Cocycles of CCR flows (Q2704685)

The author's particular interest is in the study of partially ordered set of quantum dynamical semigroups (QDSs) dominated by a semigroup on the algebra \({\mathcal B}({\mathcal H})\) of all bounded operators on a complex separable Hilbert space \({\mathcal H}\). In this book the author explains that for semigroups of \(*\)-endomorphisms the above-mentioned set can be described in terms of cocycles, proves a factorization theorem for dilations by taking advantage of this, and shows that minimal dilations of QDSs with bounded generators can be obtained via Hudson-Parthasarathy cocycles. NEWLINENEWLINENEWLINEThis book is organized as follows. Section 1 is the Introduction. The quantum dynamical semigroup (QDS) on \({\mathcal B}({\mathcal H})\) means a one parameter semigroup \(\alpha =\) \(\{ \alpha_t\); \(t \geq 0 \}\) of linear completely positive maps mapping \({\mathcal B}({\mathcal H})\) to itself, and also \(\alpha\) is assumed to be contractive, normal, and continuous in the weak operator topology. NEWLINENEWLINENEWLINEIn Section 2 notions of conjugacy and cocycle conjugacy are precisely defined. In particular Theorem 2.5 is an important result which asserts that the above two notions coincide for primary dilations of unital QDSs. Let \(\theta = \{ \theta_t\); \(t \geq 0 \}\) denote an \(E_0\)-semigroup of \({\mathcal B}({\mathcal H})\). NEWLINENEWLINENEWLINEIn Section 3 an important notion of induced semigroup is introduced. Roughly speaking, once you know the induced semigroup, you can know how far the given dilation is from being minimal. Theorem 3.7 asserts that the primary dilation \(\theta\) is a minimal dilation for all derived semigroups, implying that in some specific cases dilations are automatically minimal. NEWLINENEWLINENEWLINEThe partial order of domination for quantum dynamical semigroups is introduced in Section 4. In fact, a QDS \(\alpha =\) \(\{ \alpha_t\); \(t \geq 0 \}\) is said to be dominated by a QDS \(\tau =\) \(\{ \tau_t\); \(t \geq 0 \}\) (denoted by \(\alpha \leq \tau\)), if \(\tau_t - \alpha_t\) is completely positive for every \(t\). One interesting fact here is that positivity implies complete positivity, which is a rare thing to happen in non-commutative contexts (cf. Proposition 4.2). Theorem 4.3 is the main result of this section, which says that in case of \(E_0\)-semigroups dominated semigroups can be described through positive, contractive local cocycles. This is a key result in the sense that the collection of such cocycles forms a cocycle conjugacy invariant for the \(E_0\)-semigroup, and also that deminated semigroups can be thought of as abstract Feynman-Kac perturbations of the \(E_0\)-semigroup. Suppose that the projection \(P\) from \({\mathcal H}\) onto a subspace \({\mathcal H}_0\) (\(\subset {\mathcal H}\)) is increasing for an \(E_0\)-semigroup \(\theta\) of \({\mathcal B}({\mathcal H})\). When \(\tau\) is the unital QDS obtained by compressing \(\theta\) by \(P\), the symbol \({\mathcal D}_{\tau}\) denotes the set of all (not necessarily unital) QDSs dominated by \(\tau\). NEWLINENEWLINENEWLINESection 5 is devoted to the topic of compression under domination, where the author studies how dominated semigroups behave under compression or dilation. The important observation is that minimality of the dilation can be detected by looking at the action of compression map on the partially ordered set of dominated semigroups. As a matter of fact, the author shows first that compression by projection \(P\) maps \({\mathcal D}_{\theta}\) surjectively to \({\mathcal D}_{\tau}\). The main result here is that this compression map is injective if and only if \(\theta\) is the minimal dilation of \(\tau\), which provides with a completely algebraic characterization of minimality. NEWLINENEWLINENEWLINEOn the other hand, the concept of units is very basic for the classification theory of \(E_0\)-semigroups, and several descriptions of units are given in Section 6. In addition the author studies as to what happens to units under dilation. According to \textit{W. Arveson} [J. Funct. Anal. 146, No. 2, 557-588 (1997; Zbl 0923.46065)], we already know that units of QDSs can be lifted to their dilations. Here an explicit formula Eq.(6.2) for such a lifting is given. It is also shown in Theorem 6.6 that there is a one-to-one correspondence between normalized units of \(\tau\) and one-dimensional unital extensions of \(\tau\). NEWLINENEWLINENEWLINEThe principal theme of Section 7 is cocycle computation for CCR (Canonical Commutation Relation) flows, and QDSs dominated by CCR flows are described by computing all positive, contractive local cocycles. In particular, the Heisenberg motion group is the group of unitary local cocycles of CCR flows and this group acts on the set of positive, contractive local cocycles. NEWLINENEWLINENEWLINESection 8 treats factorization theorems. An \(E_0\)-semigroup is said to be amenable if it has sufficient number of unitary local cocycles, and amenable dilations of elementary semigroups factorize uniquely in certain fashion (cf. Theorem 8.6). While, Theorem 8.7 asserts that on the supposition that a primary dilation \(\theta\) of non-elementary unital QDS \(\tau\) is cocycle conjugate to CCR flow of index 1, \(\theta\) is the unique minimal dilation of \(\tau\). It is also proved that type I dilations factorize uniquely as minimal dilation tensored with some other \(E_0\)-semigroup in standard form (cf. Theorem 8.9), which is the most important result in this book. An immediate corollary of this is that Powers' standard form is unique for amenable \(E_0\)-semigroups. NEWLINENEWLINENEWLINESection 9 exhibits the bridge between \(E_0\)-semigroup theory and quantum stochastic calculus, where dilations of unital QDSs with bounded generators can be realized through Hudson-Parthasarathy cocycles. In fact, the deficiency index measuring how far the given dilation is from being minimal can be computed, with the result that an explicit description of the set of generators of QDSs dominated by a fixed semigroup with bounded generator can be obtained. Also it is shown that Heisenberg motion group of appropriate dimension has a natural action on it. NEWLINENEWLINENEWLINETwo appendices compensate for discussions on continuity of QDSs and cocycles of \(E_0\)-semigroups and also on decomposition of non-minimal dilations in discrete time. In References 67 papers and monographs are listed in total, and they include almost all principal research papers published during 1972-1999 in line with aformentioned themes and topics. NEWLINENEWLINENEWLINEStated below is the relationship between the main theme of this book and its historical background. Through this book the author builds a bridge between two subjects: (i) Quantum Stochastic Process; (ii) \(E_0\)-semigroups (i.e., \(*\)-endomorphisms) of the algebra \({\mathcal B}({\mathcal H})\), which have been developing independently since 1980s. NEWLINENEWLINENEWLINEAs to (i), the foundation stones of the theory of quantum stochastic processes were laid by the introduction of abstract notion of non-commutative processes as families of \(*\)-homomorphisms by \textit{L. Accardi, A. Frigerio} and \textit{J. T. Lewis} [Publ. Res. Inst. Math. Sci. 18, 97-133 (1982; Zbl 0498.60099)]. The subject was highlighted by the discovery of quantum Itô formula by \textit{R. L. Hudson} and \textit{K. R. Parthasarathy} [Commun. Math. Phys. 93, 301-323 (1984; Zbl 0546.60058)]. This field has strong interactions with Probability Theory, Operator Theory and Mathematical Physics. \textit{K. R. Parthasarathy} [``An Introduction to Quantum Stochastic Calculus'', Birkhäuser, Basel (1992; Zbl 0751.60046)] is a basic reference for this subject. NEWLINENEWLINENEWLINEAs to (ii), the research was initiated by \textit{R. T. Powers} in late 1980s [see, e.g., Publ. Res. Inst. Math. Sci. 23, 1053-1069 (1987; Zbl 0651.47025)]. CCR flows (= a part of the title of this book) are paradigm examples of \(E_0\)-semigroups, and they are obtained by considering second quantized shift operators on the Boson Fock space \(\Gamma( L_2({\mathbb R}_+, {\mathcal K}))\) of Hilbert space \({\mathcal K}\) valued square-integrable functions. A definite link between two subjects was found by the author [see, e.g., \textit{B. V. R. Bhat}, Trans. Am. Math. Soc. 348, No. 2, 561-583 (1996; Zbl 0842.46045)], where the theory of weak Markov flows developed in \textit{B. V. R. Bhat} and \textit{K. R. Parthasarathy} [Ann. Inst. Henri Poincaré, Probab. Stat. 31, No. 4, 601-651 (1995; Zbl 0832.46060)] played an essential role in showing that semigroups of completely positive maps get dilated to semigroups of \(*\)-endomorphisms. Note that semigroups of completely positive maps are known as QDS [cf. \textit{E. B. Davies}, J. Funct. Anal. 34, 421-432 (1979; Zbl 0428.47021)]. In particular, \textit{W. Arveson} [Int. J. Math. 10, No. 7, 791-823 (1999; Zbl 1110.46313)] has shown that QDS on \({\mathcal B}({\mathcal H})\) with bounded generators gets dilated to \(E_0\)-semigroups cocycle conjugate to CCR flows. Thanks to e.g. \textit{G. Lindblad} [Commun. Math. Phys. 48, 119-130 (1976; Zbl 0343.47031)] and \textit{E. Christensen} and \textit{D. E. Evans} [J. Lond. Math. Soc. 20, 358-368 (1979; Zbl 0448.46040)], the structure of these generators are known explicitly. However, the next question is to ask whether one can write down the cocycles explicitly. Now the author shows that this is possible using the theory of quantum stochastic differential equations of \textit{R. L. Hudson} and \textit{K. R. Parthasarathy} [op. cit. (1984; Zbl 0546.60058)]. This links the two subjects even more closely. Actually Hudson-Parthasarathy theory permits many candidates for dilation of a given QDS. The problem is to identify the right one, namely, the minimal one. Since checking minimality analytically seems to be rather hard, the author here adopts a completely algebraic method. Consequently the structure of CCR flows can be observed even better (cf. Theorem 8.7). In the context of quantum stochastic calculus it becomes possible to answer some fundamental questions. For example to consider dilations of a given QDS why should one be tensoring the initial space with a Fock space. The main result of this book may be a natural answer to this question, for through this book we observe that it is more or less automatic that such a factorization exists (= Theorem 8.9).