Grothendieck space

A Banach space $ X $ with the property that for all separable Banach spaces $ Y $( cf. Separable space), every bounded linear operator $ T $ from $ X $ to $ Y $ is weakly compact (i.e., $ T $ sends bounded subsets of $ X $ into weakly compact subsets of $ Y $).

The above property is equivalent to each of the following assertions (see [a4], [a5], [a9]).

1) Every weak- $ * $ convergent sequence in the dual space $ X ^ {*} $ of $ X $ is weakly convergent.

2) Every bounded linear operator $ T $ from $ X $ to $ c _ {0} $ is weakly compact.

3) For all Banach spaces $ Y $ such that $ Y ^ {*} $ has a weak- $ * $ sequentially compact unit ball, every bounded linear operator from $ X $ to $ Y $ is weakly compact.

4) For all weakly compactly generated Banach spaces $ Y $ (i.e., $ Y $ is the closed linear span of a relatively weakly compact set), every bounded linear operator from $ X $ to $ Y $ is weakly compact.

5) For an arbitrary Banach space $ Y $, the limit of any weakly convergent sequence of weakly compact operators from $ X $ to $ Y $ is also a weakly compact operator.

6) For any Banach space $ Y $, the limit of any strongly convergent sequence of weakly compact operators from $ X $ to $ Y $ is also a weakly compact operator.

Hence, besides the definition given at the beginning, either 1) or 2) can also be used as the definition of a Grothendieck space. Quotient spaces and complemented subspaces of a Grothendieck space are also Grothendieck spaces.

Reflexive Banach spaces are obvious examples of Grothendieck spaces (cf. Reflexive space). Every separable quotient space of a Grothendieck space is necessarily reflexive. The first non-trivial example of a Grothendieck space is the space $ C ( \Omega ) $ of continuous functions on a compact Stonean space $ \Omega $( i.e., a compact Hausdorff space in which each open set has an open closure) [a6].

Other examples of Grothendieck spaces are: $ C ( \Omega ) $, where $ \Omega $ is a compact $ \sigma $- Stonean space (each open $ F _ \sigma $- set has an open closure) or a compact $ F $- space (any two disjoint open $ F _ \sigma $- sets have disjoint closures) (see [a1], [a10]); $ L ^ \infty ( \mu ) $, where $ \mu $ is a positive measure; $ B ( S, \Sigma ) $, where $ \Sigma $ is a $ \sigma $- algebra of subsets of $ S $; injective Banach spaces; the Hardy space $ H ^ \infty ( D ) $ of all bounded analytic functions on the open unit disc $ D $[a2]; and von Neumann algebras [a8].

A uniformly bounded $ C _ {0} $- semi-group of operators (cf. Semi-group of operators) on a Grothendieck space is strongly ergodic if and only if the weak- $ * $ closure and the strong closure of the range of the dual operator of the generator $ A $ coincide [a11]. If $ C ( K ) $ is a Grothendieck space, then every sequence $ \{ T _ {n} \} $ of contractions on $ C ( K ) $ which converges to the identity in the strong operator topology actually converges in the uniform operator topology (see [a3], [a7]). In particular, this implies equivalence of strong continuity and uniform continuity for contraction $ C _ {0} $- semi-groups on $ C ( K ) $.

References

[a1]	T. Ando, "Convergent sequences of finitely additive measures" Pacific J. Math. , 11 (1961) pp. 395–404
[a2]	J. Bourgain, "$H^\infty$ is a Grothendieck space" Studia Math. , 75 (1983) pp. 193–216
[a3]	Th. Coulhon, "Suites d'operateurs sur un espace $C(K)$ de Grothendieck" C.R. Acad. Sci. Paris , 298 (1984) pp. 13–15
[a4]	J. Diestel, "Grothendieck spaces and vector measures" , Vector and Operator Valued Measures and Applications , Acad. Press (1973) pp. 97–108
[a5]	J. Diestel, J.J. Uhl, Jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977)
[a6]	A. Grothendieck, "Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$" Canadian J. Math. , 5 (1953) pp. 129–173
[a7]	H.P. Lotz, "Uniform convergence of operators on $L^\infty$ and similar spaces" Math. Z. , 190 (1985) pp. 207–220
[a8]	H. Pfitzner, "Weak compactness in the dual of a $C^*$-algebra is determined commutatively" Math. Ann. , 298 (1994) pp. 349–371
[a9]	F. Rábiger, "Beiträge zur Strukturtheorie der Grothendieck-Räume" Sitzungsber. Heidelberger Akad. Wissenschaft. Math.-Naturwiss. Kl. Abh. , 4 (1985)
[a10]	G. L. Seever, "Measures on $F$-spaces" Trans. Amer. Math. Soc. , 133 (1968) pp. 267–280
[a11]	S.-Y. Shaw, "Ergodic theorems for semigroups of operators on a Grothendieck space" Proc. Japan Acad. , 59 (A) (1983) pp. 132–135