Week 1 — C*-algebras, States, and the GNS Construction

We could have started this course with operators on a Hilbert space, defined a “physical theory” as a Hilbert space plus a Hamiltonian, and proceeded as in any quantum mechanics class. We do not. The reason — which will become precise in Block C — is that quantum field theory has no natural Hilbert space: it has many, and they are mutually inequivalent. The algebraic language is the one that survives.

How to use this chapter

• In class: move from the concrete algebras $C(X)$ and $M_n(\mathbb{C})$ to abstract positivity and states, then build the GNS representation in full. Keep the finite-dimensional examples on the board while each abstract definition is introduced.
• For self-study: verify every claim first in $M_2(\mathbb{C})$, then repeat it for $C(X)$. Do not leave the chapter until you can construct the GNS null ideal, representation, and cyclic vector without looking ahead.
• Instructor checkpoint: ask students to distinguish an abstract state from a density matrix in a chosen representation, and a normal pure state on $\mathcal{B}(\mathcal{H})$ from a singular pure state.

0. Reading

Primary: Bratteli & Robinson Vol. I, §2.1–2.3 (definitions of C*-algebras and states), §2.3.3 (GNS construction).

Secondary:

• Murphy, C*-algebras and Operator Theory, ch. 1–3 — gentler than B&R; recommended for first reading.
• Takesaki Vol. I, ch. I — terse but authoritative.

For the curious: Connes, Noncommutative Geometry, ch. I.1 for the Gelfand–Naimark theorem in geometric perspective.

Proof-status labels. Theorems in this note carry inline labels in the standard course convention (introduced systematically in Week 3 and codified in the note-quality template and conventions.md):

• [Proved.] Full proof given in the notes.
• [Sketched.] Main ideas given; some analytic details skipped.
• [Stated only — refs.] Theorem cited from the literature with reference.

1. Motivation: why algebras?

In a finite-dimensional quantum system, the observables are Hermitian matrices in $M_n(\mathbb{C})$, the states are density matrices, and there is essentially one Hilbert space, ${\mathbb{C}}^n$. The algebra of observables, the Hilbert space, and the state space are interconvertible, and we usually do not distinguish them.

In quantum field theory this neat picture fails, for three reasons that motivate the algebraic approach:

1. Representations are non-unique. The Stone–von Neumann theorem fails for field systems: distinct regular representations of the canonical commutation relations need not be unitarily equivalent. Minkowski and Fulling–Rindler quasifree states, for example, lead to inequivalent representations in the relevant global setting. Their Hilbert spaces may be abstractly isomorphic as separable Hilbert spaces; what fails is a unitary intertwiner preserving the field representation and state. Picking one representation is therefore a choice, not a structural feature of the abstract algebra.

2. Local algebras need not admit an intrinsic trace. As we will see in Week 12, broad classes of relativistic QFT satisfying explicit factoriality, phase-space, and scaling hypotheses have bounded-region algebras isomorphic to the hyperfinite type III${}_1$ factor. Such an algebra has no nonzero normal semifinite trace and hence no intrinsic density-matrix description relative to a local trace. An ambient representation may still express a vector state through a rank-one operator on the global Hilbert space; that is not a reduced density matrix inside the local algebra. The algebraic framework supplies the representation-independent substitute: states are positive linear functionals.

3. Modular theory needs algebras. The whole content of the Tomita–Takesaki theorem (Block B) is a structural statement about von Neumann algebras together with a cyclic and separating vector. Without the algebra, there is no statement.

This week sets up the foundations. We define C*-algebras (the algebraic object), states (the linear functionals that play the role of “expectation values”), and the GNS construction (the canonical bridge from a state on an algebra to a Hilbert-space representation). The whole rest of the course is built on this.

2. Banach algebras

2.1 Definitions

Definition 2.1. A Banach algebra is a complex vector space $\mathcal{A}$ equipped with:

• An associative bilinear product $\mathcal{A}\times \mathcal{A}\to \mathcal{A}$, $(a,b)\mapsto ab$;
• A norm $\Vert \cdot \Vert$ for which $\mathcal{A}$ is a Banach space (i.e., complete);
• The submultiplicativity property $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$ for all $a,b\in \mathcal{A}$.

If there exists $1\in \mathcal{A}$ with $1\cdot a=a\cdot 1=a$ for all $a$, we say $\mathcal{A}$ is unital and call $1$ the unit. We will assume unital throughout unless explicitly stated otherwise; the non-unital case can be handled by unitization (adjoining an external identity), which we touch on in Problem 2.

Definition 2.2. A *-algebra is a complex associative algebra $\mathcal{A}$ together with an involution $a\mapsto a^{\ast }$ satisfying:

1. $(a^{\ast }{)}^{\ast }=a$ (involutive),
2. $(a+\lambda b{)}^{\ast }=a^{\ast }+\overline{\lambda }{b}^{\ast }$ for $\lambda \in \mathbb{C}$ (conjugate-linear),
3. $(ab{)}^{\ast }=b^{\ast }{a}^{\ast }$ (antimultiplicative).

When the same space is also a Banach algebra and the involution is continuous, one speaks of a Banach *-algebra. In our concrete examples the involution is isometric. We do not yet impose the C*-identity; §4 adds exactly that decisive compatibility between norm and involution.

2.2 Examples

Example 2.3 (Bounded operators). Let $\mathcal{H}$ be a Hilbert space. The space $\mathcal{B}(\mathcal{H})$ of bounded linear operators on $\mathcal{H}$ is a unital Banach *-algebra under composition, the operator norm, and the adjoint operation $T\mapsto T^{\ast }$. Submultiplicativity is the standard $\Vert ST\Vert \le \Vert S\Vert \Vert T\Vert$.

Example 2.4 (Continuous functions). Let $X$ be a compact Hausdorff space. The space $C(X)$ of continuous $\mathbb{C}$-valued functions on $X$, equipped with pointwise multiplication, the supremum norm $\Vert f{\Vert }_{\infty }={\sup }_{x\in X}|f(x)|$, and pointwise complex conjugation $f^{\ast }(x)=\overline{f(x)}$, is a unital commutative Banach *-algebra.

Example 2.5 (Matrices). The space $M_n(\mathbb{C})$ of complex $n\times n$ matrices, with matrix multiplication, the operator norm (i.e., the largest singular value), and the conjugate-transpose, is a finite-dimensional Banach *-algebra. This is the special case $\mathcal{H}={\mathbb{C}}^n$ of Example 2.3.

Example 2.6 (Bounded measurable functions). Let $(X,\Sigma ,\mu )$ be a $\sigma$-finite measure space. The space $L^{\infty }(X,\mu )$ of essentially bounded measurable functions modulo equality $\mu$-almost-everywhere, with pointwise operations, the essential supremum norm, and pointwise conjugation, is a unital commutative Banach *-algebra.

These examples exhibit the two main concrete forms we will use. The Gelfand–Naimark theorems (§5) say, more precisely, that every abstract C*-algebra has a faithful representation as a norm-closed *-subalgebra of some $\mathcal{B}(\mathcal{H})$, and that every unital commutative C*-algebra is a $C(X)$. They do not say that every C*-algebra is literally one of the four examples just listed.

2.3 The spectrum

Definition 2.7. Let $\mathcal{A}$ be a unital Banach algebra and $a\in \mathcal{A}$. The resolvent set of $a$ is

$$\rho (a)=\{\lambda \in \mathbb{C}:a-\lambda 1\text{ is invertible in }\mathcal{A}\},$$

and the spectrum of $a$ is its complement, $\sigma (a)=\mathbb{C}\setminus \rho (a)$.

