Math 646 Notes

Day 1 (1/21/25)

• Some notes on John von Neumann
• History of von Neumann's work in function analysis
• Some examples of Hilbert spaces
• The Fourier Basis (Stone-Weierstrass)
• Dimension of a Hilbert space

Day 2 (1/23/25)

• Useful formulas in Hilbert spaces (parallelogram law and polarization)
• Properties of operator norm
• Examples of operators
• Embedding \(L^\infty\) in \(L^2\)
• Important classes of operators

Day 3 (1/28/25)

• Important classes of operators (isometry, unitary, projection, partial isometry)
• Properties of isometries
• Properties of projections
• Properties of partial isometries

Day 4 (1/30/25)

• Equivalent definitions for a partial isometry
• Polar decomposition
• Continuous Functional Calculus
• Understanding continuous functional calculus in finite dimensions
• The exponential of an operator
• A theorem of Fuglede

Day 5 (2/4/25)

• Topologies on \(B(H)\) (norm, strong operator, weak operator)
• The strong operator topology
• The weak operator topology
• Banach-Alaoglu
• Topologies are distinct in general
• Definition of \(*\)-algebra, \(C^*\)-algebra, and von Neumann algebra
• Examples of von Neumann algebras
• Definition of the commutant
• Properties of the commutant

Day 6 (2/6/25)

• Comparing the three topologies
• \(L^\infty\) is a (commutative) von Neumann algebra
• The Left Regular Representation of a group
• Definition of a group algebra \(\mathbb{C}[G]\), \(C^*\)-algebra of a group \(C^*(G)\), and von Neumann algebra of a group \(L(G)\)
• Example of group algebra coming from a cyclic group, \(\mathbb{C}[\mathbb{Z}_n]\) (the circulant matrices)
• Example of group algebra coming from the integers, \(\mathbb{C}[\mathbb{Z}]\)

Day 7 (2/11/25):

• Review of examples of algebras (\(C^*\) and von Neumann)
• Discussing the homework problems
• \(B(H)\) has no trace
• \(DX - XD = I\) in \(B(H)\)
• The commutator subspace of \(L(G)\) and \(R(G)\) is zero
• The group algebra coming from the integers \(\mathbb{C}[\mathbb{Z}]\) (in further detail)
• Definition of the trace on \(L(G)\)
• Properties of the trace
• Embedding \(L(G)\) in \(\ell^2(G)\)

Day 8 (2/13/25):

• Review of the group algebra coming from the integers \(\mathbb{C}[\mathbb{Z}]\)
• Proof that the trace on \(L(G)\) is indeed a trace
• Working with elements of \(L(G)\)
• Embedding \(L(G)\) in \(\ell^2(G)\)
• What can we say about \(c_g\) for \(\sum_{g \in G} c_g u_g \in L(G)\)?
• \(\sum c_g u_g\) takes \(\sum d_h \delta_h\) to \(\sum (c * d)_h \delta_h\) (so we need convolutions)
• For square-summable \((c_g)\), we have \(\sum c_g u_g \in L(G)\) iff the map \(d \mapsto c * d\) is bounded on \(\ell^2(G)\)
• When is \(L(G)\) a factor?

Day 9 (2/18/25):

• If \(G\) is an ICC group, then \(L(G)\) is a factor
• Examples of ICC groups
• Review of positive operators (definition and properties)
• The GNS Construction
• Start with a unital \(*\)-algebra \(A\) with a faithful state \(\varphi\) (some examples of these)
• Definition of inner product on \(A\) and Hilbert space \(H\)
• Defining embedding \(\pi : A \to B(H)\) (\(\pi(a)\) is defined on \(A\) via left multiplication, then extended to all of \(H\))

Day 10 (2/25/25):

• Review of GNS Construction from last time
• Definition of a pre-\(C^*\)-algebra, an abstract \(C^*\)-algebra, and a concrete \(C^*\)-algebra
• Gelfand's Theorem: Any abstract \(C^*\)-algebra is a concrete \(C^*\)-algebra
• Important examples of an abstract \(C^*\)-algebra and a pre-\(C^*\)-algebra
• The GNS Construction continued in more detail
• Start with \(A\) a unital pre-\(C^*\)-algebra with \(\varphi\) a faithful state (note that norm conditions on \(A\) are needed)
• Define Hilbert space \(H\) with embedding \(\pi : A \to B(H)\)
• Proof of the GNS Construction
• Example of GNS Construction when \(A = C([0, 1])\)
• Example of GNS construction when \(B = \cup_{k \geq 1} M_{2^k}(\mathbb{C})\)
• Comparing \(\cup_{k \geq 1} M_{2^k}(\mathbb{C})\) and \(\cup_{k \geq 1} M_{3^k}(\mathbb{C})\)
• After closing with the norm, these are not isomorphic (as \(C^*\)-algebras)
• Even without the closure, these are not isomorphic (as \(*\)-algebras)
• Use the trace and look at the set \(\{\tau(x) : x \text{ projection}\}\) (related to \(K\)-theory)

