# 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
How to Cite This Entry:
Grothendieck space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Grothendieck_space&oldid=51669
This article was adapted from an original article by S.-Y. Shaw (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article