Physical picture. The spectrum is the algebraic substitute for “the set of possible measurement outcomes.” In $M_n(\mathbb{C})$ it is the set of eigenvalues; in $C(X)$ it is the range of the function; in $\mathcal{B}(\mathcal{H})$ it contains the eigenvalues plus the continuous spectrum that physicists handle with “generalized eigenstates.” The definition through invertibility is what makes the notion representation-independent: whether $a-\lambda 1$ has an inverse is a question about the algebra alone, asked without ever mentioning vectors or wave functions. In Week 12 this same notion enters the Connes invariants extracted across all faithful modular data. The spectrum of one preferred modular operator is useful evidence, but by itself it does not determine the algebra’s type.

Theorem 2.8. [Proved.] For any $a\in \mathcal{A}$, the spectrum $\sigma (a)$ is a non-empty compact subset of $\mathbb{C}$ contained in the closed disk of radius $\Vert a\Vert$.

Proof. Boundedness. If $|\lambda |>\Vert a\Vert$, the Neumann series

$$R(\lambda ):=\sum _{n=0}^{\infty }{\lambda }^{-n-1}{a}^n$$

converges absolutely in $\mathcal{A}$, since $\Vert {\lambda }^{-n-1}{a}^n\Vert \le |\lambda {|}^{-1}(\Vert a\Vert /|\lambda |{)}^n$ is geometric with ratio $<1$ and $\mathcal{A}$ is complete. Multiplying term by term and telescoping, $(\lambda 1-a)R(\lambda )=R(\lambda )(\lambda 1-a)=1$. Hence $\lambda \in \rho (a)$, so $\sigma (a)$ lies in the closed disk of radius $\Vert a\Vert$.

Closedness. Invertible elements form an open set: if $x$ is invertible and $\Vert y-x\Vert <\Vert x^{-1}{\Vert }^{-1}$, then $y=x(1-x^{-1}(x-y))$ with $\Vert x^{-1}(x-y)\Vert <1$, so the Neumann series for $(1-x^{-1}(x-y){)}^{-1}$ converges and $y$ is invertible. Applying this to $x=\lambda 1-a$ shows $\rho (a)$ is open.

Non-emptiness. On $\rho (a)$ the resolvent $\lambda \mapsto (\lambda 1-a{)}^{-1}$ is holomorphic as a Banach-space-valued function: the identity

$$(\lambda 1-a{)}^{-1}-(\mu 1-a{)}^{-1}=(\mu -\lambda )(\lambda 1-a{)}^{-1}(\mu 1-a{)}^{-1}$$

(the resolvent identity) shows the difference quotient converges in norm as $\mu \to \lambda$. For $|\lambda |>\Vert a\Vert$ the Neumann series gives $\Vert (\lambda 1-a{)}^{-1}\Vert \le (|\lambda |-\Vert a\Vert {)}^{-1}\to 0$ at infinity. If $\sigma (a)$ were empty, then for every bounded functional $\phi \in {\mathcal{A}}^{\ast }$ the scalar function $\phi ((\lambda 1-a{)}^{-1})$ would be entire and vanish at infinity, hence identically zero by Liouville. Since bounded functionals separate points (Hahn–Banach), $(\lambda 1-a{)}^{-1}=0$ for all $\lambda$ — contradicting $(\lambda 1-a{)}^{-1}(\lambda 1-a)=1$. $\square$

The number

$$r(a)=\sup \{|\lambda |:\lambda \in \sigma (a)\}$$

is called the spectral radius. We will need one classical fact about it.

Theorem 2.9 (Beurling–Gelfand spectral radius formula). [Sketched.] In any unital Banach algebra,

$$r(a)=\underset{n\to \infty }{\lim }\Vert a^n{\Vert }^{1/n},$$

and the limit exists.

Sketch of proof. For the lower bound: if $\lambda \in \sigma (a)$ then ${\lambda }^n\in \sigma (a^n)$ (factor ${\lambda }^{n}1-a^n=(\lambda 1-a)q(a)$ with $q$ a polynomial; if ${\lambda }^{n}1-a^n$ were invertible, $\lambda 1-a$ would have a one-sided inverse on each side, hence be invertible). Then $|\lambda {|}^n\le \Vert a^n\Vert$ by Theorem 2.8, so $r(a)\le {\mathrm{liminf}}_n\Vert a^n{\Vert }^{1/n}$. For the upper bound: the resolvent is holomorphic on $\{|\lambda |>r(a)\}$, so its Laurent expansion ${\sum }_n{\lambda }^{-n-1}{a}^n$ — valid a priori only for $|\lambda |>\Vert a\Vert$ — converges on the whole region $|\lambda |>r(a)$ by Banach-valued holomorphy. Convergence of the series forces $\Vert {\lambda }^{-n-1}{a}^n\Vert \to 0$, hence ${\mathrm{limsup}}_n\Vert a^n{\Vert }^{1/n}\le |\lambda |$ for every $|\lambda |>r(a)$. Existence of the limit follows from comparing the two bounds. $\square$

This formula is the engine behind the rigidity of C*-algebras: it expresses a purely algebraic quantity (the spectral radius) as a purely metric limit. We use it in Lemma 4.3 below, where it forces the C*-norm to be an algebraic invariant.

3. Self-adjoint, normal, unitary, positive

In a *-algebra $\mathcal{A}$, certain elements are distinguished by their relation to the involution.

Definition 3.1. An element $a\in \mathcal{A}$ is:

• Self-adjoint (or Hermitian) if $a=a^{\ast }$;
• Normal if $a{a}^{\ast }=a^{\ast }a$;
• Unitary if $a{a}^{\ast }=a^{\ast }a=1$;
• Positive if $a=b^{\ast }b$ for some $b\in \mathcal{A}$ (the order-theoretic force of this definition will be used once $\mathcal{A}$ is a C*-algebra).

Self-adjoints generalize Hermitian operators (real spectrum, real expectation values). Unitaries generalize unitary operators. Normals generalize operators that admit a spectral decomposition.

Physical picture. The involution is where the physics enters the algebra. Self-adjoint elements are the observables — the things a lab can measure, with real outcomes (Theorem 4.5 below makes “real outcomes” a theorem, not an axiom). Unitaries are the symmetries and time evolutions — the operations that can be undone. Positive elements are the intensities: counts, probabilities, energies-above-ground-state, anything that can only come out non-negative. The decomposition $a=h+ik$ of Lemma 3.2 says the full algebra is just the complexification of its observables, so nothing physical is lost by keeping the non-self-adjoint elements around — and much computational convenience is gained, exactly as with raising/lowering operators in quantum mechanics.

Lemma 3.2. [Proved — direct verification.] In a *-algebra, any element decomposes uniquely as $a=h+ik$ with $h,k$ self-adjoint, namely $h=(a+a^{\ast })/2$ and $k=(a-a^{\ast })/(2i)$.

This is the algebraic analogue of the real-imaginary decomposition of complex numbers. Consequently, any *-algebra is the complex linear span of its self-adjoint elements, and any positive linear functional (defined below) is determined by its values on self-adjoints.

