\[ \omega:\Phi_E(X)\equiv \widetilde{E\otimes X}\to E\widehat{\otimes}X^* . \]

We say that the functor \(\Phi_E\) satisfies the condition of one-to-one correspondence if the mapping \(\omega\) is one-to-one for every \(X\) in the category of Banach spaces under consideration.

Theorem 3. Let the \(KB\)-lineal \(E\) satisfy condition \((***)\), and let \(\Phi_E\) satisfy the condition of one-to-one correspondence. Then the functor \(\Phi_E\) is exact, i.e., every exact triple

\[ X \xrightarrow{\alpha} Y \xrightarrow{\beta} Z \]

is carried by it into the exact triple

\[ \Phi_E(X)\xrightarrow{\Phi_E(\alpha)}\Phi_E(Y)\xrightarrow{\Phi_E(\beta)}\Phi_E(Z). \]

In \((^{1})\) a realization of the space \(\{\Phi_E\to\Phi_F\}\) was described in the form of the space \((E\to F)_r\) of regular mappings of \(E\) into \(F\). The space \(\{\Phi_E\to\Phi_F\}\) admits a somewhat different description under other conditions on the \(KB\)-lineals \(E\) and \(F\).

* \(\widetilde{E\otimes X}\) denotes, as usual, the completion of \(E\otimes X\) with respect to the minimal crossnorm.

Denote by \(c_0^n\) the \(n\)-dimensional coordinate space of elements \(x=(\xi_1,\ldots,\xi_n)\) with norm \(\|x\|_{c_0^n}=\max_{1\le k\le n}|\xi_k|\). Let \(E,F\) be \(KB\)-linear spaces. A mapping \(\alpha\in(E\to F)\) will be called \(K\)-bounded if it sends every sequence \((e_k)\) bounded in the sense of the ordering in \(E\) into a sequence \((\alpha e_k)\) of the same kind in \(F\).

We shall say that a \(KB\)-linear space \(E\) has property (N) if, for every monotonically increasing and bounded above sequence of elements \(e_n\ge 0\) \((n=1,2,\ldots)\) from \(E\) and every number \(\delta>0\), there exists an element \(e_\delta\in E\) majorizing all the \(e_n\) and such that

\[ \|e_\delta\|\le \lim_{n\to\infty}\|e_n\|+\delta . \]

For a conditionally \(\sigma\)-complete \(KB\)-linear space, conditions (N) and () are equivalent; in the general case (N) implies ().

Theorem 4. Let \(E,F\) be \(KB\)-linear spaces satisfying conditions () and (N); let \(\mathscr K\) be a category of Banach spaces containing spaces \(X_n\) for which there exist isometric embeddings \(i_n:c_0^n\to X_n\) \((n=1,2,\ldots)\). Then, for the functors \(\Phi_E\) and \(\Phi_F\) considered in \(\mathscr K\), the space \(\{\Phi_E\to\Phi_F\}\) is isometric to the space of \(K\)-bounded mappings \(\alpha:E\to F\), endowed with the norm*

\[ \|\alpha\|_0=\sup \left[\left\|\sup_{1\le k\le n} |\alpha e_k|\right\|_F \bigg/ \left\|\sup_{1\le k\le n}|e_k|\right\|_E\right], \]

where the supremum is taken over all finite sets of elements \(e_1,\ldots,e_n\in E\).

Remark. Theorem 3 in \((^1)\) is formulated with a superfluous restriction on the category \(\mathscr K\); for its validity it is enough to impose on \(\mathscr K\) the same condition as in Theorem 4 of the present paper. Similarly, Theorem 4 in \((^1)\) holds if one requires only that \(\mathscr K\) contain spaces \(X_n\) for which there exist isometric embeddings \(i_n:l_n^1\to X_n\), where \(l_n^1\) is the \(n\)-dimensional coordinate space of elements

\[ x=(\xi_1,\ldots,\xi_n) \quad\text{with norm}\quad \|x\|_{l_n^1}=\sum_{k=1}^{n}|\xi_k|. \]

Moscow State University
named after M. V. Lomonosov

Received
16 IV 1965

REFERENCES

\(^1\) V. L. Levin, DAN, 162, No. 2 (1965).
\(^2\) B. S. Mityagin, A. S. Shvarts, UMN, 19, issue 2 (116), 65 (1964).
\(^3\) S. Kakutani, Ann. Math., (2), 42, 523 (1941).
\(^4\) S. Kakutani, Ann. Math. (2), 42, 994 (1941).
\(^5\) L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semiordered Spaces, Moscow–Leningrad, 1950.
\(^6\) B. Z. Vulikh, Introduction to the Theory of Semiordered Spaces, Moscow, 1961.