Day 11 (2/27/25):

• Embedding \(M_{2}(\mathbb{C})\) in \(M_{4}(\mathbb{C})\) using the tensor product
• \(\cup_{k \geq 1} M_{2^k}(\mathbb{C})\) is the same as \(\otimes_{k \geq 1} M_2(\mathbb{C})\)
• Theorem 1: \(\otimes_{k \geq 1} M_2(\mathbb{C})\) is not isomorphic to \(\otimes_{k \geq 1} M_3(\mathbb{C})\) when closed in norm
• These are called AFD \(C^*\)-algebras (approximately finite dimensional)
• Theorem 2: These are isomorphic when closed in the strong operator topology!
• This is called the hyperfinite type \(\text{II}_1\) factor \(R\)
• This is the "simplest" infinite dimensional nonabelian von Neumann algebra
• Proof of Theorem 1
• Both algebras have a unique trace; applying the trace to the set of projections provides an invariant \(K^1(A)\)
• Computing \(K^1(A)\) by showing that two projections that are close have the same trace
• Lemma: if two projections are close, then they are unitarily equivalent (proof involves using an intertwiner and polar decomposition)

Day 12 (3/4/25):

• Homework problem: If \(T_i\) and \(T\) are normal operators such that \(T_i\) converges to \(T\) in the strong operator topology, then \({T_i}^*\) also converges to \(T^*\)
• This is not true for operators in general
• Finishing the proof of Theorem 1 from last time
• First, show that if two projections are close enough, then they are unitarily equivalent and thus have the same trace (using intertwiner and polar decomposition)
• Second, show that we can approximate any projection with a projection in the matrices (we can approximate a projection with a matrix, then show that this matrix is almost a projection)
• Definition of a \(\text{II}_1\) factor and the hyperfinite \(\text{II}_1\) factor \(R\)
• Classification of factors
• Theorem (for later): Any \(\text{II}_1\) factor contains \(R\)
• Theorem: \(R\) is a factor
• First, take \(x\) in the center and approximate it in \(2\)-norm by a matrix \(a\)
• From here, use the fact that \(x\) commutes with unitaries

Day 13 (3/6/25):

• Finishing the proof that \(R\) is a factor
• For any \(x \in R\), we can approximate \(x\) by a matrix \(y\) in \(2\)-norm
• Since \(x\) commutes with all unitaries, it follows that \(y\) almost commutes with all unitaries
• Since \(y\) almost commutes with all unitaries, it is almost a scalar
• To prove this, look at the average of all \(uyu^*\) in one of two ways:
• Use the Haar measure on the compact topological group \(\mathcal{U}(M_{2^n}(\mathbb{C}))\) and integrate \(uyu^*\) over the group
• Look at the closed convex hull of the set \(\{uyu^* : u \in \mathcal{U}(M_{2^n}(\mathbb{C}))\}\) and take the unique element of minimal norm
• In either case, you get something that commutes with all unitaries (hence a scalar) that is still close to \(y\)
• Review of examples of von Neumann algebras so far
• Theorem: Borel Functional Calculus
• Example: Using Borel functional calculus to show that if \(0 \leq x \leq 1\) for \(x \in B(H)\), then \(x^n\) converges s.o. to a projection

Day 14 (3/11/25):

• Borel functional calculus continued
• Applications of Borel functional calculus
• \(U \in B(H)\) is unitary if and only if \(U = e^{i H}\) for \(H\) Hermitian
• The set \(G(B(H))\) of invertible operators in \(B(H)\) is pathwise connected
• If \(A\) is a \(C^*\)-algebra, then any element of \(A\) can be written as a linear combination of:
• 2 Hermitian elements
• 4 positive elements
• 8 unitaries
• Furthermore, if \(A\) is a von Neumann algebra, then \(\operatorname{span}\mathcal{P}(A)\) closed in norm is equal to \(A\)
• In fact, if \(0 \leq \|x\| \leq 1\), then you can write \(x\) as a dyadic sum of projections