4. C*-algebras

4.1 The C*-identity

We have not yet related the norm to the involution. The single axiom doing all the work is:

Definition 4.1. A Banach *-algebra $\mathcal{A}$ is a C*-algebra if

$$\boxed{\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2\text{for all }a\in \mathcal{A}.}$$

This is called the C*-identity.

The C*-identity is innocuous-looking but extraordinarily restrictive. Its consequences propagate everywhere.

Physical picture. The C*-identity is the statement that the norm of an observable is the largest value it can be measured to take. For a self-adjoint $a$, Lemma 4.3 below gives $\Vert a\Vert =r(a)=\sup \{|\lambda |:\lambda \in \sigma (a)\}$ — the norm is the extreme measurement outcome, not an arbitrary metric bookkeeping device. Corollary 4.4 then says the metric structure of the theory is determined by the physics (which combinations of observables are invertible), with no freedom left over. Contrast a generic Banach *-algebra, where many inequivalent norms can coexist on the same algebra: there, “size” is a convention. In a C*-algebra, size is spectroscopy.

Lemma 4.2. [Proved.] In a C*-algebra, $\Vert a^{\ast }\Vert =\Vert a\Vert$ for all $a$.

Proof. From $\Vert a{\Vert }^2=\Vert a^{\ast }a\Vert \le \Vert a^{\ast }\Vert \Vert a\Vert$ we get $\Vert a\Vert \le \Vert a^{\ast }\Vert$. Replacing $a$ by $a^{\ast }$ gives the reverse, so $\Vert a^{\ast }\Vert =\Vert a\Vert$. $\square$

Lemma 4.3. [Proved.] If $a$ is self-adjoint in a C*-algebra, then $\Vert a\Vert =r(a)$.

Proof. Since $a^{\ast }=a$, the C*-identity gives $\Vert a^2\Vert =\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$. By induction, $\Vert a^{2^n}\Vert =\Vert a{\Vert }^{2^n}$. Theorem 2.9 (the limit exists, so any subsequence computes it): $r(a)={\lim }_n\Vert a^n{\Vert }^{1/n}={\lim }_n\Vert a^{2^n}{\Vert }^{1/2^n}=\Vert a\Vert$. $\square$

Corollary 4.4. [Proved.] The norm on a C*-algebra is uniquely determined by the algebraic structure.

Proof. For any $a$, the element $a^{\ast }a$ is self-adjoint, so $\Vert a^{\ast }a\Vert =r(a^{\ast }a)$ by Lemma 4.3. The spectrum $\sigma (a^{\ast }a)$ is purely algebraic (depends only on which $a^{\ast }a-\lambda 1$ are invertible), and $\Vert a{\Vert }^2=\Vert a^{\ast }a\Vert =r(a^{\ast }a)$ then determines $\Vert a\Vert$ uniquely. $\square$

This corollary is striking: in a Banach algebra one might equip the same underlying *-algebra with many different submultiplicative norms; the C*-identity rules this out.

4.2 Examples are C*-algebras

It is straightforward to verify that Examples 2.3–2.6 are all C*-algebras:

• $\mathcal{B}(\mathcal{H})$: the C*-identity is the standard identity $\Vert T^{\ast }T\Vert =\Vert T{\Vert }^2$ on bounded operators (proved via $\Vert T{\Vert }^2={\sup }_{\Vert \xi \Vert =1}\Vert T\xi {\Vert }^2=\sup ⟨\xi ,T^{\ast }T\xi ⟩=\Vert T^{\ast }T\Vert$ when $T^{\ast }T$ is self-adjoint).
• $C(X)$: $\Vert f^{\ast }f{\Vert }_{\infty }=\sup |f{|}^2=\Vert f{\Vert }_{\infty }^2$.
• $M_n(\mathbb{C})$: special case of $\mathcal{B}(\mathcal{H})$.
• $L^{\infty }(X,\mu )$: as for $C(X)$.

4.3 Spectrum and self-adjointness

Theorem 4.5. [Proved.] In a C*-algebra, the spectrum of a self-adjoint element lies in $\mathbb{R}$.

Proof. Let $a=a^{\ast }$ and $\lambda =\alpha +i\beta \in \sigma (a)$ with $\alpha ,\beta \in \mathbb{R}$. The element $a+it\cdot 1$ for real $t$ is normal, with spectrum $\sigma (a)+it$. Hence $\alpha +i(\beta +t)\in \sigma (a+it)$. Therefore

$$|\alpha +i(\beta +t){|}^2\le \Vert a+it{\Vert }^2=\Vert (a+it{)}^{\ast }(a+it)\Vert =\Vert a^2+t^2\Vert \le \Vert a{\Vert }^2+t^2.$$

Expanding the LHS: ${\alpha }^2+(\beta +t{)}^2\le \Vert a{\Vert }^2+t^2$, i.e., ${\alpha }^2+{\beta }^2+2\beta t\le \Vert a{\Vert }^2$. As $t\to \pm \infty$, the inequality fails unless $\beta =0$. $\square$

Corollary 4.6. [Proved.] In a C*-algebra, the spectrum of a unitary lies in the unit circle $\{|\lambda |=1\}$.

Proof. If $u$ is unitary, then $\Vert u\Vert =\Vert u^{-1}\Vert =1$. Theorem 2.8 gives $|\lambda |\le 1$ for $\lambda \in \sigma (u)$. Spectral mapping for inversion gives ${\lambda }^{-1}\in \sigma (u^{-1})=\sigma (u^{\ast })$, so $|\lambda {|}^{-1}\le 1$. Hence $|\lambda |=1$. $\square$

4.4 Positive elements: an order structure

Theorem 4.7. [Sketched — refs: Bratteli–Robinson Vol. I §2.2.4.] In a unital C*-algebra, the following are equivalent for a self-adjoint $a$: (i) $a=b^{\ast }b$ for some $b\in \mathcal{A}$; (ii) $\sigma (a)\subset [0,\infty )$; (iii) $a=c^2$ for some self-adjoint $c$.

We write $a\ge 0$ when these hold and call such $a$ positive. The set ${\mathcal{A}}_{+}$ of positive elements is a closed convex cone and gives ${\mathcal{A}}_{sa}$ (self-adjoints) the structure of an ordered real Banach space: $a\le b$ means $b-a\in {\mathcal{A}}_{+}$.

Physical picture. The equivalence (i) $\iff$ (ii) is the algebraic version of a familiar quantum-mechanical fact: an observable whose measured values are always non-negative is exactly one that can be written as “something dagger something” — an intensity, $|\mathrm{amplitude}{|}^2$. The order structure $a\le b$ is operational: it means every state assigns $a$ a smaller expectation value than $b$ (once states are defined in §6, check this as an exercise). Everything probabilistic in the algebraic framework — positivity of states, Cauchy–Schwarz, the GNS inner product — flows through this cone. It is also the structure that survives in type III algebras when density matrices do not: we can always say which observables dominate which, even when we cannot write $\mathrm{Tr}(\rho a)$.

Sketch: (iii) $\Rightarrow$ (i) is trivial ($b=c$). (i) $\Rightarrow$ (ii) requires showing $b^{\ast }b$ has non-negative spectrum, which follows from polar decomposition arguments. (ii) $\Rightarrow$ (iii) uses the continuous functional calculus: when $\sigma (a)\subset [0,\infty )$ we can take $\sqrt{a}$ as a self-adjoint element via $f(a)$ for $f(t)=\sqrt{t}$. The continuous functional calculus is the assignment $f\mapsto f(a)$ for $f\in C(\sigma (a))$ and self-adjoint $a$, satisfying $(fg)(a)=f(a)g(a)$, $\overline{f}(a)=f(a{)}^{\ast }$, etc.; we will use it freely in the rest of the course. (See Bratteli–Robinson Thm 2.1.13.)

