Observable algebra

Let $\mathcal{H}$ be the system Hilbert space (see finite-dimensional complex Hilbert space ). An observable algebra is a unital $*$-algebra $\mathcal{A}$ of operators acting on $\mathcal{H}$, meaning:

1. $\mathcal{A}$ is closed under addition and scalar multiplication.
2. If $A,B\in\mathcal{A}$ then $AB\in\mathcal{A}$ (closure under multiplication).
3. If $A\in\mathcal{A}$ then $A^\dagger \in \mathcal{A}$ (closure under adjoint).
4. The identity operator $I$ belongs to $\mathcal{A}$.

In the standard finite-dimensional setting one takes $\mathcal{A} = \mathcal{B}(\mathcal{H})$, the algebra of all bounded operators on $\mathcal{H}$ (equivalently, all complex matrices in a chosen basis; see matrix ).

An observable is a self-adjoint element $A \in \mathcal{A}$; its spectral projections encode measurement outcomes.

A state on $\mathcal{A}$ can be defined abstractly as a linear map $\omega:\mathcal{A}\to\mathbb{C}$ that is positive ($\omega(A^\dagger A)\ge 0$) and normalized ($\omega(I)=1$). In finite dimensions, every such state is represented by a density-operator state $\rho$ via

using the trace .

Physical interpretation

• $\mathcal{A}$ is the mathematical container for “what you can measure” and “how measurements combine.”
• Noncommutativity ($AB\neq BA$) encodes incompatible observables and the impossibility of assigning sharp values to all observables simultaneously.
• Commutative subalgebras behave like classical algebras of random variables, describing compatible families of measurements.

Key properties

• $*$-structure and reality: Self-adjoint elements correspond to real-valued measurement outcomes.
• Positivity: Operators of the form $A^\dagger A$ are positive; states must assign them nonnegative expectation values.
• Expectation values: The operational content of the algebra is summarized by the expectation functional $A\mapsto \omega(A)$.
• Subsystem structure: For composite systems, observables typically include tensor-product forms; reduced descriptions use the partial trace to produce subsystem states.