1978/01/01 by S. P. Gudder, Stanley Gudder, R. L. Hudson · 1 citation
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Quantum Mechanics and Applications #Mathematical and Theoretical Analysis
paper · doi:10.1090/s0002-9947-1978-0511398-0
A noncommutative probability theory is developed in which no boundedness, finiteness, or “tracial” conditions are imposed. The underlying structure of the theory is a “probability algebra” <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script a comma omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">a</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathcal a,ω )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script a"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a *-algebra and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">ω</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a faithful state on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script a"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.