5. Two foundational theorems (statements only)

We mention these as motivation; we do not prove them in this course.

Theorem 5.1 (Gelfand–Naimark, commutative case). [Stated only — refs: Bratteli–Robinson Vol. I §2.3.2; Murphy ch. 2.] Every unital commutative C*-algebra $\mathcal{A}$ is isometrically *-isomorphic to $C(X)$ for some compact Hausdorff space $X$, namely the spectrum (Gelfand spectrum) of $\mathcal{A}$.

This says: commutative C*-algebras and compact Hausdorff spaces are the same thing, viewed two ways. The “noncommutative space” interpretation underlying noncommutative geometry follows: a non-commutative C*-algebra is interpreted as the algebra of “continuous functions on a non-existent space.”

Theorem 5.2 (Gelfand–Naimark, general case). [Stated only — proof essentially the GNS construction of §7 applied to a separating family of states; refs: Bratteli–Robinson Vol. I §2.3.5.] Every C*-algebra is isometrically *-isomorphic to a norm-closed *-subalgebra of $\mathcal{B}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$.

This says: any abstract C*-algebra is concretely a subalgebra of bounded operators on some Hilbert space. The proof is the GNS construction (§7) applied to a “rich enough” family of states.

These two theorems frame the course: the first says C*-algebras generalize topology; the second says they always come from operators on Hilbert spaces. Modular theory (Block B) and type classification (Block C) live in the second world.

6. States

6.1 Definition

Definition 6.1. A state on a unital C*-algebra $\mathcal{A}$ is a linear functional $\omega :\mathcal{A}\to \mathbb{C}$ satisfying:

1. Positivity: $\omega (a^{\ast }a)\ge 0$ for all $a\in \mathcal{A}$;
2. Normalization: $\omega (1)=1$.

The state space, denoted $\mathcal{S}(\mathcal{A})$, is a convex subset of the dual ${\mathcal{A}}^{\ast }$.

Lemma 6.1a (Cauchy—Schwarz for a positive functional). [Proved.] If $\omega$ is positive, then

$$|\omega (a^{\ast }b){|}^2\le \omega (a^{\ast }a)\omega (b^{\ast }b).$$

Proof. Positivity first implies $\omega (x^{\ast })=\overline{\omega (x)}$: decompose $x$ into self-adjoint parts and each self-adjoint part into a difference of positive elements. Put $A=\omega (a^{\ast }a)$, $B=\omega (b^{\ast }b)$, and $z=\omega (a^{\ast }b)$. For every $\lambda \in \mathbb{C}$,

$$0\le \omega \left((a+\lambda b{)}^{\ast }(a+\lambda b)\right)=A+\lambda z+\overline{\lambda }\overline{z}+|\lambda {|}^{2}B.$$

If $B>0$, choose $\lambda =-\overline{z}/B$ to obtain $0\le A-|z{|}^2/B$. If $B=0$, the same quadratic expression is $A+2\mathrm{Re}(\lambda z)$; unless $z=0$, a suitable multiple of $-\overline{z}$ makes it negative. Thus $z=0$ in the zero-denominator case, and the inequality holds in both cases. $\square$

It is a non-trivial fact that any state is automatically continuous, with $\Vert \omega {\Vert }_{{\mathcal{A}}^{\ast }}=\omega (1)=1$. The key mechanism is worth seeing: for self-adjoint $a$ with $\Vert a\Vert \le 1$, the element $1-a$ is positive (its spectrum lies in $[0,2]$ by Lemma 4.3 and Theorem 4.5, and positivity is spectral by Theorem 4.7), so $\omega (1-a)\ge 0$, i.e. $\omega (a)\le 1$; the same for $1+a$ gives $|\omega (a)|\le 1$. The general element is then controlled by Cauchy–Schwarz, $|\omega (a){|}^2=|\omega (1^{\ast }a){|}^2\le \omega (1)\omega (a^{\ast }a)\le \Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$. Continuity of states is thus a consequence of positivity plus the spectral control the C*-identity provides — nothing needs to be assumed about $\omega$ beyond algebra. (Problem 7** asks for the careful version.)

Physical picture. A state is a preparation procedure, described entirely by the statistics it produces. The lab never hands you a vector in a Hilbert space; it hands you a list of expectation values of the observables you measured. Definition 6.1 takes that operational data — a positive, normalized assignment of numbers to observables — as the primitive notion. Vectors and density matrices are derived objects, reconstructed (when possible) by the GNS construction of §7. This inversion of logic is precisely what lets the formalism survive in QFT, where a single preparation (say, the Minkowski vacuum restricted to a Rindler wedge) admits no density-matrix description at all, yet remains a perfectly good state in the present sense.

6.2 The role of $\omega$ as expectation value

Given an observable $a\in {\mathcal{A}}_{sa}$, the number $\omega (a)$ is the expectation value of $a$ in the state $\omega$. The variance is $\omega (a^2)-\omega (a{)}^2$, and so on. All quantum-mechanical statistics is recovered from these moments.

This is exactly the role density matrices play in quantum mechanics: ${\omega }_{\rho }(a)=\mathrm{Tr}(\rho a)$. The algebraic framework abstracts this away from a specific Hilbert space.

6.3 Examples

Example 6.2 (Vector states). Let $\pi :\mathcal{A}\to \mathcal{B}(\mathcal{H})$ be a representation and $\xi \in \mathcal{H}$ a unit vector. Then

$${\omega }_{\xi }(a):=⟨\xi ,\pi (a)\xi ⟩$$

is a state. Positivity: ${\omega }_{\xi }(a^{\ast }a)=\Vert \pi (a)\xi {\Vert }^2\ge 0$. Normalization: ${\omega }_{\xi }(1)=\Vert \xi {\Vert }^2=1$.

Example 6.3 (Density-matrix states). Same setting, $\rho$ a positive trace-class operator with $\mathrm{Tr}(\rho )=1$. Then ${\omega }_{\rho }(a)=\mathrm{Tr}(\rho \pi (a))$ is a state. This generalizes Example 6.2: vector states are the case of rank-one $\rho$.

Example 6.4 (Probability measures on $C(X)$). Let $\mu$ be a probability measure on a compact Hausdorff space $X$. Then

$${\omega }_{\mu }(f)={\int }_{X}fd\mu$$

is a state on $C(X)$, and conversely (Riesz representation theorem): every state on $C(X)$ is of this form for a unique Borel probability measure $\mu$. The state space of $C(X)$ is the set of probability measures on $X$, classical probability theory inside the algebraic framework.

Example 6.5 (Tracial state on $M_n(\mathbb{C})$). The functional $\tau (a)=\frac{1}{n}\mathrm{Tr}(a)$ is a state on $M_n(\mathbb{C})$. It is tracial: $\tau (ab)=\tau (ba)$. We will see in Block A (Week 3) that the existence of a tracial state forces an algebra to be of type $\mathrm{I}$ or $\mathrm{II}$, never type $\mathrm{III}$.

