C*- and von Neumann Algebras

Instead of starting from a Hilbert space of states, one can start from the algebra of observables and recover the states as functionals on it. This algebraic viewpoint — C*-algebras and von Neumann algebras — is the natural language for systems with infinitely many degrees of freedom, where inequivalent representations proliferate: quantum statistical mechanics, and the Wightman / Osterwalder–Schrader axiomatizations of QFT. This page builds on bounded-operators.md.

C*-algebras

A C*-algebra $\mathcal{A}$ is a complex algebra with an involution $a\mapsto a^{\ast }$ and a norm, complete, satisfying the C*-identity $$\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2.$$ The motivating example is $\mathcal{B}(\mathcal{H})$ (bounded operators, with adjoint as involution — the C*-identity was noted in bounded-operators.md), and any norm-closed $\ast$-subalgebra. The Gelfand–Naimark theorem says these are all of them: every C*-algebra is isometrically a $\ast$-algebra of operators on some Hilbert space. Commutative C*-algebras are exactly $C_0(X)$, continuous functions on a locally compact space — so a commutative C*-algebra "is" a topological space, and non-commutative C*-algebras are "non-commutative topology", the observables of a quantum system.

States and the GNS construction

A state on $\mathcal{A}$ is a positive, normalized linear functional $\omega :\mathcal{A}\to \mathbb{C}$ ($\omega (a^{\ast }a)\ge 0$, $\omega (1)=1$) — the algebraic notion of an expectation value $⟨a⟩$. The bridge back to Hilbert space is:

GNS construction. Every state $\omega$ on a C*-algebra gives a Hilbert space ${\mathcal{H}}_{\omega }$, a representation ${\pi }_{\omega }:\mathcal{A}\to >\mathcal{B}({\mathcal{H}}_{\omega })$, and a cyclic vector $|\Omega ⟩$ with $\omega (a)=⟨\Omega |{\pi }_{\omega }(a)|\Omega ⟩$.

So states and representations are two views of the same data: fix the observables algebraically, and each expectation-value assignment builds a Hilbert-space representation with a distinguished vacuum. This is precisely how a QFT vacuum $\omega$ generates its Fock space, and why different vacua (e.g. inequivalent phases, or accelerated observers — the Unruh effect) give unitarily inequivalent representations.

von Neumann algebras

A von Neumann algebra is a $\ast$-subalgebra of $\mathcal{B}(\mathcal{H})$ closed in the weak operator topology — equivalently (von Neumann's bicommutant theorem) equal to its own double commutant $\mathcal{M}={\mathcal{M}}^{\prime \prime }$. They carry more structure than C*-algebras (they contain the spectral projections of their self-adjoint elements, from spectral-theorem.md) and are classified into types I, II, III by their projections. The type is physical: ordinary QM lives in type I; local algebras of relativistic QFT and thermal systems are type III — a structural fact behind the absence of a particle-number operator for local regions and the entanglement structure of the QFT vacuum.

The algebraic axioms of QFT

The algebraic viewpoint makes the axiomatizations of QFT precise:

• Wightman axioms — a QFT is a set of operator-valued tempered distributions (distributions.md) on a Hilbert space with a Poincaré- invariant vacuum, satisfying locality (spacelike-separated fields commute) and spectral positivity. This is the framework in which the QFT postulates are stated rigorously.
• Osterwalder–Schrader axioms — the Euclidean counterpart (after Wick rotation), with reflection positivity the condition guaranteeing a physical Hilbert space can be reconstructed.
• Haag–Kastler / algebraic QFT — assign a von Neumann algebra to each spacetime region; observables, not states, are primary.

These are the mathematically complete statements of "what a quantum field theory is", and the reason distribution theory and operator algebras are the true foundations of the subject.

References

• Bratteli & Robinson, Operator Algebras and Quantum Statistical Mechanics, Vols. 1–2.
• Haag, Local Quantum Physics.
• Streater & Wightman, PCT, Spin and Statistics, and All That.