Conditional expectation in an operator algebra. IV. Entropy and information

Introduction. The theory of information, created by Shannon 23, is developed by Feinstein, Kullback, MacMillan, Wiener and other American statisticians (e. g. , cf. 10), and also advanced into the ergodic theory by Gelfand, Khinchin, Kolmogorov, Yaglom and other Russian probabilists (e. g. , cf. 8). Through recent years, the theory is regarded as a new chapter in the theory of probability. Recently, Segal 22 gave a mathematical formulation of the entropy of state of a von Neumann algebra, which contains both the cases for the theory of information and the theory of quantum statistics. Segal's theorem was reformulated in operator algebraic form by Nakamura and Umegaki 16 and independently by Davis 3. Since the summer in 1954, Nakamura and Umegaki have investigated the concept of the conditional expectation in von Neumann algebra as a noncommutative extension of probability theory (cf. for example 13~18 and 25 ~28), and in the most recent paper 18 it was applied to the theory of measurements of quantum statistics which is regarded as a non-commutative case of the theory of entropy and information. Furthermore, it may be very intersesting to develope the theory of information under functional-analysistic and operator-theoretic methods. From these points of views, we shall discuss the measure of information of integrable operators or of normal states of a von Neumann algebra. Davis 3 has independently studied on the almost same theme with Nakamura-Umegaki 16 and 18, in which he developed the theory of entropy and he simplified the proof of the theorem relative to the operatorentropy. Now, we shall give the basic notations and describe the fundamental concepts in a von Neumann algebra which will be used throughout the present paper. Let A be a von Neumann algebra, that is, A is a weakly closed self-adjoint algebra of bounded operators acting over a complex Hubert space H, which contains the identity operator 7. A linear functional p of A is said to be positive if p (aa*) ^ 0 for every a ε A. Such p is said to be state if p (I) = 1, to be normal in the terminology of Dixmier 4 if p (a a) tp (a) for a a ᵃ, and to be trace if p (ab) = p (ba) for every pair α, b ε A. The normality of state is equivalent to the complete additivity: Σ p (p a) = p (Σ Pa) for any disjoint family of projections p a aA (cf. Dixmier 4).