6.4 Pure and mixed states

Definition 6.6. A state $\omega$ is pure if it cannot be written as a non-trivial convex combination of states: $\omega =t{\omega }_1+(1-t){\omega }_2$ with ${\omega }_1,{\omega }_2\in \mathcal{S}(\mathcal{A})$ and $t\in (0,1)$ implies ${\omega }_1={\omega }_2=\omega$.

The pure states are the extreme points of the convex set $\mathcal{S}(\mathcal{A})$. Krein–Milman ensures there are enough of them to recover the full state space by closure-of-convex-hull.

Fact 6.7. [Stated only — refs: Bratteli–Robinson Vol. I, §2.3.] On $\mathcal{B}(\mathcal{H})$, the normal pure states are exactly the vector states. If $\mathcal{H}$ is infinite-dimensional, however, $\mathcal{B}(\mathcal{H})$ also has singular pure states, which are not given by density operators or vectors in the defining representation. On a commutative algebra $C(X)$, the pure states are exactly the Dirac measures ${\delta }_x$, $x\in X$.

The qualifier “normal” is invisible in finite-dimensional quantum mechanics because every state is normal there. It becomes essential in infinite dimensions. The algebraic statement that survives without choosing a representation is Theorem 7.3 below: a state is pure precisely when its own GNS representation is irreducible. Thus pure states of a commutative algebra recover classical points, whereas pure states of a noncommutative algebra produce irreducible quantum representations, not necessarily vectors in a representation fixed in advance.

7. The GNS construction

We come to the centerpiece of the week. The GNS construction takes a state on a C*-algebra and produces a Hilbert-space representation in which the state is realized as a vector state. It is functorial, canonical, and the foundation of essentially everything in the algebraic framework. Tomita–Takesaki theory in Block B is built on it.

Physical picture. GNS is the rigorous version of a construction every field theorist already knows: build the Hilbert space by acting with fields on the vacuum. In canonical QFT one populates Fock space as $\varphi (f_1)\cdots \varphi (f_n)|0⟩$; the inner products of such vectors are the vacuum correlation functions $⟨0|\varphi (f_1{)}^{†}\cdots |0⟩$. GNS says: that procedure needs nothing but the correlation functions themselves. Hand me the expectation functional $\omega$ (the full set of correlation functions of the algebra), and I will manufacture the Hilbert space, the represented operators, and the cyclic vector, canonically up to unitary equivalence. This is the same logic as the Wightman reconstruction theorem, stripped to its algebraic skeleton. It also explains the central moral of the week: the state picks the representation. Different phases, temperatures, or vacua can give inequivalent GNS representations of the same algebra. The abstract algebra is kinematical; the concrete Hilbert-space realization also remembers the state.

7.1 Setup

Fix a C*-algebra $\mathcal{A}$ and a state $\omega$ on $\mathcal{A}$. We will construct:

• A Hilbert space ${\mathcal{H}}_{\omega }$;
• A *-representation ${\pi }_{\omega }:\mathcal{A}\to \mathcal{B}({\mathcal{H}}_{\omega })$;
• A unit vector ${\Omega }_{\omega }\in {\mathcal{H}}_{\omega }$ that is cyclic (the linear span of ${\pi }_{\omega }(\mathcal{A}){\Omega }_{\omega }$ is dense in ${\mathcal{H}}_{\omega }$);

such that

$$\omega (a)=⟨{\Omega }_{\omega },{\pi }_{\omega }(a){\Omega }_{\omega }⟩\text{for all }a\in \mathcal{A}.$$

The triple $({\pi }_{\omega },{\mathcal{H}}_{\omega },{\Omega }_{\omega })$ is the GNS triple associated to $\omega$.

7.2 The construction

Step 1: a sesquilinear form. Define on $\mathcal{A}$ a sesquilinear form by

$$⟨a,b{⟩}_{\omega }:=\omega (a^{\ast }b).$$

This is positive semidefinite: $⟨a,a{⟩}_{\omega }=\omega (a^{\ast }a)\ge 0$. (Note: with this convention the form is $\mathbb{C}$-linear in the second slot, conjugate-linear in the first, matching the physics convention.)

The Cauchy–Schwarz inequality proved in Lemma 6.1a applies to this form:

$$|⟨a,b{⟩}_{\omega }{|}^2\le ⟨a,a{⟩}_{\omega }⟨b,b{⟩}_{\omega }.$$

Step 2: factor out the null space. Define

$${\mathcal{N}}_{\omega }:=\{a\in \mathcal{A}:\omega (a^{\ast }a)=0\}.$$

By Cauchy–Schwarz, $a\in {\mathcal{N}}_{\omega }$ iff $\omega (b^{\ast }a)=0$ for all $b\in \mathcal{A}$. Hence ${\mathcal{N}}_{\omega }$ is a left ideal in $\mathcal{A}$: if $a\in {\mathcal{N}}_{\omega }$ and $c\in \mathcal{A}$, then $\omega ((ca{)}^{\ast }(ca))=\omega (a^{\ast }{c}^{\ast }ca)\le \Vert c^{\ast }c\Vert \omega (a^{\ast }a)=0$, so $ca\in {\mathcal{N}}_{\omega }$. (The inequality $\omega (a^{\ast }{c}^{\ast }ca)\le \Vert c^{\ast }c\Vert \omega (a^{\ast }a)$ uses that $c^{\ast }c$ is positive and bounded by $\Vert c^{\ast }c\Vert \cdot 1$; rearranging gives a positive functional applied to a positive element. Careful: this uses positivity of the state crucially.)

Physical picture. The null space ${\mathcal{N}}_{\omega }$ consists of the operations the state cannot see: $a\in {\mathcal{N}}_{\omega }$ means the “excited vector” $a\Omega$ has zero norm, i.e. acting with $a$ on this preparation produces nothing detectable in any correlation function. Quotienting by ${\mathcal{N}}_{\omega }$ is the statement that two operators which produce identical statistics on top of $\omega$ define the same physical excitation of this state — even if they are very different elements of the abstract algebra. This is why the same algebra can yield wildly different Hilbert spaces from different states: each state erases a different set of distinctions. (For a faithful state, nothing is erased: ${\mathcal{N}}_{\omega }=0$, as in the tracial example of §8.2. Faithfulness becomes the “separating” condition in Week 5.)

The quotient $\mathcal{A}/{\mathcal{N}}_{\omega }$ inherits a positive-definite inner product from $⟨\cdot ,\cdot {⟩}_{\omega }$:

$$⟨[a],[b]⟩:=\omega (a^{\ast }b),$$

where $[a]$ denotes the equivalence class of $a$. Well-definedness uses the left-ideal property: if $a\sim a^{\prime }$ and $b\sim b^{\prime }$ then $\omega (a^{\ast }b)=\omega (a^{\prime \ast }{b}^{\prime })$.

Step 3: complete to a Hilbert space. Let ${\mathcal{H}}_{\omega }$ be the completion of $\mathcal{A}/{\mathcal{N}}_{\omega }$ with respect to this inner product.

Step 4: define the representation. For $a\in \mathcal{A}$, define ${\pi }_{\omega }(a):\mathcal{A}/{\mathcal{N}}_{\omega }\to \mathcal{A}/{\mathcal{N}}_{\omega }$ by left multiplication:

$${\pi }_{\omega }(a)[b]:=[ab].$$

Well-definedness uses, again, that ${\mathcal{N}}_{\omega }$ is a left ideal. Boundedness: $\Vert {\pi }_{\omega }(a)[b]{\Vert }^2=\omega (b^{\ast }{a}^{\ast }ab)\le \Vert a{\Vert }^2\omega (b^{\ast }b)=\Vert a{\Vert }^2\Vert [b]{\Vert }^2$, so ${\pi }_{\omega }(a)$ extends uniquely to a bounded operator on ${\mathcal{H}}_{\omega }$ with $\Vert {\pi }_{\omega }(a)\Vert \le \Vert a\Vert$.