Day 15 (3/13/25):

• Proving that \(0 \leq x \leq 1\) can be written as a dyadic sum of projections
• If \(x\) is self-adjoint, then \(\chi_{\{t\}}(x)\) is the projection onto the eigenspace of \(t\)
• Proving Borel functional calculus
• Uniqueness of \(\phi\) (using a theorem of Baire)
• Existence of \(\phi\) (using the Riesz-Markov-Kakutani Representation Theorem)

Day 16 (3/25/25):

• Restatement of Borel functional calculus
• Note that norm convergence is also preserved
• Continuing the proof from last time
• Define a functional which essentially gets the matrix entries of a continuous function \(f\)
• By the Riesz-Markov-Kakutani Representation Theorem, this functional is given by an integral with respect to a certain measure
• In a sense, this integral defines a sesquilinear form, which must be of the form \(\langle T \xi, \eta \rangle\) (define \(f(x)\) to be this \(T\))
• The strong operator convergence of \(\phi\) follows from the Dominated Convergence Theorem
• From here it also follows that \(f(x)\) is in the von Neumann algebra generated by \(x\)
• Finally, show that \(\phi\) is a \(*\)-morphism with \(\|\phi\| \leq 1\) using the definition of \(T\)

Day 17 (4/1/25):

• Finishing the proof of Borel functional calculus
• Showing that \(\phi\) is multiplicative
• Statement of von Neumann's Bicommutant Theorem
• Finding the commutant and bicommutant for some examples
• Lemma involving reducing subspaces (with proof)
• Proving the Bicommutant Theorem
• One inclusion is immediate from the properties of the commutant
• To show that \(M'' \subset M\), we show that \(x \in M''\) implies that \(x \in \overline{M}^{\text{ s.o.}}\)
• In other words, we show that any open neighborhood of \(x\) intersects \(M\)

Day 18 (4/3/25):

• Finishing the proof of the Bicommutant Theorem
• Use a basis of neighborhoods of the s.o. topology to show that \(x \in \overline{M}^{\text{ s.o.}}\)
• Start with only one \(\xi\) for the basis
• Consider \(K = \overline{M \xi}\), and use the reducing subspace lemma
• The projection \(P_K\) commutes with all of \(M\), hence also with \(x\), and this implies that \(x \xi \in K\)
• For the general case, treat \(\xi_1, \ldots, \xi_n\) as a single vector in \(H^n\)
• We can identify \(y\) in \(M \subset B(H)\) with \(\tilde{y}\) in \(\tilde{M} \subset B(H^n) \cong M_n(B(H))\)
• Applying the first part to the new \(\xi\) gives the desired result
• Definition of the map \(\omega_{\xi, \eta}\)
• For a linear functional \(\omega\), weak operator continuity and strong operator continuity are equivalent to \(\omega\) being a finite sum of some \(\omega_{\xi_i, \eta_i}\)
• Some implications are relatively easy to show
• To show that s.o. continuity implies that \(\omega\) can be written as a sum, begin by bounding \(\omega(x)\) by looking at the preimage of the unit disk

Day 19 (4/8/25):

• Finishing the proof from last time: assuming \(\omega\) is s.o. continuous, we want to write it as a sum of \(\omega_{\xi_i, \eta_i}\)
• Since the s.o. topology is given by seminorms and \(\omega\) is s.o. continuous, it can be bounded by some seminorm
• Define a functional \(\varphi\) using \(\omega\) on the set of all \((x\xi_1, \ldots, x\xi_n)\) (where the \(\xi_i\) are given by the seminorm)
• Extend \(\varphi\) to the whole \(B(H)\) using the Hahn-Banach Theorem
• Using the Riesz Representation Theorem, we conclude that \(\varphi\) is given by an inner product, which gives the desired result
• Remark: if two locally convex topologies have the same continuous linear functionals, then they have the same closed convex sets
• So, a convex set \(K \subset B(H)\) is s.o. closed iff it is w.o. closed
• Similarly, if \(M \subset B(H)\) is a von Neumann algebra and \(\omega : M \to \mathbb{C}\) is linear and s.o. continuous, then we can write it as a sum of \(\omega_{\xi_i, \eta_i}\)
• As before, we can bound \(\omega\) by a seminorm and use Hahn-Banach to extend to the entire \(B(H)\), at which point we can apply the previous result
• Intro to the geometry of projections
• Some equivalent formulations of what \(p \leq q\) means for \(p, q\) projections
• Definition of the intersection and union of projections
• Note that \(\mathcal{P}(B(H))\) forms a complete lattice
• For a von Neumann algebra \(M\), an arbitrary union and intersection of projections in \(M\) is still in \(M\)
• Therefore, \(\mathcal{P}(M)\) also forms a complete lattice
• Definition of what it means for two projections to be equivalent (using partial isometries)
• Definition of what it means for \(p \prec q\) for \(p, q\) projections

