Appendix A — Functional-Analysis Survival Kit

Functional-Analysis Survival Kit

This appendix collects the analytic facts used repeatedly in the course. It is not a replacement for a functional-analysis course. It is the minimum toolkit a student needs in order to read the lecture notes without stopping every page.

A.1 Hilbert-Space Conventions

Inner products are linear in the second slot:

$$⟨\xi ,\lambda \eta ⟩=\lambda ⟨\xi ,\eta ⟩,⟨\lambda \xi ,\eta ⟩=\overline{\lambda }⟨\xi ,\eta ⟩.$$

For a bounded operator $T$ on a Hilbert space $\mathcal{H}$, the adjoint $T^{\ast }$ is defined by $⟨\xi ,T\eta ⟩=⟨T^{\ast }\xi ,\eta ⟩$.

Key classes:

• $T$ is self-adjoint if $T=T^{\ast }$.
• $T$ is positive if $⟨\xi ,T\xi ⟩\ge 0$ for all $\xi$.
• $U$ is unitary if $U^{\ast }U=U{U}^{\ast }=1$.
• $P$ is a projection if $P=P^{\ast }=P^2$.

When $P$ is a projection, $P\mathcal{H}$ is a closed subspace; every closed subspace has an orthogonal projection.

A.2 Bounded and Unbounded Operators

The course mostly uses bounded operator algebras, but QFT fields are unbounded before smearing or exponentiation.

A bounded operator $T$ satisfies $\Vert T\xi \Vert \le C\Vert \xi \Vert$ for all $\xi$; the smallest such $C$ is the operator norm $\Vert T\Vert$.

An unbounded operator $A$ is a pair $(A,\mathrm{Dom}A)$ where $\mathrm{Dom}A$ is a dense subspace and $A$ is linear on it. Domain questions matter for fields $\varphi (f)$ and for logarithms like $\log \Delta$, but the von Neumann algebra itself is built from bounded operators.

Course rule:

• Use unbounded fields as motivation and for formal calculations.
• Use bounded Weyl operators $W(f)=e^{i\varphi (f)}$, spectral projections, modular unitaries ${\Delta }^{-it}$, and von Neumann algebras for exact algebraic statements.

A.2.1 Graphs, closedness, and closability

For an unbounded operator, the formula for the action is only half of the definition. The domain is part of the operator. Thus two operators can act by the same differential expression and still be different because their domains, boundary conditions, or closures differ.

The graph of $A$ is

$$\Gamma (A):=\{(\xi ,A\xi ):\xi \in \mathrm{Dom}A\}\subset \mathcal{H}\oplus \mathcal{H}.$$

The operator is closed when $\Gamma (A)$ is closed. Equivalently, whenever ${\xi }_n\to \xi$ and $A{\xi }_n\to \eta$, one has $\xi \in \mathrm{Dom}A$ and $A\xi =\eta$.

The operator is closable when the closure of its graph is still the graph of an operator, denoted $\overline{A}$. The practical test is

$${\xi }_n\to 0,A{\xi }_n\to \eta ⟹\eta =0.$$

If this implication failed, the closed graph would assign both $0$ and $\eta \ne 0$ to the zero vector, so it could not define an operator.

Physical picture. Closing an operator adds limits that were already forced by the original action. It does not choose arbitrary new boundary conditions. In Week 5 the preliminary Tomita map $S_0$ is first defined on the algebraic domain $\mathcal{M}\Omega$. Proving that $S_0$ is closable is what licenses the closed operator $S=\overline{S_0}$ and hence its polar decomposition. Writing $S=J{\Delta }^{1/2}$ before that step would hide the main domain question.

A.2.2 Adjoints, including conjugate-linear operators

For a densely defined linear operator $A$, a vector $\eta$ belongs to $\mathrm{Dom}{A}^{\ast }$ when there is a vector $\zeta$ such that

$$⟨\eta ,A\xi ⟩=⟨\zeta ,\xi ⟩\text{for every }\xi \in \mathrm{Dom}A.$$

Then $A^{\ast }\eta :=\zeta$. The representing vector is unique because the domain of $A$ is dense. The adjoint is always closed, and a densely defined linear operator is closable exactly when $\mathrm{Dom}{A}^{\ast }$ is dense; in that case $\overline{A}=A^{\ast \ast }$.

The Tomita map is conjugate-linear. With the course convention that the inner product is linear in its second slot, the conjugate-linear adjoint is defined by

$$⟨\eta ,S\xi ⟩=⟨\xi ,S^{\ast }\eta ⟩.$$

This reversal of the two vectors is not decoration: importing the linear-adjoint formula without changing it produces the wrong conjugations. Week 5 carries out the Tomita calculation in this convention.

