PMATH 950: Topics in Analysis

Estimated study time: 38 minutes

Table of contents

These notes synthesize the analytic strands of PMATH 950 (Topics in Analysis) as taught at the University of Waterloo across several recent offerings. The course rotates topic each year; the chapters below stitch together a coherent path through tensor products of operator algebras, dynamics on operator algebras, compact quantum groups, convex geometric analysis, II\(_1\) factors, Choquet theory, and operator systems. All material is paraphrased from public textbooks and published lecture notes.

Public sources used. V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press; R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vols. I–IV; G. Pisier, Tensor Products of C*-algebras and Operator Spaces, LMS Lecture Note Series; N. P. Brown and N. Ozawa, C*-Algebras and Finite-Dimensional Approximations, AMS Graduate Studies in Mathematics; A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Springer; E. G. Effros and Z.-J. Ruan, Operator Spaces, LMS Monographs; R. R. Phelps, Lectures on Choquet’s Theorem, Springer Lecture Notes in Mathematics; M. Brannan’s public lecture notes on compact quantum groups (formerly Texas A&M); selected publicly available lecture notes from MIT OCW, Stanford, and Cambridge Part III.

Chapter 1: Tensor Products of C*-algebras

Taught at UW as PMATH 950 in Fall 2020 by Vern Paulsen.

The theory of tensor products is the analytic counterpart of multilinear algebra: where finite-dimensional algebra admits a unique tensor product, the categories of Banach spaces, operator spaces, and C*-algebras admit a whole spectrum of inequivalent norms on the algebraic tensor product. The starting point is Grothendieck’s Résumé, in which he classified the reasonable cross-norms on \(E \otimes F\) for Banach spaces, identifying two extremes — the projective norm \(\|\cdot\|_\pi\) and the injective norm \(\|\cdot\|_\varepsilon\) — and proving the celebrated Grothendieck inequality relating them. Lifting this picture to the operator-algebraic setting is the main objective of this chapter.

1.1 Algebraic and analytic tensor products

Let \(A\) and \(B\) be C*-algebras. The algebraic tensor product \(A \odot B\) is generated by elementary tensors \(a \otimes b\) modulo bilinearity; it carries a natural \(*\)-algebra structure \((a_1 \otimes b_1)(a_2 \otimes b_2) = a_1 a_2 \otimes b_1 b_2\) and \((a \otimes b)^* = a^* \otimes b^*\). The question is which C*-norms it admits.

Two distinguished cross-norms always exist. The spatial (or minimal) norm uses faithful representations \(\pi_A: A \to B(H)\), \(\pi_B: B \to B(K)\) and embeds \(A \odot B \hookrightarrow B(H \otimes K)\); the result is independent of the chosen representations. The maximal norm is obtained by taking a supremum over all \(*\)-representations \(\pi: A \odot B \to B(L)\).

Definition 1.2 (Min and max norms). For \(x \in A \odot B\), set \[ \|x\|_{\min} = \|(\pi_A \otimes \pi_B)(x)\|_{B(H \otimes K)}, \qquad \|x\|_{\max} = \sup\{\|\pi(x)\| : \pi \text{ a }*\text{-rep of } A \odot B\}. \]

The completions are denoted \(A \otimes_{\min} B\) and \(A \otimes_{\max} B\).

Every C*-cross-norm \(\gamma\) satisfies \(\|\cdot\|_{\min} \leq \|\cdot\|_\gamma \leq \|\cdot\|_{\max}\), so the moniker “minimal/maximal” is faithful. The fundamental dichotomy of the chapter is whether these two norms agree.

Definition 1.3 (Nuclear C\*-algebra). A C\*-algebra \(A\) is nuclear if \(A \otimes_{\min} B = A \otimes_{\max} B\) for every C\*-algebra \(B\).

Example 1.4. Commutative C\*-algebras \(C_0(X)\) are nuclear: the tensor product \(C_0(X) \otimes_{\min} B \cong C_0(X, B)\) coincides with the maximal one. Type I algebras and AF algebras are nuclear. The reduced group C\*-algebra \(C^*_r(\mathbb{F}_2)\) of the free group on two generators is not nuclear — this is the prototypical counterexample.

