# «by Andre Val Kornell A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in ...»

Operator algebras in Solovay’s model

by

Andre Val Kornell

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

Graduate Division

of the

University of California, Berkeley

**Committee in charge:**

Professor Marc A. Rieﬀel, Chair

Professor Leo A. Harrington

Professor Raphael Bousso

Spring 2015

Operator algebras in Solovay’s model

Copyright 2015

by

Andre Val Kornell

Abstract Operator algebras in Solovay’s model by Andre Val Kornell Doctor of Philosophy in Mathematics University of California, Berkeley Professor Marc A. Rieﬀel, Chair The ultraweak topology on bounded operators on a Hilbert space is given by functionals of the form x → ∞ ηn |xξn for ∞ ηn 2 ∞ and ∞ ξn 2 ∞. By analogy with n=0 n=0 n=0 the ultraweak topology, we deﬁne the continuum-weak topology to be given by functionals ∞ ∞ ∞ of the form x → 0 ηt |xξt dt for 0 η 2 dt ∞ and 0 ξt 2 dt ∞. In order to make ∞ sense of the integral 0 ηt |xξt dt for arbitrary bounded operators x, we work in a model of set theory where every set of real numbers is Lebesgue measurable: Solovay’s model.

Solovay’s construction produces transitive models of set theory that satisfy a number of axioms that are convenient for analysis. In any such model, every set of real numbers satisﬁes Lebesgue measurability, the Baire property, and the perfect set property. By Vitali’s theorem, such a model cannot satisfy the full axiom of choice, but it does satisfy the axiom of dependent choices, which allows us to make choices during a countable recursive construction. If we specify that the input model for Solovay’s construction satisﬁes the axiom of constructibility, then the output model also satisﬁes the axiom of choice almost everywhere, which allows us to make choices for any family of sets indexed by the real numbers, at almost all indices.

Many, but not all, familiar theorems continue to hold in our Solovay model N. In general, the proof of such a theorem relies only on the axiom of dependent choices, and not on the full axiom of choice. However, it is impractically time consuming to scrutinize the proof of every needed result down to ﬁrst principles, so we develop an alternative approach. The Solovay model N is obtained as an inner model of a forcing extension M[G], and it is closed under countable unions, so many properties are absolute for N and M[G], that is, their truth value is the same in both. We show how this observation can be leveraged to establish sophisticated results in the Solovay model N.

We develop the properties of the continuum-weak topology by analogy with those of the ultraweak topology. We then deﬁne a V*-algebra to be a ∗-algebra of operators that is closed in the continuum-weak topology, by analogy with the deﬁnition of von Neumann algebras. For each self-adjoint operator x in some V*-algebra, and each bounded complexvalued function f on the spectrum of x, we can deﬁne the operator f (x), which is also in

Acknowledgments The research project that led to this dissertation is of the kind that I hoped to have when I entered graduate school. Conducting this research has been exciting and meaningful. I do not think that I could have worked on this project with another advisor. Marc Rieﬀel inspired me to embrace the noncommutative point of view that motivates so much of my thinking about operator algebras, and he gave me the freedom to pursue my intuition and my curiosity, to make this point of view my own. I thank him for his patient encouragement, and for his generous support. I also thank him for many valuable comments on the text of this dissertation.

This research owes a lot to Andrew Marks, John Steel, and Hugh Woodin. They have patiently answered my many, often misguided questions about set theory. In doing so, they introduced me to beautiful mathematics that I might have otherwise never seen. This dissertation incorporates some of their suggestions and advice. I also thank Leo Harrington for his help with appendix B.

I thank the faculty, students, and staﬀ of the UC Berkeley mathematics department.

The department has been my academic home for many years, and I will miss it. The graduate students there have been both my comrades and my teachers. In particular, I thank Alexandru Chirvasitu, Michael Hartglass, Gregory Igusa, Andrew Marks, and Sridhar Ramesh.