A.2.3 Graph norm and cores

On $\mathrm{Dom}A$, define the graph norm

$$\Vert \xi {\Vert }_A:=(\Vert \xi {\Vert }^2+\Vert A\xi {\Vert }^2{)}^{1/2}.$$

The operator $A$ is closed exactly when its domain is complete in this norm. A subspace ${\mathcal{D}}_0\subset \mathrm{Dom}A$ is a core for $A$ if it is graph-norm dense in $\mathrm{Dom}A$. Equivalently, closing the restriction $A{|}_{{\mathcal{D}}_0}$ recovers $A$.

This is the precise meaning of statements such as “prove the identity first on $\mathcal{M}\Omega$ and then extend by closure.” Ordinary Hilbert-space density is not enough: one must approximate both $\xi$ and $A\xi$.

A.2.4 A domain checklist

Before manipulating unbounded operators, ask:

1. What is the domain of each operator?
2. Is the proposed product defined on a dense common domain?
3. Is the operator closed or at least closable?
4. Is the identity being proved on a core, or only on an unspecified dense set?
5. Does the functional calculus use a bounded function, or is a new domain required?

The bounded modular unitaries ${\Delta }^{it}$ are defined everywhere even when $\Delta$ and $\log \Delta$ are unbounded. This is why exact modular-flow statements are normally written with ${\Delta }^{it}$, while entropy expressions involving $\log \Delta$ carry explicit domain or integrability conditions.

A.3 The Spectral Theorem

If $A=A^{\ast }$ is self-adjoint, there is a projection-valued measure $E_A$ on $\mathbb{R}$ such that

$$A={\int }_{\mathbb{R}}\lambda d{E}_A(\lambda ),f(A)={\int }_{\mathbb{R}}f(\lambda )d{E}_A(\lambda )$$

for bounded Borel $f$; unbounded Borel functions are also allowed on their natural domains. If a symmetric operator is only essentially self-adjoint, this theorem applies to its unique self-adjoint closure. The spectrum is $\sigma (A)=\{\lambda \in \mathbb{R}:E_A(B)\ne 0\text{ for every open }B\ni \lambda \}$.

Used throughout the course for:

• ${\rho }^{it}$ via $\rho >0$, ${\rho }^{it}=e^{it\log \rho }$ (functional calculus).
• ${\Delta }^{-it}$ via $\Delta$ positive self-adjoint.
• $e^{-\beta H}$ for the Gibbs density.
• $K_{\mathrm{GNS}}=-\log \Delta$ as the modular Hamiltonian on standard form, with $\log \Delta$ itself appearing in relative-entropy formulas.

A.4 Topologies on $\mathcal{B}(\mathcal{H})$

We use four topologies. They are written with sequences for readability; nets are required for the general topological statements.

Norm topology. $T_i\to T$ iff $\Vert T_i-T\Vert \to 0$. Strongest of the four.

Strong operator topology (SOT). $T_i\to T$ iff $T_i\xi \to T\xi$ for every $\xi \in \mathcal{H}$. It is weaker than norm (and strictly weaker in infinite dimension).

Weak operator topology (WOT). $T_i\to T$ iff $⟨\eta ,T_i\xi ⟩\to ⟨\eta ,T\xi ⟩$ for every $\xi ,\eta$. It is weaker than SOT.

$\sigma$-weak (ultraweak) topology. $T_i\to T$ iff $\mathrm{Tr}(C{T}_i)\to \mathrm{Tr}(CT)$ for every trace-class $C$. On norm-bounded subsets of $\mathcal{B}(\mathcal{H})$ it agrees with WOT; as a topology on the whole space it is finer. It is the weak-* topology defined by the predual and therefore the natural topology for normal functionals.

A von Neumann algebra is a unital $\ast$-subalgebra of $\mathcal{B}(\mathcal{H})$ closed in WOT (equivalently, SOT, equivalently $\sigma$-weak). Norm-closure gives only a C*-algebra, not generally a vN algebra. The equivalence concerns which *-algebras are closed: for a unital *-algebra all three closures equal the bicommutant (Week 2, Theorem 4.1 and §4.2), while the three topologies themselves remain different. Week 2 §3 explains why the definition is stated with WOT.

