We denote all scalar products by h., .i and identify operators and their ampliations if no confusion arises.
Filtered random variables, bialgebras and convolutions
Xk (σ) ⊗ 1⊗∞)(P (σ)⊗(l−1) ⊗ 1 ⊗ P (σ)⊗∞) of an ampliation of Xk (σ) into bB⊗∞ and a pro jection indexed by σ .
Filtered random variables, bialgebras and convolutions
X = X(l, k) is the (l, k)-th ampliation of x ∈ Al into bA1 and P = P(l, σ), where l ∈ L, k ∈ N, σ ∈ P (N).
Filtered random variables, bialgebras and convolutions
This is exactly what is not true of ampliative inference, and it is what has led some writers (e.g., [Morgan, 1998]) to deny that there is any such thing as a nonmonotonic logic.
Evaluating Defaults
For m ≥ 0, let us consider the ampliation of the maps Θ, b and β as maps from A N B(Γfr( ˆk0 )) into itself.
Quantum random walks and their convergence
B ) given by the generators of the ν : B 7→ B δµ quantum Itˆo equation (3.2) and ıµ ν is the ampliation of B .
Quantum Stochastic Positive Evolutions: Characterization, Construction, Dilation
The w*-representation : B → L (G ) of B = L (H) is always an ampliation (B ) = B ⊗ J, where J is an orthopro jector onto a subspace K1 ⊆ K, corresponding to the minimal dilation in G1 = H ⊗ K1 .
Quantum Stochastic Positive Evolutions: Characterization, Construction, Dilation
The algebra B = L (H) is represented on G by the ampliation (B ) = B ⊗ J, where J = 1 ⊕ J ⊕ 1, and (B ) Lgη ∈ G ◦, where the pre-Hilbert space D ⊕ D• is isometrically embedded into D ⊕ D• ⊕ D as g (η ⊕ η• ) = 0 ⊕ η• ⊕ η .
Quantum Stochastic Positive Evolutions: Characterization, Construction, Dilation
It is shown that although the spectrum of the analytic generator of a one– parameter group of isometries of a Banach space may be equal to C (cf [VD] and [ElZs]), a simple operation of ampliating the analytic generator onto its graph locates its spectrum in IR+ .
A remark on the spectrum of the analytic generator
We denote by Un the natural ampliation of U to H0 ⊗ T Φ where Un acts as U on the tensor product of H0 and the n-th copy of H and U acts as the identity of the other copies of H.
Repeated Quantum Interactions and Unitary Random Walks
We denote by ai j (n) their natural ampliation to T Φ acting on the n-th copy of H only.
Repeated Quantum Interactions and Unitary Random Walks
Let H(∞) = H ⊗ ℓ2 be the inﬁnite ampliation of H.
Conjugate Dynamical Systems on C*-algebras
When the random-walk generator acts by ampliation and multiplication or conjugation by a unitary operator, necessary and suﬃcient conditions are given for the quantum stochastic cocycle which arises in the limit to be driven by an isometric, co-isometric or unitary process.
Quantum random walks and thermalisation II
It suﬃces to show that PxLGPx is unitarily equivalent to an ampliation of Ln, such that generators are mapped to generators.
Isomorphisms of algebras associated with directed graphs
If α is a cardinal number, we let H α denote the direct sum of α copies of H, and for x ∈ B(H ), we let x ⊗ 1α be the ampliation of x acting on H α .
Local Operator Multipliers and Positivity
***