1.2 Completely positive maps and the Choi–Effros characterization

The intrinsic description of nuclearity uses completely positive (cp) maps. Recall \(\varphi: A \to B\) is n-positive if \(\varphi \otimes \mathrm{id}_{M_n}: M_n(A) \to M_n(B)\) sends positives to positives, and completely positive if it is \(n\)-positive for every \(n\).

Theorem 1.5 (Choi–Effros). A C\*-algebra \(A\) is nuclear if and only if there exist nets of completely positive contractions \(\varphi_\lambda: A \to M_{n(\lambda)}(\mathbb{C})\) and \(\psi_\lambda: M_{n(\lambda)}(\mathbb{C}) \to A\) such that \(\psi_\lambda \circ \varphi_\lambda \to \mathrm{id}_A\) point-norm.

This completely positive approximation property (CPAP) shows nuclearity is fundamentally a finite-dimensional approximation phenomenon. The proof uses Stinespring dilation to convert positive maps into representations and a Hahn–Banach separation argument to distinguish min and max when no such approximation exists.

1.3 Exactness and the Kirchberg–Wassermann theory

Exactness is a weaker, more flexible cousin of nuclearity. A C*-algebra \(A\) is exact if tensoring with \(A\) preserves short exact sequences in the spatial norm: whenever \(0 \to J \to B \to B/J \to 0\) is exact, so is

\[ 0 \to A \otimes_{\min} J \to A \otimes_{\min} B \to A \otimes_{\min} (B/J) \to 0. \]

Theorem 1.6 (Kirchberg). A separable C\*-algebra \(A\) is exact if and only if it embeds (as a C\*-subalgebra) into a nuclear C\*-algebra.

Exactness is preserved under subalgebras, while nuclearity is not — this is the entire point of the distinction. The reduced free group C*-algebras \(C^*_r(\mathbb{F}_n)\) are exact but not nuclear, since they sit inside the nuclear Cuntz algebra \(\mathcal{O}_2\).

Theorem 1.7 (Kirchberg's tensor product theorem). For any C\*-algebra \(A\), one has \(A \otimes_{\max} \mathcal{O}_\infty \cong A \otimes_{\min} \mathcal{O}_\infty\), and \(\mathcal{O}_2 \otimes \mathcal{O}_2 \cong \mathcal{O}_2\).

These absorption identities are at the heart of the Kirchberg–Phillips classification of purely infinite simple nuclear C*-algebras, one of the great theorems of late twentieth-century operator algebras.

1.4 Operator spaces and the Haagerup tensor product

Beyond min and max lies a third natural norm, the Haagerup tensor norm \(\|\cdot\|_h\), defined for \(x = \sum_i a_i \otimes b_i \in A \odot B\) by

\[ \|x\|_h = \inf \left\{ \left\|\sum_i a_i a_i^*\right\|^{1/2} \left\|\sum_i b_i^* b_i\right\|^{1/2} \right\}, \]

the infimum over representations of \(x\). The Haagerup norm is the universal cross-norm for completely bounded bilinear maps and is the right tensor product in the category of operator spaces. Effros–Ruan’s abstract characterization of operator spaces — concrete subspaces of \(B(H)\) up to complete isometry are exactly those satisfying the matrix-norm axioms (M1) and (M2) — provides the framework.

Theorem 1.8 (Christensen–Effros–Sinclair / Pisier). An operator space \(V\) admits a multiplication making it completely isometrically isomorphic to a unital operator algebra if and only if there is an associative completely contractive multiplication \(V \otimes_h V \to V\).

This abstract characterization of operator algebras is the operator-space analogue of the Gelfand–Naimark theorem and underpins applications to similarity problems: Pisier’s solution of the similarity degree problem characterizes when a bounded representation of an operator algebra is similar to a completely bounded one.

Chapter 2: Operator Algebras and Dynamics

Taught at UW as PMATH 950 in Fall 2021 by Matthew Kennedy.

Group dynamics on C*-algebras and von Neumann algebras forms one of the most active branches of modern operator algebra theory. The basic input is a topological group \(G\), often discrete and countable, acting on a topological space, a measure space, or directly on an algebra by automorphisms. The output is a wealth of constructions — crossed products, groupoid algebras, boundary algebras — whose structural properties encode subtle features of the group.