Linearity, multiplicativity, and the *-property ${\pi }_{\omega }(a^{\ast })={\pi }_{\omega }(a{)}^{\ast }$ are direct calculations.

Step 5: identify the cyclic vector. Let ${\Omega }_{\omega }:=[1]\in \mathcal{A}/{\mathcal{N}}_{\omega }\subset {\mathcal{H}}_{\omega }$. Then:

• $\Vert {\Omega }_{\omega }{\Vert }^2=\omega (1^{\ast }\cdot 1)=\omega (1)=1$, so ${\Omega }_{\omega }$ is a unit vector.
• ${\pi }_{\omega }(a){\Omega }_{\omega }=[a]$ for all $a$, so ${\pi }_{\omega }(\mathcal{A}){\Omega }_{\omega }=\mathcal{A}/{\mathcal{N}}_{\omega }$, which is dense in ${\mathcal{H}}_{\omega }$ by construction. Hence ${\Omega }_{\omega }$ is cyclic.
• $⟨{\Omega }_{\omega },{\pi }_{\omega }(a){\Omega }_{\omega }⟩=⟨[1],[a]⟩=\omega (1^{\ast }a)=\omega (a)$. $\square$

We have proved:

Theorem 7.1 (GNS). [Proved — see §7.1 construction above.] For any state $\omega$ on a unital C*-algebra $\mathcal{A}$, there exists a Hilbert space ${\mathcal{H}}_{\omega }$, a *-representation ${\pi }_{\omega }:\mathcal{A}\to \mathcal{B}({\mathcal{H}}_{\omega })$, and a cyclic unit vector ${\Omega }_{\omega }\in {\mathcal{H}}_{\omega }$ such that $\omega (a)=⟨{\Omega }_{\omega },{\pi }_{\omega }(a){\Omega }_{\omega }⟩$ for all $a$.

7.3 Uniqueness

Theorem 7.2 (Uniqueness of GNS). [Proved.] Suppose $(\pi ,\mathcal{H},\Omega )$ is any *-representation with cyclic unit vector $\Omega$ such that $\omega (a)=⟨\Omega ,\pi (a)\Omega ⟩$. Then there is a unique unitary $U:{\mathcal{H}}_{\omega }\to \mathcal{H}$ such that $U{\Omega }_{\omega }=\Omega$ and $U{\pi }_{\omega }(a)U^{-1}=\pi (a)$ for all $a$.

Proof. Define $U$ on the dense subspace ${\pi }_{\omega }(\mathcal{A}){\Omega }_{\omega }$ by $U{\pi }_{\omega }(a){\Omega }_{\omega }:=\pi (a)\Omega$. To show this is well-defined and isometric, compute:

$$⟨\pi (a)\Omega ,\pi (b)\Omega {⟩}_{\mathcal{H}}=⟨\Omega ,\pi (a^{\ast }b)\Omega ⟩=\omega (a^{\ast }b)=⟨{\Omega }_{\omega },{\pi }_{\omega }(a^{\ast }b){\Omega }_{\omega }⟩=⟨{\pi }_{\omega }(a){\Omega }_{\omega },{\pi }_{\omega }(b){\Omega }_{\omega }⟩.$$

So $U$ extends to an isometry ${\mathcal{H}}_{\omega }\to \mathcal{H}$. Surjectivity: $U({\pi }_{\omega }(\mathcal{A}){\Omega }_{\omega })=\pi (\mathcal{A})\Omega$, which is dense in $\mathcal{H}$ by cyclicity of $\Omega$. The intertwining property is direct. $\square$

This uniqueness justifies calling $({\pi }_{\omega },{\mathcal{H}}_{\omega },{\Omega }_{\omega })$ “the” GNS representation.

Remark 7.2a (faithfulness of the representation). The GNS representation need not be injective: $\ker {\pi }_{\omega }=\{a:\omega (b^{\ast }{a}^{\ast }ab)=0\forall b\}$ can be a nonzero two-sided ideal. If $\omega$ is faithful ($\omega (a^{\ast }a)=0\Rightarrow a=0$), then ${\pi }_{\omega }$ is injective, and since an injective *-homomorphism of C*-algebras is automatically isometric (a consequence of Corollary 4.4: both norms compute the same spectral radius), ${\pi }_{\omega }$ embeds $\mathcal{A}$ isometrically into $\mathcal{B}({\mathcal{H}}_{\omega })$. This is the mechanism behind the Gelfand–Naimark Theorem 5.2: take a separating family of states $\{{\omega }_i\}$ (one exists by Hahn–Banach), form ${⨁}_i{\pi }_{{\omega }_i}$, and the direct sum is injective, hence isometric. The abstract-versus-concrete distinction for C*-algebras thus evaporates — which is why we will move freely between the two pictures all course.

7.4 Pure states and irreducible representations

Theorem 7.3. [Stated only — refs: Bratteli–Robinson Vol. I §2.3.8; proof uses Schur’s lemma in vN-algebra form.] The GNS representation ${\pi }_{\omega }$ is irreducible if and only if $\omega$ is pure.

(A representation is irreducible if it has no closed invariant subspaces other than $\{0\}$ and the whole space.) The proof uses Schur’s lemma in von Neumann algebra form; we will not give it here.

This is a clean structural correspondence: pure states ↔ irreducible representations. In the commutative case (Example 6.4), pure states are points $x\in X$, and the corresponding irreducible representations are 1-dimensional, given by ${\pi }_x(f)=f(x)\in \mathbb{C}$. In the non-commutative case, irreducible representations can be arbitrarily large.

8. Worked examples of GNS

8.1 Vector states on $\mathcal{B}(\mathcal{H})$

Let $\mathcal{A}=\mathcal{B}(\mathcal{H})$ with $\mathcal{H}$ a Hilbert space, and let ${\omega }_{\xi }(a)=⟨\xi ,a\xi ⟩$ for a unit vector $\xi \in \mathcal{H}$.

Identifying the GNS data: take ${\mathcal{H}}_{\omega }=\mathcal{H}$, ${\pi }_{\omega }=\mathrm{id}$, ${\Omega }_{\omega }=\xi$. The cyclicity of $\xi$ is automatic when $\xi$ is a cyclic vector for $\mathcal{B}(\mathcal{H})$, which it is whenever $\mathcal{H}$ is the closed span of $\mathcal{B}(\mathcal{H})\xi =\mathcal{H}$ — trivially true. So GNS recovers the original Hilbert space.

If $\xi$ is not cyclic for $\mathcal{B}(\mathcal{H})$ (impossible for $\mathcal{B}(\mathcal{H})$ acting on the whole space, but possible for sub-algebras), then ${\mathcal{H}}_{\omega }$ is the cyclic subspace $\overline{\pi (\mathcal{A})\xi }$.

8.2 The tracial state on $M_2(\mathbb{C})$

Let $\mathcal{A}=M_2(\mathbb{C})$ and $\tau (a)=\frac{1}{2}\mathrm{Tr}(a)$. We construct GNS explicitly.

