WCG space

weakly compactly generated space

A Banach space possessing a weakly compact subset $K$ (cf. Weak topology) whose linear span is dense. These spaces have regularity properties not found in a general Banach space. Examples of WCG spaces are all separable spaces (cf. Separable space; pick a sequence $( x _ { n } )$ which is dense in the unit ball and take $ K = \{ x _ { n } / n : n \in \mathbf{N} \} \cup \{ 0 \}$), all reflexive spaces (cf. Reflexive space; take $K$ to be the unit ball), all spaces $L _ { 1 } ( \mu )$ if $\mu$ is a finite or $\sigma$-finite measure (if $\mu$ is finite, take $K = \left\{ f : \int | f | ^ { 2 } \leq 1 \right\}$, i.e., the unit ball of $L _ { 2 } ( \mu )$ considered as a subset of $L _ { 1 } ( \mu )$) and certain spaces $C ( \Omega )$, see below. Counterexamples are the non-separable spaces $\text{l} _ { \infty }$ and $L _ { \infty } [ 0,1 ]$ (here, a weakly compact set can be shown to be norm separable, and hence so is its closed linear span) and ${\bf l}_1 ( \Gamma )$ if $\Gamma$ is uncountable (here, a weakly compact set is even norm compact and thus norm separable as well).

The study of WCG spaces was initiated by D. Amir and J. Lindenstrauss [a1], building on previous work by Lindenstrauss on reflexive spaces [a11]. Their key lemma establishes the existence of a projectional resolution of the identity in a WCG space. Denote the density character of a Banach space $X$ by $\operatorname{dens} (X )$, i.e., $\operatorname{dens} (X )$ is the smallest cardinal number $\kappa$ for which $X$ has a dense subset of cardinality $\kappa$. Let $\mu$ be the smallest ordinal number of cardinality $\operatorname{dens} (X )$, and let $\omega_0$ denote the smallest infinite ordinal number. Then a projectional resolution of the identity is a family of projections $P _ { \alpha }$ on $X$, $\omega _ { 0 } \leq \alpha \leq \mu$, satisfying:

1) $\| P _ { \alpha } \| = 1$ for all $\alpha$;

2) $P _ { \mu } = \operatorname{Id}$ and $P _ { \alpha } P _ { \beta } = P _ { \beta } P _ { \alpha } = P _ { \alpha }$ if $\alpha < \beta$;

3) $\operatorname { dens } ( P _ { \alpha } ( X ) ) \leq \operatorname { card } ( \alpha )$ for all $\alpha$;

4) $\overline { \cup _ { \alpha < \beta } P _ { \alpha } ( X ) } = P _ { \beta } ( X )$ if $\beta$ is a limit ordinal number. It then follows that $\alpha \mapsto P _ { \alpha } ( x )$ is continuous in the order topology of $[ \omega _ { 0 } , \mu ]$ and the norm topology of $X$, for each $x \in X$. Properties of Banach spaces admitting a projectional resolution of the identity can often be investigated by means of transfinite induction arguments over the index set $[ \omega _ { 0 } , \mu ]$, starting from the separable case.

The most important results from [a1] are the following:

a) For a WCG space $X$ there exist a set $\Gamma$ and a continuous linear one-to-one operator from $X$ into $c_0 ( \Gamma )$, the sup-normed space of all functions $f$ on $\Gamma$ such that for each $\varepsilon > 0$ the set of $\gamma$ satisfying $| f ( \gamma ) | \geq \varepsilon$ is finite.