The research reported here was supported by National Science Foundation grant DMSChapter 1 Working in Solovay’s model In this chapter, we explain how to verify that familiar theorems hold in the Solovay model by appealing to absoluteness, and we brieﬂy survey functional analysis in this new setting.

Our primary reference for Solovay’s construction is his original paper [27], from which we have taken the notation M and N1. Jech’s Set Theory [11] is our reference for set theory in general. Schechter’s Handbook of Analysis and its Foundations is an excellent reference for the role of the axiom of choice in analysis. Some notation and terminology is reviewed in appendix A.

1.1 The role of set theory It is sometimes convenient to assume that every subset of R is Lebesgue measurable. For example, it is diﬃcult to make sense of deﬁnition 2.1.2 without this assumption. However, Vitali’s theorem implies that this assumption is inconsistent with the standard development of mathematics. Therefore, before proceeding with this assumption, we argue that many familiar mathematical results are compatible with it. Thus, we are led to examine the foundations of mathematics. We work with Zermelo-Fraenkel set theory because it is the established foundational system for mathematics, and because its extensions and fragments are the objects of modern research into consistency.

The role of set theory as a foundational system for mathematics may be explained with a simile. Computers handle ﬁnite mathematical structures by storing their data as strings of bits. Conceptually, a ﬁnite graph is not literally a string of bits, but an ideal computer can search through ﬁnite graphs by searching through strings of the appropriate kind. Similarly, the set {{}, {{}}} is not literally the number 2, but it is a set that is commonly used to represent the number 2. Thus, a set should be thought of as a block of information that can be used to represent a mathematical structure.

Set theory typically considers only the hereditary sets; a set is said to be hereditary iﬀ all of its elements are sets, all of the elements of its elements are sets, etc. The class V of all hereditary sets is assumed to satisfy the Zermelo-Fraenkel axioms of set theory, denoted CHAPTER 1. WORKING IN SOLOVAY’S MODEL 2 ZF. These axioms formalize our intuition of sets as classes of limited size. The class V is also assumed to satisfy the axiom of choice, denoted AC. The theory ZF + AC, often abbreviated ZFC, is the usual foundation for mathematics.

In summary: The class V of hereditary sets satisﬁes the theory ZFC, Zermelo-Fraenkel set theory with the axiom of choice.

1.2 The interpretation of mathematics in set theory Whatever one’s views on the foundation of mathematics, within standard mathematical discourse we assume an external mathematical reality, which consists of mathematical objects which may or may not have various mathematical properties, independent of our ability to determine whether or not a particular mathematical object has a particular mathematical property. For example, it is a simple consequence of classical logic that either there exists an uncountable subset of R not equinumerous to R or there does not, despite the fact that this proposition, the continuum hypothesis, cannot be decided from the usual mathematical assumptions.

Thus, we imagine a multitude of mathematical objects, which form our universe of discourse, and each mathematical property yields a function from our universe to the set {, ⊥} of truth-values, true and false ⊥. We also often consider properties that apply jointly to a ﬁnite sequence of objects O1,..., On. For example, group isomorphism is a property of two groups. Each such property yields a function that assigns a truth-value to each n-tuple of objects. The propositions are mathematical properties that do not refer to any variable objects; these are true or false of the universe as whole. For example, the continuum hypothesis is a proposition.

We use the word class to refer to the totality of objects satisfying a particular mathematical property (of a single variable object). For us, classes are not mathematical objects, but are simply properties that are identiﬁed if they hold of exactly the same objects.

In summary: The universe of mathematical discourse consists of mathematical objects.

Each property holds or fails for all values of variable objects O1,..., On that it references, where n depends on the property. A proposition is a property that references no variable objects; it simply holds or fails.