Day 20 (4/10/25):

• Recall the definition of \(p \sim q\) from last time
• Example of two projections that are not equivalent
• If \((p_i)\) and \((q_i)\) are both mutually orthogonal, with \(p_i \sim q_i\), then \(\sum p_i \sim \sum q_i\)
• Note that the sum is the s.o. limit of finite sums here, since the projections are mutually orthogonal
• For the proof, if \(p_i \sim q_i\) via \(v_i\), then \(\sum p_i \sim \sum q_i\) via \(\sum v_i\)
• Recall the definition of \(p \prec q\) from last time
• Example of two projections that are not comparable (so \(\prec\) is not a total order in general)
• Heading towards the Comparison Theorem: if \(M\) is a factor, then any two projections are comparable
• Showing that \(\prec\) is essentially a partial order on projections (where we have \(\sim\) instead of \(=\))
• Reflexivity is trivial
• To show transitivity, compose the two given partial isometries (drawing out a diagram is helpful)
• To show antisymmetry, we essentially mimic the Cantor-Schröder-Bernstein Theorem
• If \(p \prec q\) and \(q \prec p\), then \(p \sim q_1 \leq q\) and \(q \sim p_1 \leq p\)
• Since \(q_1 \leq q\), it follows that \(q_1 \sim p_2 \leq p_1\) (similarly \(p_1 \sim q_2 \leq q_1\))
• Repeating this process allows us to break \(p\) into infinitely many subprojections (a diagram helps here)
• Breaking up \(p\) into these subprojections and using the diagram, we can show that \(p \sim p_1\), from which we get \(p \sim q\)
• Comments on a homework problem
• von Neumann's Ergodic Theorem: if \(\|T\| \leq 1\), then \((1 + T + \cdots + T^{n-1}) / n \overset{\text{s.o.}}{\to} p\), where \(p\) is the projection onto the fixed points of \(T\)
• A special case: \(H = \operatorname{span}\{e_1, \ldots, e_n\}\) and \(T\) permutes \(\{e_1, \ldots, e_n\}\)

Day 21 (4/15/25):

• Recall the polar decomposition for \(x \in B(H)\)
• If \(M\) is a von Neumann algebra with \(x \in M\), then the \(v\) from the partial decomposition of \(x\) is in \(M\)
• From here, it follows that the projections \(v^*v\) and \(vv^*\) are also in \(M\)
• For the proof, to show that \(v\) is in \(M\), show that \(v\) is in \(M''\), or that \(v\) commutes with \(M'\) (work with unitaries \(u\) in \(M'\))
• Since \(u\) commutes with \(x\) and \(|x|\), it follows that \(uvu^*\) also works in the polar decomposition of \(x\)
• By uniqueness, it follows that \(uvu^* = v\), which finishes the proof
• Definition of the left and right support
• \(L(x)\) is the smallest projection such that \(px = x\), and \(R(x)\) is the smallest projection such that \(xq = x\)
• In particular, the left and right support are equivalent and remain inside the von Neumann algebra
• Factors have corners: if \(M\) is a factor and \(p, q\) are nonzero projections in \(M\), then \(pMq \ne 0\)
• Example of what this looks like and how this can fail outside of a factor
• Proof of why this always fails outside of a factor using central projections
• The proof of the theorem is done by contradiction, where you look at the union of all \(upu^*\) and show that it must be the identity
• The Comparison Theorem in factors: if \(M\) is a factor and \(p, q\) are projections in \(M\), then \(p \prec q\) or \(q \prec p\)
• The idea is to break \(p\) and \(q\) into families of subprojections (satisfying certain conditions)
• Considering all such pairs of families, use Zorn's Lemma to obtain a maximal element
• This maximal element shows that \(p \prec q\) or \(q \prec p\)

Day 22 (4/22/25):

