Grothendieck spaces, operators, and beyond Tomasz Kania Academy of Sciences of the Czech Republic, Praha Zimní škola z abstraktní analýzy Svratka, 13 leden 2019 joint work with K. Beanland & N. J. Laustsen 1 I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I call such spaces Grothendieck. This class is I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. 2 I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I call such spaces Grothendieck. This class is I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. 2 I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is 2 I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! 2 I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), 2 I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! 2 s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. 2 So is it one of those ‘non-separable’ talks? Well... Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. 2 Background Theorem (Grothendieck, 1953). Every weak* convergent sequence (fn) in the dual of `1(Γ) converges with respect to the weak topology. Overview ∗ I Grothendieck’s motivation: measure-theoretic as `1(Γ) comprises finitely additive measures on }(Γ); strengthening certain measure theoretic principles. I call such spaces Grothendieck. This class is I closed under surjective linear images (hence complemented subspaces) –just apply Hahn–Banach! I not closed under taking subspaces (c0 ⊂ `1), I closed under forming `p-sums with p 2 (1; 1) but not for p = 1! I A separable space is G. if and only if it is reflexive–use Eberlein–Šmulian. s I Ergodic theory of op on G. spaces works rather well. So is it one of those ‘non-separable’ talks? Well... 2 C(K) for K Stonean, σ-Stonean, F -space, all indecomposable spaces C(K) constructed by Koszmider, Plebanek, and others. No topological characterisation of when a C(K)-space is Grothendieck exists. If C(K) is Grothendieck, K does not have non-trivial convergent sequences. 1 H (D), bdd analytic functions (Bourgain 1983) Dual spaces with Pełczyński’s property (V ). (X has prop. (V) whenever for any space Y every unconditionally converging operator T : X ! Y is weakly compact.) I C(K)-spaces have prop. (V). I C*-algebras have prop. (V) (Pfitzner, 1994); all von Neumann algebras are Grothendieck spaces B(H), the space of ops on a Hilbert space H, is Grothendieck. This answered a question from Diestel–Uhl’s Vector measures: Finally, there is some evidence (Akemann [1967], [1968]) that the space L (H; H) of bounded linear operators on a Hilbert space is a Grothendieck space and that more generally the space L (X ; X ) is a Grothendieck space for any reflexive Banach space. Grothendieck spaces are rare animals Some examples. 3 1 H (D), bdd analytic functions (Bourgain 1983) Dual spaces with Pełczyński’s property (V ). (X has prop. (V) whenever for any space Y every unconditionally converging operator T : X ! Y is weakly compact.) I C(K)-spaces have prop. (V). I C*-algebras have prop. (V) (Pfitzner, 1994); all von Neumann algebras are Grothendieck spaces B(H), the space of ops on a Hilbert space H, is Grothendieck. This answered a question from Diestel–Uhl’s Vector measures: Finally, there is some evidence (Akemann [1967], [1968]) that the space L (H; H) of bounded linear operators on a Hilbert space is a Grothendieck space and that more generally the space L (X ; X ) is a Grothendieck space for any reflexive Banach space. Grothendieck spaces are rare animals Some examples. C(K) for K Stonean, σ-Stonean, F -space, all indecomposable spaces C(K) constructed by Koszmider, Plebanek, and others. No topological characterisation of when a C(K)-space is Grothendieck exists. If C(K) is Grothendieck, K does not have non-trivial convergent sequences. 3 Dual spaces with Pełczyński’s property (V ). (X has prop. (V) whenever for any space Y every unconditionally converging operator T : X ! Y is weakly compact.) I C(K)-spaces have prop. (V). I C*-algebras have prop. (V) (Pfitzner, 1994); all von Neumann algebras are Grothendieck spaces B(H), the space of ops on a Hilbert space H, is Grothendieck. This answered a question from Diestel–Uhl’s Vector measures: Finally, there is some evidence (Akemann [1967], [1968]) that the space L (H; H) of bounded linear operators on a Hilbert space is a Grothendieck space and that more generally the space L (X ; X ) is a Grothendieck space for any reflexive Banach space.