The above picture is quite inconvenient for mathematical logic because of the diversity of properties that appear in mathematical research. The foundation of mathematics in set theory reduces the study of general mathematical properties to the study of the ﬁrst-order properties of the hereditary sets, which can be described succinctly and precisely. A ﬁrstorder property is one that can be expressed by a formula in the language of set theory, i. e., by a grammatically correct string using parantheses, variables, the membership symbol ∈, and the logical symbols =, ¬, ∧, ∨, ∀, and ∃.

If hereditary sets are the possible strings of bits in our imaginary computer, then the language of set theory is our programming language. The interpretation of mathematics in set theory is essentially a function that assigns formulas of the language of set theory to CHAPTER 1. WORKING IN SOLOVAY’S MODEL 3 mathematical properties, in a way that preserves logical structure and truth. In particular, each property of n variable objects is interpreted as an n-ary formula of the language of set theory, that is, a formula of the language of set theory that has n free variables. The preservation of truth refers to the the requirement that if p is a proposition that is interpreted as the 0-ary formula π, then p is true iﬀ π is true in the class V of all hereditary sets.

The preservation of logical structure refers to the requirement that if a property p is interpreted as the formula π, then the property “it is not the case that p” is interpreted as the formula ¬π, the property “there is an object Om with property p” is interpreted as the formula ∃xm : π, and so on. This requirement implies that any mathematical proof can be formalized, i. e, transformed into a sequence of 0-ary set-theoretic formulas that can be mechanically veriﬁed to be a proof, provided that the interpretations of our mathematical assumptions themselves have formal proofs from our set-theoretic axioms, ZFC.

Due to the diversity of properties that appear in mathematical research, it is diﬃcult to expound an interpretation of mathematics in set theory here. The interpretations that appear in the literature diﬀer, for example, in their construction of the real numbers, i. e., in their interpretation of the property of being a real number. These diﬀerences are not important for us, but for concreteness, we mention the interpretation given in Bourbaki’s classic texts [4]. The one unacceptable feature of this interpretation is that it does not respect equality, that is, there exists a hereditary set that codes both a set and a natural number in this interpretation, even though there is no object that is both a natural number and a set. This aspect of Bourbaki’s interpretation can be easily corrected, essentially by encoding each object as a pair, with the ﬁrst element indicating its type, and the second element being its code in Bourbaki’s interpretation. This is the interpretation that we accept as our standard interpretation.

In summary: We ﬁx a standard interpretation of mathematics in set theory. The standard interpretation sends each property referring to n variable objects to a set-theoretic formula with n free variables, in a way that preserves logical structure. Each proposition is true iﬀ the corresponding 0-ary set-theoretic formula is true of the hereditary sets.

One advantage of Bourbaki’s interpretation is that all hereditary sets code themselves, i. e., any property of hereditary sets that is expressible by a formula of the language of set theory, is interpreted as that formula. This is not the case for the standard interpretation. Indeed, suppose that the standard interpretation of the property p of hereditary sets expressed by the formula ¬x0 ∈ x0 is ¬x0 ∈ x0. Then, the standard interpretation of the proposition “all objects have property p” is ∀x0 : ¬x0 ∈ x0, which is true in the class of hereditary sets, so all objects have property p. But, p is a property of hereditary sets, so we conclude that all objects are hereditary sets, contrary to our conception of mathematical reality. Nevertheless, the standard interpretation of any proposition expressed by a set-theoretic formula is provably equivalent to that formula, from the axioms of Zermelo-Fraenkel set theory.

Mathematical properties are not mathematical objects, but set-theoretic formulas are.

We use boldface for the former and lightface for the latter. For example, AC denotes the axiom of choice as a proposition about hereditary sets, while AC denotes the set-theoretic formula expressing this proposition, or equivalently, the standard interpretation of this propoCHAPTER 1. WORKING IN SOLOVAY’S MODEL 4 sition. Tarski’s undeﬁnability theorem implies that the truth of a set-theoretic formula in the class V of all hereditary sets is not a mathematical property.