Sesquilinear form. For $a,b\in M_2(\mathbb{C})$:

$$⟨a,b{⟩}_{\tau }=\tau (a^{\ast }b)=\frac{1}{2}\mathrm{Tr}(a^{\ast }b).$$

This is the (rescaled) Hilbert–Schmidt inner product.

Null space. $\tau (a^{\ast }a)=\frac{1}{2}{\sum }_{i,j}|a_{ij}{|}^2=0$ implies $a=0$. Hence ${\mathcal{N}}_{\tau }=\{0\}$.

Hilbert space. ${\mathcal{H}}_{\tau }=M_2(\mathbb{C})$ with the inner product $⟨a,b{⟩}_{\tau }=\frac{1}{2}\mathrm{Tr}(a^{\ast }b)$. As a Hilbert space, ${\mathcal{H}}_{\tau }\cong {\mathbb{C}}^4$.

Representation. ${\pi }_{\tau }(a)$ acts on ${\mathcal{H}}_{\tau }$ by left multiplication. As an operator on ${\mathbb{C}}^4\cong {\mathbb{C}}^2\otimes {\mathbb{C}}^2$, it acts as $a\otimes 1$.

Cyclic vector. ${\Omega }_{\tau }=1\in M_2(\mathbb{C})$. Identifying $1$ with $\frac{1}{\sqrt{2}}(|0⟩\otimes |0⟩+|1⟩\otimes |1⟩)\in {\mathbb{C}}^2\otimes {\mathbb{C}}^2$, this is exactly the maximally entangled Bell state on a doubled system.

The GNS representation of the tracial state is “left multiplication” on the algebra, regarded as a Hilbert space via Hilbert–Schmidt; the cyclic vector is the maximally entangled state on the doubled Hilbert space. This will reappear in Week 6 as the prototype for KMS states and modular theory: ${\Omega }_{\tau }$ is both cyclic and separating for ${\pi }_{\tau }(M_2(\mathbb{C}))$, and the modular operator turns out to be the identity (the state is tracial). The tracial structure is the toy model for type $\mathrm{II}$ algebras.

8.3 A point-evaluation state on $C(X)$

Let $\mathcal{A}=C(X)$ for $X$ compact Hausdorff, and pick $x_0\in X$. The state $\omega ={\delta }_{x_0}$ is $\omega (f)=f(x_0)$.

Null space. $\omega (f^{\ast }f)=|f(x_0){|}^2=0$ iff $f(x_0)=0$. So ${\mathcal{N}}_{\omega }=\{f:f(x_0)=0\}$ — a maximal ideal of codimension 1 in $C(X)$.

Quotient. $C(X)/{\mathcal{N}}_{\omega }\cong \mathbb{C}$ via $f\mapsto f(x_0)$.

Hilbert space. ${\mathcal{H}}_{\omega }=\mathbb{C}$.

Representation. ${\pi }_{\omega }(f)=f(x_0)\cdot 1$ on $\mathbb{C}$. 1-dimensional, irreducible. (Consistent with Theorem 7.3: ${\delta }_{x_0}$ is a pure state.)

Cyclic vector. ${\Omega }_{\omega }=1\in \mathbb{C}$.

The whole representation collapses to the evaluation map $f\mapsto f(x_0)$. This illustrates how, in commutative algebras, GNS tends to be “degenerate” — irreducible representations are 1-dimensional. The interesting examples are non-commutative.

8.4 A non-pure state on $M_2(\mathbb{C})$ (preview of type $\mathrm{II}$)

Recall the Pauli matrices ${\sigma }_x,{\sigma }_y,{\sigma }_z$. Consider the state $\omega (a)=\frac{1}{2}⟨e_1,a{e}_1⟩+\frac{1}{2}⟨e_2,a{e}_2⟩=\frac{1}{2}\mathrm{Tr}(a)=\tau (a)$. We already constructed this in §8.2.

Note that this state is not pure — it is the average of the two pure vector states ${\omega }_{e_1}$ and ${\omega }_{e_2}$, where $e_1,e_2$ are the standard basis vectors. Yet by Theorem 7.3 the corresponding GNS representation is reducible. Indeed, the Hilbert space ${\mathcal{H}}_{\tau }\cong {\mathbb{C}}^4$ decomposes under ${\pi }_{\tau }$ as ${\mathbb{C}}^2\oplus {\mathbb{C}}^2$ (left multiplication by $a$ acts on each column independently), with ${\pi }_{\tau }(a)$ acting as $a$ on each ${\mathbb{C}}^2$.

This decomposition is a useful finite-dimensional preview of the standard representation of a finite algebra: left multiplication has a nontrivial right-multiplication commutant, and the trace vector is cyclic and separating. It is not itself type II—the represented algebra is still $M_2(\mathbb{C})$, hence type I${}_2$. Week 3 will show which additional infinite-dimensional feature, namely the absence of minimal projections, distinguishes a genuine type II${}_1$ factor.

9. What to take away

Three structural facts run the rest of the course:

1. The algebraic framework. Observables form a C*-algebra; states are positive normalized linear functionals; everything is defined without picking a Hilbert space first.

2. GNS is the bridge. Any state on any C*-algebra produces a canonical Hilbert space representation. This is how we move between the algebraic and the Hilbert-space pictures whenever convenient.

3. The non-commutativity is the physics. In commutative algebras (classical probability), pure states are points; states are probability measures; everything is determined by an underlying space. In non-commutative algebras (quantum), pure states are vectors in a representation; the representation depends on the state (different states can give non-equivalent GNS representations). This is the source of all subtlety in QFT, including the type-III phenomena we will meet in Block C.

10. Looking ahead: Week 2

Next week we strengthen “C*-algebra” to “von Neumann algebra” by closing in the weak operator topology rather than the norm topology. That closure is taken inside a chosen representation, so the definition we give is concrete. Nothing is lost by this choice: Sakai’s theorem (Week 2 §6) characterizes von Neumann algebras abstractly, as the C*-algebras that are dual Banach spaces, and the σ-weak topology and the normal states are intrinsic to the algebra. The benefit is a much richer structure: every vN algebra has a wealth of projections that generate it, and the order structure on projections leads to the Murray–von Neumann classification (Week 3). Tomita–Takesaki (Block B) and the type-III classification of QFT algebras (Block C) all live in the von Neumann setting.

11. Problem set

Core problems (everyone).

1. Banach-algebra warm-ups. (a) Show that in any Banach algebra, the spectrum of $a$ is contained in the closed disk of radius $\Vert a\Vert$ centered at $0$. Where does completeness enter? (b) Show that for $a\in M_n(\mathbb{C})$, the spectrum coincides with the set of eigenvalues, and the spectral radius $r(a)$ equals $\max \{|\lambda |:\lambda \text{ eigenvalue}\}$.

2. Unitization (a hands-on encounter with the non-unital case). Let $\mathcal{A}$ be a non-unital Banach *-algebra. Define $\tilde{\mathcal{A}}=\mathcal{A}\oplus \mathbb{C}$ as a vector space, with multiplication $(a,\lambda )(b,\mu )=(ab+\lambda b+\mu a,\lambda \mu )$ and involution $(a,\lambda {)}^{\ast }=(a^{\ast },\overline{\lambda })$. (a) Verify that $\tilde{\mathcal{A}}$ is a unital *-algebra and embed $\mathcal{A}$ into it as $a\mapsto (a,0)$. (b) Show that with the norm $\Vert (a,\lambda )\Vert =\Vert a\Vert +|\lambda |$, $\tilde{\mathcal{A}}$ is a unital Banach *-algebra. (c) (Harder.) When $\mathcal{A}$ is a C*-algebra, the norm in (b) does not give $\tilde{\mathcal{A}}$ the C*-property in general. Construct the minimal-unitization norm by representing $(a,\lambda )$ as the adjointable left multiplier $b\mapsto ab+\lambda b$ of the standard Hilbert $\mathcal{A}$-module $\mathcal{A}$. Explain why merely writing $\mathcal{B}(\mathcal{A})$ for bounded Banach-space operators would not supply the needed adjoint operation.