Physical reading (Week 2 §2): norm convergence controls all unit-vector matrix elements uniformly, whereas WOT tests each chosen pair of vectors. The comparison is operationally suggestive, but WOT closure should not be defined as “what no experiment can distinguish”: finite experimental precision is a different notion. Mathematically, WOT closure supplies the spectral projections and bounded limits needed for an algebra of observables.

Key fact (Sakai). A C*-algebra $\mathcal{A}$ is *-isomorphic to a vN algebra iff it has a predual: there is a Banach space ${\mathcal{A}}_{\ast }$ with $\mathcal{A}=({\mathcal{A}}_{\ast }{)}^{\ast }$. Such C*-algebras are called W*-algebras. The definition above is concrete, but the $\sigma$-weak topology and the normal states are intrinsic to the algebra and do not depend on the chosen faithful normal representation.

A.5 States, Normal States, Predual

A state on a C*-algebra $\mathcal{A}$ is a positive linear functional $\omega :\mathcal{A}\to \mathbb{C}$ with $\omega (1)=1$.

A state $\omega$ on a vN algebra $\mathcal{M}$ is normal if it is $\sigma$-weakly continuous; equivalently if it is order-continuous: $\omega ({\sup }_{\alpha }{a}_{\alpha })={\sup }_{\alpha }\omega (a_{\alpha })$ for increasing nets of positive operators.

The predual ${\mathcal{M}}_{\ast }$ is the space of normal linear functionals; $\mathcal{M}=({\mathcal{M}}_{\ast }{)}^{\ast }$.

For $\mathcal{M}=\mathcal{B}(\mathcal{H})$: ${\mathcal{M}}_{\ast }\cong \mathcal{T}(\mathcal{H})$, and every normal state is represented by a unique trace-class density operator. General type-I von Neumann algebras have the corresponding direct-integral version of this statement.

For type III: ${\mathcal{M}}_{\ast }$ is still the Banach space of normal functionals, but there is no intrinsic density $h\in \mathcal{M}$ relative to a faithful normal semifinite trace, because no such trace exists. In a concrete embedding $\mathcal{M}\subset \mathcal{B}(\mathcal{H})$ a normal functional may be extended and represented by an ambient trace-class operator; that representation is nonunique and is not a reduced density matrix belonging to the local algebra.

A.5.1 Weights and semifiniteness

A weight on a von Neumann algebra $\mathcal{M}$ is a map

$$\phi :{\mathcal{M}}_{+}⟶[0,+\infty ]$$

that is additive and positively homogeneous on the positive cone. A state is a finite normalized weight; the point of allowing $+\infty$ is that an infinite algebra may have a useful integration theory without a finite trace.

The weight is:

• faithful if $x\ge 0$ and $\phi (x)=0$ imply $x=0$;
• normal if $x_i\uparrow x$ implies $\phi (x_i)\uparrow \phi (x)$ for increasing nets in ${\mathcal{M}}_{+}$;
• semifinite if, for every $x\in {\mathcal{M}}_{+}$, $$\phi (x)=\sup \{\phi (y):0\le y\le x,\phi (y)<\infty \};$$
• tracial if $\phi (u^{\ast }xu)=\phi (x)$ for every unitary $u\in \mathcal{M}$ and $x\in {\mathcal{M}}_{+}$.

Thus a faithful normal semifinite trace is an invariant integration rule that is locally finite on enough positive elements, even when $\tau (1)=+\infty$. Type III factors admit faithful normal weights but no faithful normal semifinite trace. Their continuous cores do admit such a trace.

A.5.2 Affiliated and measurable operators

A closed densely defined operator $T$ is affiliated with $\mathcal{M}$ if it commutes with the commutant in the domain-sensitive sense

$$u^{\prime }T\subset T{u}^{\prime }\text{for every unitary }{u}^{\prime }\in {\mathcal{M}}^{\prime }.$$

For self-adjoint $T$, this is equivalent to requiring all spectral projections of $T$ to lie in $\mathcal{M}$. Affiliation is the correct meaning of saying that an unbounded observable or density “belongs to” a von Neumann algebra.

Given a faithful normal semifinite trace $\tau$, an affiliated operator $T$ is $\tau$-measurable when its domain is $\tau$-dense: for every $ϵ>0$ there is a projection $e\in \mathcal{M}$ such that

$$e\mathcal{H}\subset \mathrm{Dom}T,\tau (1-e)<ϵ.$$

The noncommutative space $L^1(\mathcal{M},\tau )$ consists of the appropriate $\tau$-measurable affiliated operators with finite trace norm. A normal state on a semifinite algebra has a positive density $h\in L^1(\mathcal{M},\tau )$ with $\tau (h)=1$; $h$ need not be a bounded element of $\mathcal{M}$. This is the setting of the core densities and entropies used in Weeks 13—14.

