DIPARTIMENTO DI MATEMATICA \FEDERIGO ENRIQUES" Corso di Laurea Magistrale in Matematica Stone duality above dimension zero Relatore: Laureando: Prof. Vincenzo Marra Luca Reggio Anno Accademico 2013{2014 Abstract The ordinary algebraic structures usually constitute finitary varieties: that is, they are axiomatisable by means of equations and finitary operations, as in the case of groups and rings. It was only in the sixties that the algebraic theory of the structures equipped with infinitary operations | the so-called infinitary varieties | has been developed [64, 48, 49], after the pioneering works of G. Birkhoff dating back to the thirties. Still in the thirties, M. H. Stone showed in the fundamental work [65] that the dual of the category of zero-dimensional compact Hausdorff spaces and continuous maps is equivalent to the finitary variety of Boolean algebras and their homomorphisms. This is the celebrated Stone duality. If we now lift the zero-dimensionality assumption on spaces, we are left with the category KHaus of compact Hausdorff spaces. The question arises, is there a (finitary or infinitary) variety of algebras, providing a generalisation of Boolean algebras, that is equivalent to the dual category KHausop. The answer is positive, as proved by J. Duskin in 1969 [27, 5.15.3]. However, subsequent results by B. Banaschewski [9, p. 1116] entail that every variety that is equivalent to KHausop must use an infinitary operation. On the other hand, J. Isbell had already shown [42] the existence of an infinitary variety equivalent to KHausop in which a finite number of finitary operations, together with a single infinitary operation of countable arity, suffice. Semantically, Isbell's operation is the uniformly convergent series 1 X fi : 2i i=1 The problem of providing an explicit axiomatisation of a variety equivalent to KHausop has remained open. The main result of the thesis consists in a finite axiomatisation of such a variety. ii Contents Abstract ii Symbols v Introduction vii 1 Prologue: which language suffices to capture KHausop? 1 1.1 Algebraic . .1 1.2 First-order and extensions . .4 2 Lattice-ordered groups and MV-algebras 9 2.1 Lattice-ordered groups . .9 2.1.1 The variety of `-groups . .9 2.1.2 H¨older'stheorem . 15 2.1.3 Subobjects and Quotients . 18 2.1.4 Maximal ideals . 22 2.1.5 Strong order units . 24 2.2 MV-algebras . 26 2.2.1 Basic theory . 26 2.2.2 Ideals and congruences . 29 2.2.3 Subdirect representation . 34 2.2.4 Radical and infinitesimal elements . 36 2.2.5 Representing semisimple and free MV-algebras . 39 2.3 The equivalence Γ . 47 3 Yosida duality 55 3.1 The H¨older-Yosida construction . 56 3.2 Yosida map . 60 3.3 The categorical duality . 64 4 δ-algebras 70 4.1 Summary of MV-algebraic results . 70 4.2 Definition and basic results . 72 4.3 Every δ-algebra is semisimple . 80 4.4 The representation theorem . 84 5 The Lawvere-Linton theory of δ-algebras 99 5.1 Algebraic theories . 99 iii Contents iv 5.2 The case of δ-algebras: Hilbert cubes . 101 6 C∗-algebras 110 6.1 Banach algebras . 110 6.1.1 Introduction . 110 6.1.2 Spectrum and Gelfand-Mazur theorem . 115 6.1.3 Maximal ideals and multiplicative functionals . 119 6.1.4 Gelfand transform . 124 6.1.5 Involution . 127 6.2 Gelfand-Neumark duality . 130 6.3 Monadicity of C∗ ................................ 138 7 Epilogue 146 7.1 Axiomatisability of KHausop: one negative result . 146 7.2 Axiomatisability of KHausop: one positive result . 149 Bibliography 155 Index 160 Symbols N Set of natural numbers 1; 2; 3;::: Z; Q; R; C Set of integer, rational, real and complex numbers @0 Cardinality of the set N @1 Cardinality of the set R ! First infinite ordinal !1 First uncountable ordinal =∼ Isomorphism (in the appropriate category) ' Equivalence of categories α := β α is defined as β X n Y Set-theoretic difference of the sets X and Y × Cartesian product in the category of sets Xn Cartesian product of the set X with itself n times fjK Restriction of the function f to the subset K of its domain Cop Dual of the category C Ac Completion of the semisimple MV-algebra, or archimedean `-group, A with respect to the norm induced by the unit Ad Divisible hull of the MV-algebra, or `-group, A C(X; Y ) Set of all the continuous functions from the space X to the space Y coz f Cozero-set of the function f Coz(X) Family of the cozero-sets of the space X d(x; y) Chang distance of the elements x; y of an MV-algebra Freeκ Free MV-algebra over a set of κ generators g+; g− Positive and negative parts, respectively, of the element g of an `-group jgj Absolute value of the element g of an `-group GA Chang `-group associated to the MV-algebra A GX Gleason cover of the compact Hausdorff space X ∗ HA `-group of the self-adjoint elements of the C -algebra A infinit A Set of infinitesimal elements of the MV-algebra A v Symbols vi Λ Gelfand transform Max A Maximal spectrum of the MV-algebra, or `-group, A Rad A Radical of the MV-algebra, or `-group, or Banach algebra, A ∗ ΣA Maximal spectrum of the C -algebra A σx Spectrum of the element x of a Banach algebra Y Yosida map Ωx Resolvent set of the element x of a Banach algebra Ω(X) Lattice of the open subsets of the topological space X k · k1 Uniform norm on C(X; R) or C(X; C) k · ku Seminorm induced on the unital `-group (G; u) by the unit u Categories Alexc Countably compact Alexandroff algebras and lattice homo- morphisms preserving countable joins Bool Boolean algebras and homomorphisms of Boolean algebras C∗ Commutative unital C∗-algebras and ∗-homomorphisms ∆ δ-algebras and δ-homomorphisms KHaus Compact Hausdorff spaces and continuous maps `Grp `-groups and `-homomorphisms `Grpu Unital `-groups and unital `-homomorphisms Mod T Models of the theory T and homomorphisms MV MV-algebras and MV-homomorphisms Set Sets and functions St Stone spaces and continuous maps Str Σ Σ-structures and homomorphisms YAlg Cauchy-complete, divisible, and archimedean unital `-groups and unital `-homomorphisms Introduction The main object of study of this thesis is the dual of the category KHaus of compact Hausdorff spaces and continuous maps. A celebrated result by Stone [65] shows that the full subcategory St of the category KHaus, whose objects are Stone spaces (=zero- dimensional compact Hausdorff spaces), is dually equivalent to the finitary variety of Boolean algebras. The question arises, is KHaus dually equivalent to a finitary variety. In view of results by Rosick´yand Banaschewski [61, 9] not only is the answer known to be negative, but the dual category KHausop is not axiomatisable by a wide class of first-order theories. However, Duskin had already proved in 1969 [27, 5.15.3] that the category KHaus is dually equivalent to a variety of infinitary algebras, i.e. structures with function symbols of infinite arity. Hence the problem of providing an explicit ax- iomatisation of an infinitary variety dually equivalent to KHaus arises. In 1982 Isbell proved [42] that KHausop is equivalent to a variety in which every function symbol has arity at most countable. More precisely, the signature of the latter variety consists of finitely many finitary operations, along with exactly one operation of countably infinite arity. Indeed, Isbell defined an explicit set of operations and showed that it suffices to generate the algebraic theory of KHausop, in the sense of S lomi´nsky, Lawvere, and Linton [64, 48, 49]. The algebraic theory of KHausop had been described by Negrepontis in [58], by means of Gelfand-Neumark duality between KHaus and the category of commutative unital C∗-algebras. The problem of axiomatising by equations an infinitary variety du- ally equivalent to the category KHaus has remained open. The main contribution of the thesis is to offer a solution. Using as a key tool the theory of MV-algebras | a gen- eralisation of Boolean algebras that provides the algebraic counterpart toLukasiewicz' many-valued logic | along with Isbell's basic insight on the semantic nature of the infinitary operation, in Chapter 4 we provide a finite axiomatisation. The thesis is organised as follows. Chapter 1 gives a historical account of the problem of axiomatising the dual of the category KHaus. vii Introduction viii The first two sections of Chapter 2 provide an introduction to the basic theory of lattice- ordered groups and MV-algebras. These two classes of algebraic structures are tightly related via the equivalence Γ. This connection is exploited in the third section of the chapter. The content of Chapter 2, and its exposition, are standard in the literature. Historically, an important characterisation of the dual algebra of a compact Hausdorff space has been provided by Yosida in the language of lattice-ordered vector spaces. Chapter 3 is devoted to the exposition of the related categorical duality. Here, all the results are known. However, a detailed account of Yosida duality in the case of `-groups with a strong order unit cannot be found in the literature. Chapter 4 is the core of the thesis. Here we present a finite axiomatisation of a variety of infinitary algebras, and prove that this variety forms a category that is dually equivalent to the category KHaus. The whole chapter is original, however it relies on the theory of MV-algebras introduced in Chapter 2. All the MV-algebraic results which are employed in Chapter 4 are recalled in the first section, so that the latter chapter is self-contained.