2.1 Topological and measurable actions

Definition 2.1 (Action). A (left) action of a discrete group \(G\) on a compact Hausdorff space \(X\) is a homomorphism \(\alpha: G \to \mathrm{Homeo}(X)\). The action is minimal if every orbit is dense, free if \(\alpha_g(x) = x\) implies \(g = e\), and topologically amenable if there is a sequence of continuous maps \(\mu_n: X \to \mathrm{Prob}(G)\) with \(\sup_x \|\mu_n(g \cdot x) - g_* \mu_n(x)\|_1 \to 0\) for every \(g\).

Measurable actions on a probability space \((X, \mu)\) replace homeomorphisms by measure-preserving (or non-singular) automorphisms, and topological notions by measure-theoretic analogues: ergodicity replaces minimality, essential freeness replaces freeness. The dictionary between topological and measurable dynamics is one of the recurring themes of the course.

2.2 Crossed product C*-algebras

Given a discrete group \(G\) acting on a C*-algebra \(A\) by automorphisms \(\alpha_g\), one forms a new C*-algebra encoding both \(A\) and the dynamics.

Definition 2.2 (Crossed product). The algebraic crossed product \(A \rtimes_{\mathrm{alg}} G\) consists of finite formal sums \(\sum_g a_g u_g\) with \(a_g \in A\), multiplication \(u_g a u_g^* = \alpha_g(a)\). The full crossed product \(A \rtimes G\) is its enveloping C\*-algebra; the reduced crossed product \(A \rtimes_r G\) is the closure of its image under the regular covariant representation on \(A \otimes \ell^2(G)\).

When \(A = \mathbb{C}\), one recovers the full and reduced group C*-algebras \(C^*(G)\) and \(C^*_r(G)\). When \(A = C(X)\) and \(G\) acts by homeomorphisms, the crossed product captures the topological dynamical system. The full and reduced crossed products coincide whenever the action is amenable; for free groups acting on themselves, they differ.

Theorem 2.3 (Zeller-Meier; Renault). If \(G\) acts on a unital C\*-algebra \(A\) and the action is topologically amenable, then \(A \rtimes G = A \rtimes_r G\), and this crossed product is nuclear whenever \(A\) is nuclear.

2.3 Boundary actions and C*-simplicity

A fundamental question, dating to Powers (1975) and only fully resolved by Kalantar–Kennedy (2017) and Breuillard–Kalantar–Kennedy–Ozawa (2017), is when the reduced group C*-algebra \(C^*_r(G)\) is simple — admits no proper non-trivial closed two-sided ideals.

Definition 2.4 (Furstenberg boundary). The Furstenberg boundary \(\partial_F G\) of a discrete group \(G\) is the universal minimal strongly proximal \(G\)-space: a compact Hausdorff \(G\)-space which is minimal, in which orbit closures of probability measures contain Dirac masses, and which factors onto every other such space.

Theorem 2.5 (Kalantar–Kennedy). The reduced C\*-algebra \(C^*_r(G)\) is simple if and only if \(G\) acts freely on \(\partial_F G\).

This characterization translates a question about ideals in an operator algebra into a problem in topological dynamics: simplicity of \(C^*_r(G)\) becomes a freeness property of the boundary action. A complementary result of Breuillard–Kalantar–Kennedy–Ozawa identifies a unique trace on \(C^*_r(G)\) precisely when \(G\) has trivial amenable radical.

2.4 Random walks and the Poisson boundary

For a probability measure \(\mu\) on \(G\), the random walk on \(G\) with step distribution \(\mu\) generates a measure-preserving system on the path space. The Poisson boundary \(\Pi(G, \mu)\) is the space of bounded \(\mu\)-harmonic functions, with structure as a measurable \(G\)-space. Furstenberg showed the boundary is trivial precisely when the group is amenable and the measure is well-chosen, and the recent characterization of Choquet–Deny groups — those for which every spread-out \(\mu\) gives trivial boundary — by Frisch, Hartman, Tamuz, and Vahidi Ferdowsi identifies them as exactly the virtually nilpotent groups.