3. The C*-identity has teeth. (a) Show that in a C*-algebra, $\Vert a{\Vert }^2=\Vert a^{\ast }a\Vert =\Vert a{a}^{\ast }\Vert$. (b) Deduce that if $a$ is normal, then $\Vert a^n\Vert =\Vert a{\Vert }^n$ and hence $r(a)=\Vert a\Vert$. (c) Use (b) to show that the spectrum of a unitary lies in the unit circle.

4. GNS for the tracial state, completed. (a) Verify all the steps in §8.2: that ${\mathcal{N}}_{\tau }=0$, that the Hilbert space is the rescaled Hilbert–Schmidt space, that left multiplication is bounded with $\Vert {\pi }_{\tau }(a)\Vert =\Vert a\Vert$ (operator norm). (b) Show that the cyclic vector ${\Omega }_{\tau }$ is also cyclic for the commutant ${\pi }_{\tau }(M_2(\mathbb{C}){)}^{\prime }$ — i.e., it is cyclic and separating. (The commutant ${\pi }_{\tau }(M_2(\mathbb{C}){)}^{\prime }$ acts on ${\mathcal{H}}_{\tau }$ by right multiplication.) (c) Compute the modular conjugation $J:{\mathcal{H}}_{\tau }\to {\mathcal{H}}_{\tau }$ defined by $J[a]=[a^{\ast }]$. Verify that $J$ is anti-unitary and $J^2=1$. If $L_{a}X=aX$ and $R_{a}X=Xa$, show carefully that

$$J{L}_{a}J=R_{a^{\ast }}.$$

Thus $J{\pi }_{\tau }(M_2)J={\pi }_{\tau }(M_2{)}^{\prime }$: modular conjugation exchanges left and right multiplication. (This previews Tomita–Takesaki in the simplest case, where the modular operator is trivial.)

Starred problems (Ph.D. expected; ambitious M.Sc. encouraged).

5*. Vector states are pure—and where normality enters. Let $\xi \in \mathcal{H}$ be a unit vector and ${\omega }_{\xi }(a)=⟨\xi ,a\xi ⟩$ on $\mathcal{B}(\mathcal{H})$. Show that ${\omega }_{\xi }$ is pure without assuming that arbitrary states on $\mathcal{B}(\mathcal{H})$ are trace-class. (Hint: if ${\omega }_{\xi }=t{\omega }_1+(1-t){\omega }_2$, then for the projection $q=1-|\xi ⟩⟨\xi |$ positivity gives ${\omega }_1(q)={\omega }_2(q)=0$. Use Cauchy–Schwarz for positive functionals to show ${\omega }_i(qa)={\omega }_i(aq)=0$ for every $a$, and conclude that each ${\omega }_i$ is supported on the one-dimensional corner $(1-q)\mathcal{B}(\mathcal{H})(1-q)$.) Then explain why the converse must be stated as “every normal pure state is a vector state” when $\mathcal{H}$ is infinite-dimensional.

6*. A non-pure state and its GNS decomposition. Take $\omega =\frac{1}{2}{\omega }_{e_1}+\frac{1}{2}{\omega }_{e_2}$ on $M_2(\mathbb{C})$ as in §8.4. (a) Compute the GNS representation of $\omega$ explicitly and identify the decomposition into two copies of the defining representation. (b) Find the projection in ${\pi }_{\omega }(M_2(\mathbb{C}){)}^{\prime }$ (the commutant) onto the first copy. (c) Discuss: how does this 2-fold decomposition reflect that $\omega$ is the equal-weights mixture of two pure states? Generalize to $\omega ={\sum }_i{p}_i{\omega }_{e_i}$ with $p_i>0$ and ${\sum }_i{p}_i=1$; if zero weights are allowed, explain why their summands disappear from the GNS space.

7** (Optional, very hard). Prove that every state on a C*-algebra is bounded with norm 1 (i.e., ${\sup }_{\Vert a\Vert \le 1}|\omega (a)|=1$), using only the positivity and normalization axioms. (Hint: the key step is to control $|\omega (a){|}^2\le \omega (a^{\ast }a)$ via Cauchy–Schwarz, then iterate using $a^{\ast }a$ self-adjoint and $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$.)

Self-study answer checkpoints

These are result checks, not substitute solutions. The starred problems remain extension work: use the cited theory, the supplied hints, and instructor feedback to control the harder steps.

1. Banach-algebra warm-ups. For $|\lambda |>\Vert a\Vert$, the inverse of $\lambda 1-a$ is the norm-convergent Neumann series ${\sum }_{n\ge 0}{\lambda }^{-n-1}{a}^n$; completeness is what puts its sum back in the algebra. In $M_n(\mathbb{C})$, non-invertibility is equivalent to $\det (a-\lambda 1)=0$, so the spectrum is the eigenvalue set and $r(a)$ is the largest eigenvalue modulus.

2. Unitization. The unit is $(0,1)$, and direct expansion verifies multiplication and the involution. The ${\ell }^1$ norm is submultiplicative and complete. For a non-unital C*-algebra, the C*-unitization norm is

   $$\Vert (a,\lambda ){\Vert }_{\tilde{\mathcal{A}}}=\Vert L_a+\lambda I{\Vert }_{\mathcal{L}({\mathcal{A}}_{\mathcal{A}})},$$

   where $(L_a+\lambda I{)}^{\ast }=L_{a^{\ast }}+\overline{\lambda }I$ on the standard Hilbert $\mathcal{A}$-module. This adjoint is the ingredient absent from an arbitrary Banach-operator norm.

3. The C*-identity. Applying the identity to $a$ and $a^{\ast }$ gives $\Vert a^{\ast }a\Vert =\Vert a{a}^{\ast }\Vert =\Vert a{\Vert }^2$. For normal $a$, repeated use of commutativity of $a$ and $a^{\ast }$ gives $\Vert a^n\Vert =\Vert a{\Vert }^n$, hence $r(a)=\Vert a\Vert$. For unitary $u$, $\Vert u\Vert =\Vert u^{-1}\Vert =1$, so spectral mapping for inversion forces $|\lambda |=1$ on $\sigma (u)$.

4. Tracial GNS model. Faithfulness of $\tau$ gives ${\mathcal{N}}_{\tau }=0$; the completion is $M_2$ with the rescaled Hilbert–Schmidt norm, and $\Vert L_a\Vert =\Vert a\Vert$. The commutant is the right-multiplication algebra and both left and right orbits of $1$ are all of $M_2$. Finally,

   $$JX=X^{\ast },J^2=1,J{L}_{a}J=R_{a^{\ast }},$$

   with $J$ antiunitary.

Wiki connections. Tomita–Takesaki modular theory · type III₁ von Neumann algebras

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Notes prepared for the algebraic-QFT course, Semester I Block A. Last revised 2026-09-29.