We define a class of higher-order linear maps that transform quantum operations into quantum operations and satisfy suitable requirements of normality and complete positivity. For this class of maps we prove two dilation theorems that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. A structure theorem for probability measures with values in this class of ...

In this paper we obtain almost sure convergence theorems for vectorvalued uniform amarts and C-sequences without assuming the Radon-Nikodym Property. Specifically, it is shown that if a limit exists in a weak sense for these martingale generalizations, then a.s. scalar and strong convergence follow. These results lead to some new versions of the Ito-Nisio theorem. Convergence results for random...

We prove regularity (i.e., smoothness) of measures on R d satisfying the equation L = 0 where L is an operator of type Lu = tr(Au 00) + B ru. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely-square integrable. We also treat a class of corresponding innnite dimensional cases where IR d is replaced by a locally convex topological vector space X. In this cases...

In this paper we prove two theorems relating positive definite measures to induced representations. The first shows how the injection of a positive definite measure on a topological group H into a containing locally compact group G in which H is closed gives rise to induced representations. The second is another version of Mackey's imprimitivity theorem, along the lines of Loomis' proof [5]. We...

A Banach spaceX with the property that every absolutely continuousX−valued function is almost everywhere differentiable is said to be a Radon-Nikodym space [7, pp. 217–219] or [2, 13] (see also [3]). For example, every reflexive Banach space (in particular, every Hilbert space) is a Radon-Nikodym space, but the space L∞ [0, 1] of all K−valued, essentially bounded functions defined on the interv...

The duality between Hl and BMO, the space of functions of bounded mean oscillation (see [JN]), was first proved by C. Fefferman (see [F], [FS]) and then other proofs of it were obtained . Using the atomic decomposition approach ([C], [L]) the author studied the problem of characterizing the dual space of Hl of vector-valued functions . In [B2] the author showed, for the case SZ = {Iz1 = 1}, tha...

We consider two singular limits: a fast reaction limit with non-monotone nonlinearity and regularization of the forward-backward diffusion equation. derive pointwise identities satisfied by Young measure generated these problems. As result, we obtain an explicit formula for even without non-degeneracy assumption used in previous works. The main new idea is application Radon–Nikodym theorem to d...