Theorem 2.6 (Frisch–Hartman–Tamuz–Vahidi Ferdowsi). A discrete countable group \(G\) admits a non-trivial Poisson boundary for some non-degenerate probability measure if and only if \(G\) has an ICC quotient (every nontrivial conjugacy class is infinite). Equivalently, the Choquet–Deny groups are exactly the virtually nilpotent ones.

Chapter 3: Quantum Representation Theory

Taught at UW as PMATH 950 in Winter 2023 by Michael Brannan.

The slogan of noncommutative geometry is that compact spaces correspond to commutative unital C*-algebras via Gelfand–Naimark. Compact quantum groups are obtained by relaxing commutativity while keeping the comultiplication structure that, classically, encodes a group law on the underlying space.

3.1 The Woronowicz axioms

Definition 3.1 (Compact quantum group). A compact quantum group (CQG) is a pair \((A, \Delta)\) where \(A\) is a unital C\*-algebra and \(\Delta: A \to A \otimes_{\min} A\) is a unital \(*\)-homomorphism satisfying \[ (\Delta \otimes \mathrm{id}) \circ \Delta = (\mathrm{id} \otimes \Delta) \circ \Delta \qquad (\text{coassociativity}) \]

and the cancellation conditions: \(\Delta(A)(1 \otimes A)\) and \(\Delta(A)(A \otimes 1)\) are dense in \(A \otimes A\).

The motivating example is \(A = C(G)\) for a classical compact group \(G\), with \(\Delta(f)(g, h) = f(gh)\). The cancellation conditions abstract the property that \((g, h) \mapsto (g, gh)\) and \((g, h) \mapsto (gh, h)\) are homeomorphisms of \(G \times G\). A second canonical example is \(A = C^*_r(\Gamma)\) for a discrete group \(\Gamma\), with \(\Delta(\lambda_g) = \lambda_g \otimes \lambda_g\).

Theorem 3.2 (Woronowicz; existence and uniqueness of the Haar state). Every compact quantum group \((A, \Delta)\) admits a unique state \(h: A \to \mathbb{C}\) which is bi-invariant: \((h \otimes \mathrm{id}) \Delta = h(\cdot) 1 = (\mathrm{id} \otimes h) \Delta\). The state \(h\) is faithful on the canonical dense Hopf \(*\)-subalgebra \(\mathcal{O}(\mathbb{G})\).

3.2 Representations and the Peter–Weyl theorem

A finite-dimensional representation of a CQG \(\mathbb{G} = (A, \Delta)\) on \(\mathbb{C}^n\) is an invertible matrix \(u = (u_{ij}) \in M_n(A)\) satisfying \(\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}\). It is unitary if \(u\) is unitary in \(M_n(A)\). One develops the entire representation theory — direct sums, tensor products, intertwiners, irreducibility, complete reducibility — in this language.

Theorem 3.3 (Quantum Peter–Weyl). Let \(\mathrm{Irr}(\mathbb{G})\) denote the set of equivalence classes of irreducible unitary representations. Then the linear span of all matrix coefficients \[ \mathcal{O}(\mathbb{G}) = \mathrm{span}\left\{ u^\alpha_{ij} : \alpha \in \mathrm{Irr}(\mathbb{G}),\, 1 \le i, j \le n_\alpha \right\} \]

is a dense Hopf \(*\)-subalgebra of \(A\), and the Schur orthogonality relations hold in the GNS representation of \(h\), modified by the Woronowicz character (a positive operator that fails to be the identity unless \(\mathbb{G}\) is of “Kac type”).

3.3 Tannaka–Krein reconstruction

The category \(\mathrm{Rep}(\mathbb{G})\) of finite-dimensional unitary representations of a CQG is a concrete rigid C*-tensor category equipped with a faithful unitary fiber functor to Hilbert spaces. Woronowicz’s celebrated reconstruction theorem turns this around.

Theorem 3.4 (Woronowicz–Tannaka–Krein). Every concrete rigid C\*-tensor category equipped with a unitary fiber functor to finite-dimensional Hilbert spaces arises as \(\mathrm{Rep}(\mathbb{G})\) for an essentially unique compact quantum group \(\mathbb{G}\).