A.6 GNS Construction (Recap from Week 1)

Given a state $\omega$ on a C*-algebra $\mathcal{A}$:

1. The sesquilinear form $⟨a,b{⟩}_{\omega }:=\omega (a^{\ast }b)$ on $\mathcal{A}$ is positive semi-definite.
2. Quotient by the null space ${\mathcal{N}}_{\omega }=\{a:\omega (a^{\ast }a)=0\}$ to get a pre-Hilbert space $\mathcal{A}/{\mathcal{N}}_{\omega }$.
3. Complete to a Hilbert space ${\mathcal{H}}_{\omega }$.
4. Left multiplication by $\mathcal{A}$ defines a $\ast$-representation ${\pi }_{\omega }:\mathcal{A}\to \mathcal{B}({\mathcal{H}}_{\omega })$.
5. The image of $1\in \mathcal{A}$ is a cyclic vector ${\Omega }_{\omega }\in {\mathcal{H}}_{\omega }$ with $\omega (a)=⟨{\Omega }_{\omega },{\pi }_{\omega }(a){\Omega }_{\omega }⟩$.

The GNS triple $({\mathcal{H}}_{\omega },{\pi }_{\omega },{\Omega }_{\omega })$ is unique up to unitary equivalence.

If the normal extension of $\omega$ to ${\pi }_{\omega }(\mathcal{A}{)}^{\prime \prime }$ is faithful, then ${\Omega }_{\omega }$ is separating for that von Neumann algebra, and conversely. Faithfulness on the original C*-algebra makes ${\pi }_{\omega }$ faithful but should not be substituted silently for faithfulness of the normal extension.

The state is factorial when ${\pi }_{\omega }(\mathcal{A}{)}^{\prime \prime }$ has trivial center. This is weaker than irreducibility: an irreducible representation has ${\pi }_{\omega }(\mathcal{A}{)}^{\prime }=\mathbb{C}1$, hence ${\pi }_{\omega }(\mathcal{A}{)}^{\prime \prime }=\mathcal{B}({\mathcal{H}}_{\omega })$, whereas a type II or type III factor representation is factorial but not irreducible in this sense. Pure states, not general factorial states, give irreducible GNS representations.

A.7 Trace-Class and Hilbert–Schmidt Operators

The Hilbert–Schmidt operators $\mathcal{H}\mathcal{S}(\mathcal{H})$ are those with ${\sum }_i\Vert T{e}_i{\Vert }^2<\infty$ for any orthonormal basis. They form a two-sided ideal in $\mathcal{B}(\mathcal{H})$.

The trace-class operators $\mathcal{T}(\mathcal{H})$ are those with ${\sum }_i⟨e_i,|T|e_i⟩<\infty$. They form a two-sided ideal in $\mathcal{B}(\mathcal{H})$ and $\mathcal{T}(\mathcal{H})\subset \mathcal{H}\mathcal{S}(\mathcal{H})\subset \mathcal{K}(\mathcal{H})$ (compact) $\subset \mathcal{B}(\mathcal{H})$.

The trace $\mathrm{Tr}:\mathcal{T}(\mathcal{H})\to \mathbb{C}$ is the standard sum-of-diagonal-entries, basis-independent.

In Week 7 §3: the Hilbert–Schmidt space $\mathcal{H}\mathcal{S}(\mathcal{H})=M_n(\mathbb{C})$ (finite-dim case) is used as the GNS Hilbert space for type-I states, with left multiplication as the representation. This is the natural setting for explicit modular-operator and relative-modular-operator computations.

A.8 Tensor Products

The algebraic tensor product ${\mathcal{H}}_1{\otimes }_{\mathrm{alg}}{\mathcal{H}}_2$ has a natural inner product $⟨{\xi }_1\otimes {\xi }_2,{\eta }_1\otimes {\eta }_2⟩=⟨{\xi }_1,{\eta }_1⟩⟨{\xi }_2,{\eta }_2⟩$. Completing gives the Hilbert-space tensor product ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$.

For vN algebras: ${\mathcal{M}}_1\overline{\otimes }{\mathcal{M}}_2$ is the spatial tensor product, the WOT-closure of ${\mathcal{M}}_1{\otimes }_{\mathrm{alg}}{\mathcal{M}}_2$ on ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$.

