Quasistate

Quasistate, in the context of operator algebras, refers to a generalization of the notion of a state on a C-algebra. Let A be a unital C-algebra. A quasistate on A is a map φ: A → C that is positive (φ(a a) ≥ 0 for all a ∈ A), normalized (φ(1) = 1), and linear on every abelian (commutative) C-subalgebra of A. Equivalently, φ is additive and homogeneous on any subalgebra in which all elements commute with each other. A quasistate may fail to be linear on the whole algebra if noncommutativity is involved.

Relationship to states and commutativity is straightforward: every state is a quasistate, since a state is a linear positive functional on all of A with φ(1) = 1. Conversely, if φ is linear on all of A, then φ is a state. When A is commutative, any quasistate is automatically a state, because the whole algebra is abelian and linearity on the subalgebras covers A itself.

Properties and significance: Quasistates generalize the probabilistic viewpoint of states to settings where noncommutativity prevents a single linear functional from describing all measurements uniformly. They may be non-linear on A as a whole, illustrating the difference between local (commuting) additivity and global linearity. In many standard operator-algebraic contexts, additional regularity conditions (such as boundedness or sigma-additivity on projections) can force a quasistate to be linear, thereby reducing it to a genuine state. Quasistates play a role in discussions of noncommutative probability, Gleason-type theorems, and foundational questions about quantum measurements.

See also: state (functional), C-algebra, Gleason's theorem, noncommutative probability, quasi-probability.