b) There is a continuous linear injection from the dual space $X ^ { * }$ (cf. also Adjoint space) into $c_0 ( \Gamma )$ that is continuous for the weak-${}^{*}$ topology (cf. also Topological vector space) of $X ^ { * }$ and the weak topology of $c_0 ( \Gamma )$. This has important consequences on renormings of WCG spaces (see below) and on the structure of weakly compact sets. A topological space is called an Eberlein compactum if it is homeomorphic to a weakly compact set of some Banach space. Every compact metric space is an Eberlein compactum, but the ordinal space $[ 0 , \omega ]$ is not if $\omega$ is uncountable. It follows from the above that an Eberlein compactum is even homeomorphic to a weakly compact subset of some $c_0 ( \Gamma )$-space and, consequently, Eberlein compacta embed homeomorphically into a "small" subset of $[ 0,1 ] ^ { \Gamma }$; for precision, see below.

c) If $X$ is WCG, then the dual unit ball $B _ { X } *$ in its weak-${}^{*}$ topology is an Eberlein compactum. The Eberlein–Shmul'yan theorem (cf. Banach space) implies that it is weak-${}^{*}$ sequentially compact (i.e., each bounded sequence in $X ^ { * }$ has a weak-${}^{*}$ convergent subsequence).

d) A space of continuous functions $C ( \Omega )$ on a compact Hausdorff space is WCG if and only $\Omega$ is Eberlein compact.

e) Another remarkable property of WCG spaces is the separable complementation property: If $Y$ is a separable subspace of a WCG space $X$, then there exists a separable subspace $Z \subset X$ containing $Y$ that is the range of a contractive projection.

By [a3], a Banach space $X$ is WCG if and only if there is a continuous linear operator from some reflexive space into $X$ having dense range, and an Eberlein compactum is homeomorphic to a weakly compact subset of some reflexive space. An interesting topological property is that the weak topology of a WCG space is a Lindelöf space [a17]. If $X ^ { * }$ is WCG, then every separable subspace of $X$ has a separable dual; in other words, $X$ is an Asplund space.

As for permanence properties of WCG spaces, it is clear that quotients of WCG spaces are again WCG. However, a closed subspace of a WCG space need not be WCG; the first example of this kind was constructed by H.P. Rosenthal [a15]. In certain classes of Banach spaces, the WCG-property is known to be hereditary, for example in WCG spaces with an equivalent Fréchet-differentiable norm [a9] (cf. also Fréchet derivative). An important class of hereditarily WCG spaces are Banach spaces $X$ that are $M$-ideals in $X ^ {**}$, meaning that in the canonical decomposition of $X ^ { * * * }$ into $X ^ { * }$ and $X ^ { \perp }$ the norm is additive: $\| x ^ { * } + x ^ { \perp } \| = \| x ^ { * } \| + \| x ^ { \perp } \|$ [a6]. A Banach space is isomorphic to a subspace of a WCG space if and only if its dual unit ball in the weak-${}^{*}$ topology is an Eberlein compactum [a2]. Turning to duality, it is obvious that the dual of a WCG space need not be WCG (consider $\mathbf{l}_{1}$ with its dual space $\text{l} _ { \infty }$); however, there are also examples of non-WCG spaces with WCG duals [a10]. It is an open problem (1998) whether $X$ has to be WCG whenever $X ^ {**}$ is.

As remarked above, the injection of WCG spaces into $c_0 ( \Gamma )$ leads to renorming results; for example, since $c_0 ( \Gamma )$ has a strictly convex equivalent norm (see Banach space), every WCG space can be renormed to be strictly convex. Likewise, a WCG space $X$ has a Gâteaux-differentiable equivalent norm (cf. Gâteaux derivative), whose corresponding dual norm on $X ^ { * }$ is strictly convex. A much stronger result is due to S. Troyanski [a18]: If $X$ is WCG, then $X$ has an equivalent locally uniformly rotund norm whose dual norm is strictly convex. If $X ^ { * }$ is WCG, then $X$ has an equivalent locally uniformly rotund norm whose dual norm is locally uniformly rotund, too; in particular, this norm is Fréchet differentiable (cf. also Fréchet derivative). Recall that a norm is locally uniformly rotund (or convex) if $\| x _ { n } \| _ { \rightarrow } \| x \|$ and $\| ( x _ { n } + x ) / 2 \| \rightarrow \| x \|$ imply $x _ { n } \rightarrow x$ (see Banach space).

