On isomorphisms and embeddings of C(K) spaces Grzegorz Plebanek Instytut Matematyczny, Uniwersytet Wrocławski Hejnice, January/February 2013 G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 1 / 12 K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 we then write C(K) ∼ C(L). Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 Preliminaries K and L always stand for compact Hausdorff spaces. For a given K, C(K) is the Banach space of all continuous real-valued functions f : K → R, with the usual norm: ||g|| = supx∈K |f (x)|. A linear operator T : C(K) → C(L) is an isomorphic embedding if there are M, m > 0 such that for every g ∈ C(K) m · ||g|| ¬ ||Tg|| ¬ M · ||g||. If M is the least constant with such a property then M = ||T ||, likewise m = 1/||T −1||. Isomorphic embedding T : C(K) → C(L) which is onto is called an isomorphism; we then write C(K) ∼ C(L). G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 2 / 12 Theorem Under CH, for every K of weight ¬ c,C(K) embeds isometrically into ∗ C(ω ) (which itself is isometric to l∞/c0)). Dow & Hart: Consistently, the measure algebra does not embed into P(ω)/fin, so its Stone space S is not an image of ω∗. ∗ On the other hand, C(S) ≡ L∞[0, 1] ∼ l∞ ≡ C(βω) embeds into C(ω ). Todorˇcević (2011) proved that, consistently, there is a “small” compact K such that C(K) does not embed into C(ω∗), cf. Krupski-Marciszewski (2012). C(K) spaces for nonmetrizable K G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 3 / 12 Dow & Hart: Consistently, the measure algebra does not embed into P(ω)/fin, so its Stone space S is not an image of ω∗. ∗ On the other hand, C(S) ≡ L∞[0, 1] ∼ l∞ ≡ C(βω) embeds into C(ω ). Todorˇcević (2011) proved that, consistently, there is a “small” compact K such that C(K) does not embed into C(ω∗), cf. Krupski-Marciszewski (2012). C(K) spaces for nonmetrizable K Theorem Under CH, for every K of weight ¬ c,C(K) embeds isometrically into ∗ C(ω ) (which itself is isometric to l∞/c0)). G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 3 / 12 ∗ On the other hand, C(S) ≡ L∞[0, 1] ∼ l∞ ≡ C(βω) embeds into C(ω ). Todorˇcević (2011) proved that, consistently, there is a “small” compact K such that C(K) does not embed into C(ω∗), cf. Krupski-Marciszewski (2012). C(K) spaces for nonmetrizable K Theorem Under CH, for every K of weight ¬ c,C(K) embeds isometrically into ∗ C(ω ) (which itself is isometric to l∞/c0)). Dow & Hart: Consistently, the measure algebra does not embed into P(ω)/fin, so its Stone space S is not an image of ω∗. G. Plebanek (IM UWr) Isomorphisms of C(K) spaces 2013 3 / 12 Todorˇcević (2011) proved that, consistently, there is a “small” compact K such that C(K) does not embed into C(ω∗), cf. Krupski-Marciszewski (2012). C(K) spaces for nonmetrizable K Theorem Under CH, for every K of weight ¬ c,C(K) embeds isometrically into ∗ C(ω ) (which itself is isometric to l∞/c0)). Dow & Hart: Consistently, the measure algebra does not embed into P(ω)/fin, so its Stone space S is not an image of ω∗.