• Recall from last time:
• The Comparison Theorem in factors
• Factors have no nontrivial corners (or no holes)
• The Classification of Factors (using minimal and finite projections)
• Definition of a minimal projection (no nontrivial subprojections)
• Examples of von Neumann algebras with (or without) minimal projections
• Remark: \(p\) is minimal iff \(pMp = \mathbb{C}p\)
• Theorem: If \(M\) is a factor with at least a minimal projection, then \(M \cong B(H)\)
• For the proof, we will use the matrix units \(E_{ij} : H \to H\)
• Consider the family of all \((p_i)\) such that the \(p_i\) are mutually orthogonal minimal projections, and use Zorn's Lemma to find a maximal element \((e_i)\)
• Claim: \(\sum e_i = 1\)

Day 23 (4/24/25):

• Continuing from last time: a discrete factor is isomorphic to \(B(H)\)
• Start with the case when \(M\) is finite dimensional, and take the family \((e_i)\) from last time
• Show that \(\sum e_i = 1\) by using the comparison theorem and the fact that the family is maximal
• Since the \(e_i\) are all minimal, we have \(e_1 \sim e_2 \sim \cdots \sim e_n\), so there exists a partial isometry \(v_i\) that relates \(e_i\) to \(e_1\)
• Use these \(v_i\) to define \(e_{ij}\), which act like matrix units (in particular, any \(x \in M\) can be written as a linear combination of these \(e_{ij}\))
• Define a \(*\)-morphism that sends these \(e_{ij}\) to the actual matrix units \(E_{ij} \in B(H)\) (showing that this is a \(*\)-morphism relies on properties of the \(e_{ij}\))
• For the infinite dimensional case, the proof is similar, and you'll have to extend the morphism using s.o. continuity
• Corollary: A finite dimensional factor is isomorphic to \(M_n(\mathbb{C})\) (so it must be discrete)
• In fact, any finite dimensional \(*\)-algebra is isomorphic to a direct sum of matrix algebras
• To see this, use strong induction on the dimension of the center of \(M\)
• If the dimension is one, \(M\) is a factor and we're done
• Otherwise, there is a nontrivial projection \(p\) in the center, which we can use to decompose \(M\) into \(pMp\) and \(qMq\) (where \(q = 1 - p\))
• The induction hypothesis applies to these two subalgebras, and the proof is finished
• Definition of a finite projection
• Examples of von Neumann algebras with (or without) finite projections
• Classification of factors again

Day 24 (4/29/25):

• Some notes on Kazhdan and Margulis
• Recap on group representations
• How can we compare two representations?
• The weak containment property: \(\pi \prec \rho\) if all coefficients of \(\pi\) can be approximated by a finite sum of coefficients of \(\rho\)
• The Fell topology for representations
• Describe a subbasis of neighborhoods for \(\pi\)
• Note that \(\pi \prec \rho\) implies that \(\rho\) is in any neighborhood of \(\pi\)
• What does it look like for \(1_G \prec \rho\)?
• Rewrite the basis to obtain a basis of neighborhoods of \(1_G\)
• Ultimately, for a representation to be close to \(1_G\), it must have almost invariant vectors
• Definition of almost invariant vectors
• Definition of Kazhdan's Property (T)
• Property (T) means that the trivial representation is isolated
• This is called a "rigidity" property
• Examples of groups with Property (T) (and nonexamples)

Day 25 (5/1/25):

• Recall definition of property (T) and examples from last time
• Proving that \(\operatorname{SL}(n, \mathbb{Z})\) has property (T) for \(n \geq 3\)
• Consider the case when \(n = 3\)
• There is an interesting subgroup \(G\) satisfying \(G = \mathbb{Z}^2 \rtimes \operatorname{SL}(2, \mathbb{Z})\)
• There are several ways to embed \(G\) in \(\operatorname{SL}(3, \mathbb{Z})\), and their union is the whole group
• The proof follows from the following result
• \(\mathbb{Z}^2 \subset \mathbb{Z}^2 \rtimes \operatorname{SL}(2, \mathbb{Z})\) has relative property (T)
• Using a specific \(\varepsilon\) and finite set \(Q\), we show that a unitary representation \(\pi\) with an almost invariant vector has an invariant vector
• Extend \(\pi\) to \(\text{v.N.}(\mathbb{Z}^2) \cong L^\infty(\mathbb{T}^2, \mu)\)
• Assume that \(\pi\) has no invariant vectors, and rephrase what a invariant vector means in terms of characteristic functions
• Identify \(\mathbb{T}^2\) with \((-1/2, 1/2]^2\), and translate information on the problem to information about \(\mu = \mu_{\xi, \xi}\)
• Using a probability measure defined using \(\mu\) and the information we have about \(\mu\), along with a clever partition of the space, we obtain a contradiction
• Important takeaway: changing a problem about operators to a problem about measures can be useful