Observables as Algebras

Specifying an observable algebra requires more than listing Hermitian operators. One must decide which products, limits, and spectral events belong to the model. The resulting algebra records the questions its states can answer. This page explains that construction for bounded generators and shows how unbounded observables enter through represented spectral data.

Required background. C*-Algebras supplies norm closure and continuous functional calculus. Projection-Valued Measures supplies the events associated with a sharp observable.

Self-adjoint observables generate a larger complex algebra

Given a bounded family $\mathcal S\subset\mathcal B(\mathcal H)$, define $C^*(I,\mathcal S)$ as the smallest norm-closed unital star subalgebra containing it. Equivalently, take finite linear combinations of words in the generators and their adjoints, then complete in norm.

Even if every generator is self-adjoint, products need not be. The self-adjoint elements form a real vector space, but generally not an associative algebra. For self-adjoint $A,B$,

$$AB=\frac12(AB+BA)+\frac12(AB-BA).$$

The first term is self-adjoint and the second is anti-self-adjoint. The complex algebra retains both the symmetric product and the commutator information.

Algebra multiplication should not be read as a universal rule for performing two successive measurements. Sequential probabilities depend on an instrument or state-update map, not only on the effects being multiplied. Instruments supplies that additional measurement structure.

Two binary measurements can generate every qubit operator

Take the projections

$$P=\frac{I+\sigma_z}{2},\qquad Q=\frac{I+\sigma_x}{2}.$$

From $P,Q,I$ one recovers $\sigma_z$ and $\sigma_x$, and their product gives $\sigma_z\sigma_x=i\sigma_y$. Hence

$$C^*(I,P,Q)=M_2(\mathbb C).$$

In particular, a state on the generated algebra assigns an expectation to $\sigma_y$. Knowing only the two numbers $\omega(P)$ and $\omega(Q)$ does not yet specify that state: their products carry further information.

For example, the two states polarized along opposite $y$ directions both assign probability $1/2$ to $P$ and $Q$, while assigning opposite expectations to $\sigma_y$. There is no contradiction. A generating set determines an algebra through its multiplication rules, whereas a linear functional is not multiplicative and is not determined by its values on algebra generators alone.

For commuting projections the situation changes. Their generated algebra is commutative and contains their joint Boolean events, such as $PQ$ and $P(I-Q)$. Marginal probabilities still need not fix the joint distribution. Commutativity alone does not supply statistical independence.

Unbounded observables require bounded encodings

A self-adjoint Hamiltonian or position operator usually lies outside every C*-algebra of bounded operators. One can instead specify bounded resolvents, spectral projections, or suitable bounded functions. These choices need not generate the same C*-algebra.

For a self-adjoint $T$, the resolvent $(T-iI)^{-1}$ is bounded and normal. Functional calculus identifies the nonunital algebra generated appropriately by resolvent functions with $C_0(\sigma(T))$; adjoining $I$ adds constants. Its represented von Neumann closure contains the spectral projections of $T$.

The distinction is visible for position on the line. Continuous functions vanishing at infinity describe bounded smooth approximations to position tests. A sharp interval indicator is measurable but discontinuous, so it generally enters only after passing to the represented measurable calculus.

Thus saying that an unbounded observable is affiliated with a von Neumann algebra identifies all its spectral events with elements of that algebra. It does not assert that the unbounded operator itself is bounded or that every state has a finite expectation of it.

States evaluate effects, while normality controls infinite sums

An effect is an algebra element $e$ with $0\leq e\leq1$. A state gives a probability $0\leq\omega(e)\leq1$. A finite family of effects summing to $1$ has probabilities summing to one by linearity.

For a countable measurement whose effect sum converges strongly in a von Neumann algebra, countable normalization requires normality of the state. Mere norm continuity does not pass through that strong limit. This is the same distinction made precise on Normal States.

The algebra, state class, and measurement structure therefore contribute separate pieces: available bounded quantities, expectation assignments with specified continuity, and physical outcome/update rules. Declaring a large operator algebra alone does not settle which laboratory operations can be implemented.

Exercise: generators versus informational completeness

For an arbitrary qubit state $\rho=(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z)/2$, determine the information obtained from the probabilities of $P$ and $Q$ above. Why does their generating the full algebra not make those two probabilities informationally complete?

Solution

They give $\omega(P)=(1+r_z)/2$ and $\omega(Q)=(1+r_x)/2$. The component $r_y$ remains undetermined, except for the positivity constraint $r_x^2+r_y^2+r_z^2\leq1$. Although products generate the missing algebra direction, no rule reconstructs their expectations by multiplying the measured probabilities. Informational completeness requires a set whose linear span determines every state expectation, as explained in Informational Completeness.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987, doi:10.1007/978-3-662-02520-8 — generated algebras and bounded functional calculus.
• Naaijkens, Pieter, Quantum Spin Systems on Infinite Lattices, Lecture Notes in Physics 933, Springer, 2017, arXiv:1311.2717 — observable algebras and state restrictions.