This is a vast generalization of classical Tannaka–Krein and provides the principal route by which examples are constructed: one specifies a category of “intertwiner spaces” combinatorially, and the theorem produces a quantum group. The free orthogonal and free unitary quantum groups \(O_N^+\), \(U_N^+\) of Wang and Van Daele arise this way, with intertwiner spaces given by Temperley–Lieb diagrams.

3.4 Drinfeld–Jimbo q-deformations and applications

Example 3.5 (q-deformed SU(2)). For \(0 < q \le 1\), the algebra \(C(SU_q(2))\) is the universal unital C\*-algebra generated by \(\alpha, \gamma\) with relations \[ \alpha^* \alpha + \gamma^* \gamma = 1, \quad \alpha \alpha^* + q^2 \gamma \gamma^* = 1, \quad \gamma \gamma^* = \gamma^* \gamma, \quad \alpha \gamma = q \gamma \alpha, \quad \alpha \gamma^* = q \gamma^* \alpha, \]

with comultiplication on the matrix \(u = \left[\begin{matrix} \alpha & -q \gamma^* \\ \gamma & \alpha^* \end{matrix}\right]\). At \(q = 1\) one recovers \(C(SU(2))\); for \(q < 1\) the algebra is non-commutative.

Applications discussed at the end of the course include quantum symmetry of finite graphs (Banica’s quantum automorphism groups), quantum information-theoretic interpretations of magic squares, and the role of compact quantum groups in subfactor theory via Jones’ planar algebras.

Chapter 4: Convex Geometric Analysis

Taught at UW as PMATH 950 in Winter 2025 by Kateryna Tatarko.

Convex geometric analysis lives at the crossroads of geometry, probability, and functional analysis. Its objects are convex bodies — compact convex sets with non-empty interior in \(\mathbb{R}^n\) — and its core questions concern volume, surface area, and how they behave as the dimension \(n \to \infty\). Phenomena that look ordinary in dimension three reveal subtle and counter-intuitive behavior in high dimensions, the realm of asymptotic geometric analysis.

4.1 Convex bodies and the Brunn–Minkowski inequality

Definition 4.1 (Minkowski sum). For sets \(A, B \subseteq \mathbb{R}^n\) and \(\lambda \ge 0\), the Minkowski sum is \(A + B = \{a + b : a \in A, b \in B\}\) and the dilation is \(\lambda A = \{\lambda a : a \in A\}\).

Theorem 4.2 (Brunn–Minkowski). For any non-empty Borel sets \(A, B \subseteq \mathbb{R}^n\), \[ |A + B|^{1/n} \ge |A|^{1/n} + |B|^{1/n}, \]

where \(|\cdot|\) is Lebesgue measure. Equivalently, \(t \mapsto |(1-t) A + t B|^{1/n}\) is concave on \(\left[0, 1\right]\).

The Brunn–Minkowski inequality is the geometric mother of dimension-free isoperimetry. From it one derives the isoperimetric inequality in \(\mathbb{R}^n\) — the Euclidean ball minimizes surface area among bodies of a given volume — and a host of refinements: the Prékopa–Leindler inequality (a functional Brunn–Minkowski), the Borell–Brascamp–Lieb inequality, and the log-Brunn–Minkowski conjecture.

4.2 Log-concave measures and Prékopa–Leindler

Definition 4.3 (Log-concavity). A function \(f: \mathbb{R}^n \to \left[0, \infty\right)\) is log-concave if \(f((1-t)x + ty) \ge f(x)^{1-t} f(y)^t\) for all \(x, y\) and \(t \in \left[0, 1\right]\). A Borel measure is log-concave if it is supported on a convex set and has a log-concave density.

Theorem 4.4 (Prékopa–Leindler). Let \(t \in \left[0, 1\right]\) and let \(f, g, h: \mathbb{R}^n \to \left[0, \infty\right)\) be measurable with \(h((1-t)x + ty) \ge f(x)^{1-t} g(y)^t\) for all \(x, y\). Then \[ \int h \ge \left(\int f\right)^{1-t} \left(\int g\right)^t. \]

This is the functional form of Brunn–Minkowski (recover the geometric statement by taking \(f, g, h\) indicator functions of \(A, B, (1-t)A + tB\)) and is fundamental to high-dimensional probability: Gaussians, uniform measures on convex bodies, and exponentials of concave functions are all log-concave.