Commutant theorem. $({\mathcal{M}}_1\overline{\otimes }{\mathcal{M}}_2{)}^{\prime }={\mathcal{M}}_1^{\prime }\overline{\otimes }{\mathcal{M}}_2^{\prime }$ (this works in the spatial tensor product; abstract tensor products are more delicate).

In AQFT there is generally no canonical factorization $\mathcal{H}=\mathcal{H}(\mathcal{O})\otimes \mathcal{H}({\mathcal{O}}^{\prime })$ whose first factor carries the bounded-region algebra. Type III local structure is one way this failure becomes visible. Under the split property, strictly separated regions admit an intermediate type-I factor and a corresponding tensor-product implementation; this does not create a sharp tensor split between a region and its touching causal complement. Type-I regulators also permit tensor reasoning, with regulator dependence stated explicitly.

A.9 Polar Decomposition

A closed densely-defined operator $T$ has a unique polar decomposition $T=V|T|$, where $|T|=(T^{\ast }T{)}^{1/2}\ge 0$ and $V$ is a partial isometry with initial space $\overline{\mathrm{ran}|T|}$ and final space $\overline{\mathrm{ran}T}$.

For conjugate-linear $T$ (e.g., the Tomita operator $S$), the same decomposition holds with $V$ antiunitary. In the cyclic-separating setting, $V$ extends to an antiunitary on the full Hilbert space — this is the modular conjugation $J$.

Used in:

• Tomita operator polar decomposition $S=J{\Delta }^{1/2}$ (Week 5 §3).
• Connes cocycle existence (Week 7 §4).

A.10 Analytic Vectors and the Analytic Subalgebra

For a self-adjoint $A$ generating $U(t)=e^{itA}$, a vector $\xi \in {\bigcap }_{n\ge 0}\mathrm{Dom}(A^n)$ is analytic for $A$ if ${\sum }_n|t{|}^n\Vert A^n\xi \Vert /n!<\infty$ for $|t|$ in some neighborhood of zero. Every self-adjoint operator has a dense set of analytic vectors, for example vectors with compact spectral support.

For such a vector the power series for $U(z)\xi$ converges locally in complex $z$; entire analytic vectors have convergence for all $z$. Separately, an element $a$ is analytic for an automorphism group ${\alpha }_t$ when $t\mapsto {\alpha }_t(a)$ has the required complex continuation. These analytic elements form the convenient dense subalgebra used in KMS arguments. The KMS condition itself concerns analytic continuation of correlation functions and does not say that every vector is analytic.

Physical reading: for a positive-energy Hamiltonian, spectral positivity helps control Euclidean-time factors such as $e^{-\tau H}$. Modular generators are generally not bounded below, so their strip analyticity comes instead from the Tomita—Takesaki/KMS structure and the domains of powers of $\Delta$. Thus a formal expression such as ${\Delta }^{1/2}a\Omega$ is licensed by a domain or analytic-element statement, not by a generic Wick-rotation rule.

A.11 What’s Used Where

Topic	Used in
Spectral theorem	Throughout; especially Weeks 4, 5, 6 (modular operator), 10 (boost generator)
Bounded vs. unbounded	Weeks 8 (smeared fields), 5 (Tomita operator), 10 (boost)
Closed/closable operators, graph norms, cores	Week 5 (Tomita operator), Week 7 (relative modular operator)
Topologies (WOT/SOT)	Week 2 (vN algebra definition)
Predual / normal states	Weeks 2, 12 (type III; no density matrix as operator means via predual)
Weights, affiliation, $\tau$-measurability	Weeks 3, 13, 14 (semifinite traces and core densities)
GNS	Week 1 (foundation), all subsequent weeks
Trace-class	Week 7 (Hilbert–Schmidt GNS for type I)
Tensor products	Week 5 (type-I modular), Week 9 (no factorization for AQFT), Week 13 (crossed product on $\mathcal{H}\otimes L^2(\mathbb{R})$)
Polar decomposition	Week 5 (Tomita operator), Week 7 (Connes cocycle)
Analytic vectors	Weeks 4, 6, 14 (KMS strip analytic continuations)

A.12 Suggested Reading

For students who need a full functional-analysis text:

• Reed & Simon, Methods of Modern Mathematical Physics, Vol. I (functional analysis) and Vol. II (Fourier analysis, self-adjointness). The standard reference.
• Conway, A Course in Functional Analysis. More accessible introduction.
• Pedersen, Analysis Now. Compact alternative.

For operator algebras specifically:

• Murphy, C*-algebras and Operator Theory. Friendly first text.
• Bratteli & Robinson Vol. I, ch. 2. Course’s primary reference.