Operator Algebra Minimum

Operator-algebraic quantum mechanics starts from an algebra of observables and then asks how it is represented and which states assign its expectation values. The preparation needed is modest but precise: know what multiplication and adjoint do, what positivity means, and why a state is relative to the chosen algebra. The construction theorems belong to Operator-Algebraic QM; this page prepares the reader to recognize their inputs.

Helpful background. Bounded Operators supplies operator norms and adjoints. Density Operators supplies the finite-dimensional trace pairing with observables.

Algebras retain products and adjoints

A complex unital star algebra has addition, scalar multiplication, associative multiplication, an identity, and an adjoint operation satisfying

$$\begin{gathered} (AB)^*=B^*A^*,\\ (\lambda A)^*=\overline\lambda A^*,\qquad (A^*)^*=A. \end{gathered}$$

The order reversal is consequential: observable algebras are generally noncommutative. Although sharp bounded observables are self-adjoint elements, the full algebra includes non-self-adjoint elements so that it is closed under multiplication and complex linear combinations.

A unital C*-algebra is such an algebra, complete in a submultiplicative norm, with

$$\|A^*A\|=\|A\|^2.$$

The bounded operators $B(\mathcal H)$ provide the basic example, as do norm-closed star subalgebras containing the identity. An arbitrary unbounded Hamiltonian is not an element of $B(\mathcal H)$. Its bounded resolvents, spectral projections, or unitary exponentials may instead be represented in a suitable algebra. An assertion about which of these belongs to a smaller observable algebra is additional information.

Positivity connects the algebra to probability

In a C*-algebra, an element is positive when it is of the form $B^*B$, equivalently when it is self-adjoint with nonnegative spectrum. A state on a unital C*-algebra is a linear functional $\omega$ satisfying

$$\omega(B^*B)\ge0,\qquad\omega(I)=1.$$

This definition is formulated without selecting a Hilbert-space representation. It implies continuity and $|\omega(A)|\le\|A\|$. One key estimate is the Cauchy–Schwarz inequality

$$|\omega(B^*A)|^2 \le\omega(A^*A)\,\omega(B^*B).$$

To see its origin, expand $\omega((A+zB)^*(A+zB))\ge0$ and minimize the resulting quadratic expression in the complex number $z$. If $\omega(B^*B)=0$, varying $z$ forces the cross term to vanish. This is the same positivity mechanism as for an inner product, even though some nonzero algebra elements can have zero state-dependent length.

That degeneracy is the starting point of the GNS construction: one quotients the zero-length elements and completes the resulting inner-product space. This observation explains the construction’s purpose; its quotient and representation details belong to the GNS Construction.

The chosen observable algebra changes what states can distinguish

Consider a qubit with density matrix

$$\rho=\begin{pmatrix}p&c\\\overline c&1-p\end{pmatrix}, \qquad |c|^2\le p(1-p).$$

On the full algebra $M_2(\mathbb C)$, $\omega(A)=\operatorname{Tr}(\rho A)$ determines both the populations and the coherence. Restrict the observable algebra to diagonal matrices. For $D=\operatorname{diag}(a,b)$,

$$\omega(D)=pa+(1-p)b,$$

which does not depend on $c$. Distinct full-algebra states therefore restrict to the same diagonal-algebra state. In particular, the pure vector $(1,1)/\sqrt2$ restricts to the equal mixture of the two diagonal evaluation states. Purity is relative to an observable algebra; restriction need not preserve it.

This example is a model of limited accessible observables. It does not by itself establish a fundamental superselection rule. Such a physical claim requires an argument for why the off-diagonal observables are excluded.

Representations and continuity add separate data

A representation $\pi$ maps the abstract algebra to bounded operators on a Hilbert space while preserving products and adjoints; here unital representations also satisfy $\pi(I)=I$. A faithful representation has zero kernel. An irreducible representation has no nontrivial closed subspace invariant under every represented element and its adjoint. Faithfulness and irreducibility are different conditions.

A von Neumann algebra is a unital star subalgebra of $B(\mathcal H)$ closed in the weak operator topology. Normal states preserve suprema of increasing bounded nets of positive elements. On $B(\mathcal H)$, the normal states are exactly the positive trace-class density-operator states. General states on this infinite-dimensional algebra need not be normal and need not have a density operator. Thus “state equals density matrix” needs a representation and a normality qualification.

The convergence primer gives the norm, strong, and weak tests. Remember that norm closure and weak closure are different constructions, even when the generating matrices or operators have the same symbols. Bratteli and Robinson develop the state, representation, and normality theorems summarized here.

Exercise: detect a coherence outside the smaller algebra

For the qubit above, compute the expectation of $\sigma_x$ and explain what happens when the same state is restricted to the diagonal algebra.

Solution

$\operatorname{Tr}(\rho\sigma_x)=c+\overline c=2\operatorname{Re}c$. The restricted functional contains no value for $\sigma_x$ because $\sigma_x$ is not in the diagonal algebra. This is absence of an observable from the specification, not a prediction that its expectation is zero. Multiple full-algebra extensions can agree on every diagonal observable and disagree on $\sigma_x$.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1: C- and W*-Algebras, Symmetry Groups, Decomposition of States*, 2nd ed., Springer, 1987, doi:10.1007/978-3-662-02520-8 — positive functionals, representations, and normal states.
• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — bounded operators, states, and subsystem restrictions.
• Teschl, Gerald, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, author’s edition — the C*-algebra of bounded operators and weak and strong topologies.