A Markushevich basis of a Banach space $X$ is a system $\{ ( x _ { i } , x _ { i } ^ { * } ) : i \in I \} \subset X \times X ^ { * }$ that is bi-orthogonal ($x _ { i } ^ { * } ( x _ { j } ) = \delta _ { i j }$), fundamental (the linear span of the $x_{i}$ is dense) and total (if $x _ { i } ^ { * } ( x ) = 0$ for all $i$, then $x = 0$; equivalently, the linear span of the $x _ { i } ^ { * }$ is weak-${}^{*}$ dense); it is called shrinking if the linear span of the $x _ { i } ^ { * }$ is even norm dense. Every WCG space admits a Markushevich basis, and a Banach space with a shrinking Markushevich basis is WCG; in fact, such a space is hereditarily WCG.

Generalizations.

In the 1990s, several generalizations of the concept of a WCG space have been investigated (see [a7]). One of these is the notion of a weakly countably determined space (a WCD space), introduced by L. Vašák [a19]. A Banach space $X$ is said to be WCD if there are countably many weak-${}^{*}$ compact subsets $K _ { 1 } , K _ { 2 } , \ldots$ of $X ^ {**}$ such that whenever $x \in X$ and $x ^ { * * } \in X ^ { * * } \backslash X$, then $x \in K _ { n }$ and $x ^ { * * } \notin K _ { n }$ for some $n$. Every WCG space is WCD (consider the doubly indexed countable collection $n K + m ^ { - 1 } B _ { X^{**} } $, where $K$ is a weakly compact set generating $X$), and since the latter class is hereditary, even every subspace of a WCG space is WCD. On the other hand, there are WCD spaces which are not isomorphic to subspaces of any WCG space. Essentially all the results on WCG spaces carry over to this larger class and thus to subspaces of WCG spaces: WCD spaces have projectional resolutions of the identity, they inject into $c_0 ( \Gamma )$, they enjoy the separable complementation property, their weak topology is Lindelöf, they can be renormed with locally uniformly rotund norms, and they have Markushevich bases.

The class of compact spaces that goes with WCD spaces are the Gul'ko compact spaces, or Gul'ko compacta. By definition, $\Omega$ is Gul'ko compact if $C ( \Omega )$ is WCD. This class can also be described topologically. For a set $\Gamma$, let

\begin{equation*} \Sigma ( \Gamma ) : = \left\{ f \in [ 0,1 ] ^ { \Gamma } : \begin{array} { c c } { f ( \gamma ) \neq 0 } \\ { \text { for at most countable many } \gamma } \end{array} \right\} \end{equation*}

equipped with the product topology. Then $\Omega$ is Gul'ko compact if and only if it is homeomorphic to some compact subset $\Omega ^ { \prime }$ of some $\Sigma ( \Gamma )$ so that there exist $\Gamma _ { 1 } , \Gamma _ { 2 } , \ldots \subset \Gamma$ with the property that for every $\gamma _ { 0 } \in \Gamma$ and for every $f \in \Omega ^ { \prime }$ there is an $m \in \mathbf{N}$ such that $\gamma _ { 0 } \in \Gamma _ { m }$ and $\{ \gamma \in \Gamma _ { m } : f ( \gamma ) \neq 0 \}$ is finite. By contrast, $\Omega$ is Eberlein compact if and only if it is homeomorphic to some compact subset $\Omega ^ { \prime }$ of some $\Sigma ( \Gamma )$ so that there exist $\Gamma _ { 1 } , \Gamma _ { 2 } , \ldots \subset \Gamma$ with the property that for every $\gamma _ { 0 } \in \Gamma$ there is an $m \in \mathbf{N}$ such that $\gamma _ { 0 } \in \Gamma _ { m }$ and for every $f \in \Omega ^ { \prime }$ and every $n \in \mathbf N$, the set $\{ \gamma \in \Gamma _ { n } : f ( \gamma ) \neq 0 \}$ is finite. A Banach space is WCD if and only if its dual unit ball in the weak-${}^{*}$ topology is Gul'ko compact. Note that a Corson compact space, or Corson compactum, is, by definition, a compact space homeomorphic to some compact subset of $\Sigma ( \Gamma )$ for a suitable $\Gamma$. It is known that $X$ has a projectional resolution of the identity if $( B _ { X ^ *} , w ^ { * } )$ is Corson compact.

