Conditional expectation in an operator algebra, II

CONDITIONAL EXPECTATION IN THE KOPKA'S D-POSETS

The notion of a $D$-poset was introduced in a connection withquantum mechanical models. In this paper, we introduce theconditional expectation of  random variables on theK^{o}pka's $D$-Poset and prove the basic properties ofconditional expectation on this  structure.

The Similarity Degree of an Operator Algebra, Ii

For every integer d ≥ 1, there is a unital closed subalgebra A d ⊂ B(H) with similarity degree equal precisely to d, in the sense of our previous paper. This means that for any unital homomorphism u: A d → B(H) we have u cb ≤ Ku d with K > 0 independent of u, and the exponent d in this estimate cannot be improved. The proof that the degree is larger than d − 1 crucially uses an upper bound for ...

Conditional Expectation

Let μ and λ be two positive bounded measures on the same meaurable space (Ω,F). We call μ and λ equivalent, and write μ ≡ λ, if they have the same null sets— so, if they were probability measures, the notion of “a.s.” would be the same for both. More generally, we call λ absolutely continuous (AC) w.r.t. μ, and write λ μ, if μ(A) = 0 implies λ(A) = 0, i.e., if every μ-null set is also λ-null. W...

The Measure Algebra as an Operator Algebra

Introduction. In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G). Let A denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of A in M(G)*. In § II, it is shown that P , with the Arens multiplication, is a commutative J3*-algebra with identity. Thus P = ...

Conditional expectation