4.3 Concentration of measure

Theorem 4.5 (Lévy concentration on the sphere). Let \(f: S^{n-1} \to \mathbb{R}\) be 1-Lipschitz and let \(M_f\) be its median with respect to the uniform measure \(\sigma\) on \(S^{n-1}\). Then for every \(t > 0\), \[ \sigma\left\{ x \in S^{n-1} : |f(x) - M_f| > t \right\} \le 2 \exp\!\left( -\tfrac{(n-1) t^2}{2} \right). \]

The exponential decay constant \(n - 1\) is what makes high dimensions special: a Lipschitz function on the sphere is essentially constant once \(n\) is large. Applied to coordinate projections, this is Dvoretzky’s theorem: every centrally symmetric convex body of dimension \(n\) has an almost-Euclidean section of dimension \(c \log n\). The constant \(c\) is sharp up to the “logarithmic conjecture” of Milman.

4.4 Mixed volumes, polar bodies, and the Blaschke–Santaló inequality

For convex bodies \(K_1, \ldots, K_n\) in \(\mathbb{R}^n\), the polynomial expansion

\[ |t_1 K_1 + \cdots + t_n K_n| = \sum_{i_1, \ldots, i_n = 1}^n V(K_{i_1}, \ldots, K_{i_n})\, t_{i_1} \cdots t_{i_n} \]

defines the mixed volumes \(V(K_{i_1}, \ldots, K_{i_n})\), symmetric and multilinear in their arguments. The Aleksandrov–Fenchel inequality

\[ V(K_1, K_2, K_3, \ldots, K_n)^2 \ge V(K_1, K_1, K_3, \ldots, K_n) \cdot V(K_2, K_2, K_3, \ldots, K_n) \]

generalizes Brunn–Minkowski and remains the source of many open problems.

Definition 4.6 (Polar body). For a convex body \(K \subseteq \mathbb{R}^n\) with \(0 \in \mathrm{int}(K)\), the polar body is \(K^\circ = \{ y \in \mathbb{R}^n : \langle x, y \rangle \le 1 \text{ for all } x \in K \}\).

Theorem 4.7 (Blaschke–Santaló). Among all centrally symmetric convex bodies \(K \subseteq \mathbb{R}^n\), the Mahler product \(|K|\, |K^\circ|\) is maximized by the Euclidean ball.

The lower bound — Mahler’s conjecture — remains open in general dimensions, with the cube and cross-polytope conjectured to be extremal among symmetric bodies. Bourgain’s slicing problem and the KLS conjecture of Kannan–Lovász–Simonovits are further central open questions, recently linked through breakthrough work of Chen and Klartag–Lehec.

Chapter 5: II\(_1\) Factors and Subfactor Theory

Taught at UW as PMATH 950 in Fall 2025.

Murray and von Neumann’s classification of factors — von Neumann algebras whose centre is trivial — divided them into types I, II, and III. The most subtle and combinatorially rich are the II\(_1\) factors: infinite-dimensional algebras on which a unique normalized trace exists.

Definition 5.1 (II\(_1\) factor). A II\(_1\) factor is an infinite-dimensional von Neumann algebra \(M\) with trivial centre admitting a faithful normal tracial state \(\tau: M \to \mathbb{C}\): \(\tau(xy) = \tau(yx)\), \(\tau(x^* x) > 0\) for \(x \ne 0\), and \(\tau\) continuous in the ultraweak topology.

The trace gives a notion of dimension for projections: \(\dim(p) = \tau(p) \in \left[0, 1\right]\), filling the unit interval continuously rather than the integers. This is the original surprise of Murray–von Neumann.

Theorem 5.2 (Murray–von Neumann; Connes). The hyperfinite II\(_1\) factor \(R\) — the unique (up to isomorphism) II\(_1\) factor admitting an increasing sequence of finite-dimensional unital \(*\)-subalgebras with weakly dense union — is amenable, and Connes proved every amenable II\(_1\) factor is isomorphic to \(R\).