Simpler proofs of the existence of a projectional resolution of the identity in a WCG space (in fact, in a WCD space) have been given by S.P. Gul'ko [a8], J. Orihuela and M. Valdivia [a14], and C. Stegall [a16]. The theory of WCG spaces is surveyed in [a5] and [a12]; for more recent accounts see [a4], [a7] and [a13].

References

[a1]	D. Amir, J. Lindenstrauss, "The structure of weakly compact sets in Banach spaces" Ann. of Math. , 88 (1968) pp. 35–46
[a2]	Y. Benyamini, M.E. Rudin, M. Wage, "Continuous images of weakly compact subsets of Banach spaces" Pacific J. Math. , 70 (1977) pp. 309–324
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[a4]	R. Deville, G. Godefroy, V. Zizler, "Smoothness and renormings in Banach spaces" , Longman (1993)
[a5]	J. Diestel, "Geometry of Banach spaces: Selected topics" , Lecture Notes Math. , 485 , Springer (1975)
[a6]	M. Fabian, G. Godefroy, "The dual of every Asplund space admits a projectional resolution of the identity" Studia Math. , 91 (1988) pp. 141–151
[a7]	M. Fabian, "Gâteaux differentiability of convex functions and topology" , Wiley–Interscience (1997)
[a8]	S.P. Gul'ko, "On the structure of spaces of continuous functions and their complete paracompactness" Russian Math. Surveys , 34 : 6 (1979) pp. 36–44
[a9]	K. John, V. Zizler, "Smoothness and its equivalents in weakly compactly generated Banach spaces" J. Funct. Anal. , 15 (1974) pp. 1–11
[a10]	W.B. Johnson, J. Lindenstrauss, "Some remarks on weakly compactly generated Banach spaces" Israel J. Math. , 17 (1974) pp. 219–230 (Corrigendum: 32 (1979), 382-383)
[a11]	J. Lindenstrauss, "On nonseparable reflexive Banach spaces" Bull. Amer. Math. Soc. , 72 (1966) pp. 967–970
[a12]	J. Lindenstrauss, "Weakly compact sets: their topological properties and the Banach spaces they generate" R.D. Anderson (ed.) , Symp. Infinite Dimensional Topol. , Math. Studies , 69 (1972) pp. 235–273
[a13]	S. Negrepontis, "Banach spaces and topology" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of set-theoretic topology , Elsevier Sci. (1984) pp. 1045–1142
[a14]	J. Orihuela, M. Valdivia, "Projective generators and resolutions of identity in Banach spaces" Rev. Mat. Univ. Complutense Madr. , 2 (1989) pp. 179–199
[a15]	H.P. Rosenthal, "The heredity problem for weakly compactly generated Banach spaces" Compositio Math. , 28 (1974) pp. 83–111
[a16]	Ch. Stegall, "A proof of the theorem of Amir and Lindenstrauss" Israel J. Math. , 68 (1989) pp. 185–192
[a17]	M. Talagrand, "Sur une conjecture de H.H. Corson" Bull. Sci. Math. , 99 (1975) pp. 211–212
[a18]	S.L. Troyanski, "On locally uniformly convex and differentiable norms in certain non-separable Banach spaces" Studia Math. , 37 (1971) pp. 173–180
[a19]	L. Vašák, "On one generalization of weakly compactly generated Banach spaces" Studia Math. , 70 (1981) pp. 11–19