THE ENTROPY ON F-QUANTUM SPACES

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THE ENTROPY ON F-QUANTUM SPACES

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Mathematica Slovaca

Dagmar Markechová The entropy on F -quantum spaces Mathematica Slovaca, Vol. 40 (1990), No. 2, 177--190

Persistent URL: http://dml.cz/dmlcz/129642

Terms of use: © Mathematical Institute of the Slovak Academy of Sciences, 1990 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz

Math. Slovaca 40, 1990, No. 2, 177—190

THE ENTROPY ON F-QUANTUM SPACES DAGMAR MARKECHOVA

Introduction A usual mathematical model for the quantum statistical mechanics is the quantum logic theory, i.e. the theory of orthomodular lattices [1]. A state m on an orthomodular a-complete lattice L( v , A , 1 , 0, 1) is a maping m: L -» satisfying the following two conditions: l.m(l)= 1

(

oo

\

oc

\ / a, J = ]T m(a,). /-1

/

/=i

R i e c a n and D v u r e c e n s k i j pointed out in [2] and [3] that the Piasecki P-measure has the same algebraic structure. The Piasecki P-measure m: M -> (cf. [4]) is defined on an appropriate set of real functions M cz r \ \ ir following conditions: :i

v

*'•

'* - f ) ) = \ for every feM. / 00

If feM

(n-

L 2 , .. > , f l f

(/ #y), ,/hen m I V f J

(1.5) \

oo

= I

m(f).

(1.6)

Lemma 1.1. An F-siate m on an F-quantum space (A, M) has the following properties: '11(f) + m(f') = 1 for every feM. Iff

geM,f< Iff

g, then m(g) = m(f) + m(g Af').

g e M, f < g, then m(f) < m(g).

(1.7) (1.8) (1.9)

Proof. Since flf for every feM by (1.6) we obtain 1 = m(f) + m(f'). Let f g e M , f