Beyond \(R\), the structure theory of non-amenable II\(_1\) factors is governed by Popa’s deformation/rigidity theory, which produces W\(^*\)-superrigidity results: certain II\(_1\) factors arising from group actions remember the group and the action up to isomorphism, in stark contrast to the rigidity-free world of amenable factors.

Definition 5.3 (Jones index). Given an inclusion \(N \subseteq M\) of II\(_1\) factors, the Jones index \([M : N]\) is the von Neumann dimension of \(L^2(M)\) as a left \(N\)-module: \([M : N] = \dim_N L^2(M)\).

Theorem 5.4 (Jones, 1983). The set of values of \([M : N]\) for finite-index subfactors is exactly \[ \left\{ 4 \cos^2(\pi/n) : n = 3, 4, 5, \ldots \right\} \cup \left[4, \infty\right]. \]

The discrete spectrum \(4 \cos^2(\pi/n)\) is one of the most striking quantizations in operator algebras. Iterating the basic construction \(N \subseteq M \subseteq M_1 \subseteq M_2 \subseteq \cdots\), Jones produced a tower whose relative commutants form a tower of finite-dimensional algebras — the standard invariant — encoded combinatorially by Jones’ planar algebras and (in another guise) by Ocneanu’s paragroups. The unexpected output was the Jones polynomial of knots, born from representations of the Temperley–Lieb algebra inside the tower.

Chapter 6: Choquet Theory

Taught at UW as PMATH 950 in Winter 2026.

Choquet theory is the convex-analytic generalization of the Krein–Milman theorem: where Krein–Milman represents points in a compact convex set as limits of convex combinations of extreme points, Choquet asks for an integral representation supported on the extreme points themselves. The theory is the natural language for understanding states on operator algebras, ergodic decomposition of invariant measures, and harmonic measures on boundaries.

Definition 6.1 (Extreme point). Let \(C\) be a convex subset of a topological vector space. A point \(x \in C\) is extreme if \(x = (1-t) y + t z\) with \(y, z \in C\) and \(t \in (0, 1)\) forces \(y = z = x\). Write \(\mathrm{ext}(C)\) for the set of extreme points.

Theorem 6.2 (Krein–Milman). Every non-empty compact convex subset \(C\) of a locally convex Hausdorff topological vector space equals the closed convex hull of its extreme points: \(C = \overline{\mathrm{conv}}(\mathrm{ext}(C))\).

The Choquet refinement is to replace finite convex combinations by integration against probability measures supported on \(\mathrm{ext}(C)\).

Theorem 6.3 (Choquet, metrizable case). Let \(C\) be a metrizable compact convex subset of a locally convex space and let \(x \in C\). Then there exists a Borel probability measure \(\mu\) on \(C\) supported on \(\mathrm{ext}(C)\) (which is a Borel set) such that \[ f(x) = \int_C f\, d\mu \quad \text{for every continuous affine } f: C \to \mathbb{R}. \]

In the non-metrizable case (Choquet–Bishop–de Leeuw), \(\mathrm{ext}(C)\) need no longer be Borel; one settles for measures maximal in a partial order due to Choquet, supported on \(\mathrm{ext}(C)\) in the sense of being zero on every Baire set disjoint from it.

Definition 6.4 (Choquet simplex). A compact convex set \(C\) in a locally convex space is a Choquet simplex if for every \(x \in C\) the representing measure \(\mu\) on \(\mathrm{ext}(C)\) is unique.

Example 6.5. The set of probability measures on a compact Hausdorff space \(K\) is a Choquet simplex with extreme points the Dirac masses \(\delta_x\). The state space of a unital commutative C\*-algebra \(C(K)\) is therefore a simplex; the state space of a non-commutative C\*-algebra typically is not.

The applications are pervasive. Ergodic decomposition writes any invariant probability measure as an integral over ergodic ones — a Choquet representation in the simplex of invariant measures. Bauer’s theorem identifies which compact convex sets are simplices with closed extreme boundary (precisely the state spaces of commutative C*-algebras, by Gelfand), recovering Choquet representation as integration against the Dirac masses on the spectrum. In probability, de Finetti’s theorem on exchangeable sequences is a Choquet decomposition of the simplex of exchangeable measures into i.i.d. ones. Phelps’s monograph develops the full picture, including the role of uniqueness of representing measures in classifying state spaces.

