Quantized Convolution Semigroups
نویسندگان
چکیده
We describe our construction of a continuous tensor product system in the sense of Arveson for a general W ∗-continuous completely positive semigroup of B(H) (H separable). This product system is canonically isomorphic to the product system of the minimal dilation E0-semigroup. We use this construction to show that contrary to previous speculation the minimal dilations of all quantized convolution semigroups are completely spatial. This class includes the heat flow on the CCR algebra studied recently by Arveson, which we show is cocycle conjugate to a CAR/CCR flow of index two. The analysis of these examples also involves Lévy processes and their stochastic area processes. Additionally, we prove the following fact: given a product system F and a type I product system E, if there is a bijection ψ : E → F , not necessarily measurable, that preserves fibers and multiplication and ψ is unitary fiberwise, then E and F are isomorphic.
منابع مشابه
Convolution Semigroups of States
Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective cor...
متن کاملar X iv : 0 90 5 . 12 96 v 1 [ m at h . O A ] 8 M ay 2 00 9 CONVOLUTION SEMIGROUPS OF STATES
Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective corres...
متن کاملTHE ANALOGUE OF WEIGHTED GROUP ALGEBRA FOR SEMITOPOLOGICAL SEMIGROUPS
In [1,2,3], A. C. Baker and J.W. Baker studied the subspace Ma(S) of the convolution measure algebra M, (S) of a locally compact semigroup. H. Dzinotyiweyi in [5,7] considers an analogous measure space on a large class of C-distinguished topological semigroups containing all completely regular topological semigroups. In this paper, we extend the definitions to study the weighted semigroup ...
متن کاملSelfdecomposability and Semi-selfdecomposability in Subordination of Cone-parameter Convolution Semigroups
Extension of two known facts concerning subordination is made. The first fact is that, in subordination of 1-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on R,...
متن کاملGeneralized Mehler Semigroups and Catalytic Branching Processes with Immigration
Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein-Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is est...
متن کامل