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
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