Chapter 7: Operator Systems and Completely Positive Maps

Taught at UW as PMATH 950 in Fall 2026.

Operator systems are to unital completely positive (ucp) maps what operator spaces are to completely bounded maps. They are the natural setting for non-commutative function theory and the home of the dilation theorems that lie at the heart of modern operator algebras.

Definition 7.1 (Operator system). A concrete operator system is a self-adjoint subspace \(S \subseteq B(H)\) containing the identity \(1_H\). Equipping \(M_n(S)\) with the cone of positive matrices inherited from \(M_n(B(H))\) gives a matrix order structure.

The abstract Choi–Effros characterization identifies operator systems intrinsically: a \(*\)-vector space \(V\) with an order unit and a compatible system of cones on \(M_n(V)\) (satisfying the Archimedean property and matrix-order axioms) is unitally completely order isomorphic to a concrete operator system.

Theorem 7.2 (Stinespring dilation). Let \(A\) be a unital C\*-algebra and \(\varphi: A \to B(H)\) a unital completely positive map. Then there exist a Hilbert space \(K\), an isometry \(V: H \to K\), and a unital \(*\)-representation \(\pi: A \to B(K)\) with \[ \varphi(a) = V^* \pi(a) V \qquad \text{for all } a \in A. \]

The dilation \((\pi, V, K)\) is unique up to unitary equivalence if taken minimal.

Stinespring’s theorem is the operator-algebraic generalization of GNS: every ucp map is a “compression” of a representation. It immediately implies the Schwarz inequality \(\varphi(a^* a) \ge \varphi(a)^* \varphi(a)\) for ucp \(\varphi\), and characterizes when equality holds (the “multiplicative domain”).

Theorem 7.3 (Arveson extension). Let \(S \subseteq A\) be an operator system in a unital C\*-algebra \(A\). Every unital completely positive map \(\varphi: S \to B(H)\) extends to a unital completely positive map \(\tilde{\varphi}: A \to B(H)\).

This is the operator-system analogue of the Hahn–Banach theorem; together with Stinespring it forms the cornerstone of dilation theory and of the theory of boundary representations and the C*-envelope, which assigns to each operator system a canonical minimal C*-algebra it generates.

Definition 7.4 (Matrix convex set). A matrix convex set over \(\mathbb{R}^d\) is a sequence \(K = (K_n)_{n \ge 1}\) with \(K_n \subseteq M_n(\mathbb{R})^d\) closed under direct sums and unital completely positive compressions: \(\sum_i V_i^* X^{(i)} V_i \in K_n\) whenever \(X^{(i)} \in K_{n_i}\) and \(V_i: \mathbb{C}^n \to \mathbb{C}^{n_i}\) satisfy \(\sum V_i^* V_i = 1_n\).

Theorem 7.5 (Webster–Winkler / Effros–Winkler duality). The category of matrix convex sets in \(\mathbb{R}^d\) is dual to the category of operator systems generated by \(d\) self-adjoint elements. Extreme points in the matrix-convex sense correspond to boundary representations.

Matrix convexity is the framework in which Davidson–Kennedy resolved Arveson’s boundary representation conjecture, proving every operator system is the intersection of all its boundary representations and so admits a faithful representation by its C*-envelope. The applications loop back: Stinespring dilations give quantum channels, Arveson extension gives the existence of optimal quantum error-correcting codes, and matrix convexity provides the geometric backbone for the developing theory of non-commutative function theory — free analogues of holomorphic functions, free probability, and the operator-valued moment problem.

The seven chapters above are far from exhausting PMATH 950’s range — algebraic topics (model theory of operator algebras, quantum information theory, free probability) recur in other terms — but they exhibit the analytic spine of the topic: representations and dilations, dynamics and crossed products, tensor products and approximation properties, convexity in finite and infinite dimensions, and the rich interplay between geometry and operator theory that makes this corner of mathematics so durable.