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Classification of E0–Semigroups by Product Systems

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Classification of E0–Semigroups by Product Systems Classification of E0–Semigroups by Product Systems Michael Skeide∗ January 2009, this revision May 2014; to appear in Memoirs of the AMS Abstract In his Memoir from 1989, Arveson started the modern theory of product systems. More precisely, with each E0–semigroup (that is, a unital endomorphism semigroup) on B(H) he associated a product system of Hilbert spaces (Arveson system, henceforth). He also showed that the Arveson system determines the E0–semigroup up to cocycle conjugacy. In three successor papers, Arveson showed that every Arveson system comes from an E0–semigroup. There is, therefore, a one-to-one correspondence between E0–semigroups on B(H) (up to cocycle conjugacy) and Arveson systems (up to isomorphism). In the meantime, product systems of correspondences (or Hilbert bimodules) have been constructed from Markov semigroups on general unital C∗–algebras or on von Neumann algebras. These product systems showed to be an efficient tool in the construction of dila- tions of Markov semigroups to E0–semigroups and to automorphism groups. In particular, product systems over correspondences over commutative algebras (as they arise from clas- sical Markov processes) or other algebras with nontrivial center, show surprising features that can never happen with Arveson systems. A dilation of a Markov semigroup constructed with the help of a product system always arXiv:0901.1798v4 [math.OA] 13 May 2014 acts on Ba(E), the algebra of adjointable operators on a Hilbert module E. (If the Markov semigroup is on B(H) then E is a Hilbert space.) Only very recently, we showed that every product system can occur as the product system of a dilation of a nontrivial Markov semi- group. This makes it necessary to extend the theory to the relation between E0–semigroups on Ba(E) and product systems of correspondences. In these notes we present a complete theory of classification of E0–semigroups by prod- uct systems of correspondences. As an application of our theory, we answer the fundamen- tal question if a Markov semigroup admits a dilation by a cocycle perturbations of noise: It does if and only if it is spatial. ∗This work is supported by research funds of University of Molise and Italian MIUR. 1 Contents 1 Introduction...................................... ...................................... 2 2 Morita equivalence and representations . .................................... 14 3 Stable Morita equivalence for Hilbert modules . ................................... 21 4 Ternaryisomorphisms............................... .................................... 23 5 Cocycle conjugacy of E0–semigroups........................................ ............. 26 6 E0–Semigroups, product systems, and unitary cocycles . ............................... 29 7 Conjugate E0–Semigroups and Morita equivalent product systems . .................... 34 8 Stable unitary cocycle (inner) conjugacy of E0–semigroups................................. 37 9 Aboutcontinuity................................... ..................................... 39 10 Hudson-Parthasarathy dilations of spatial Markov semigroups............................... 45 11 Von Neumann case: Algebraic classification . ................................... 55 12 Von Neumann case: Topological classification . ................................... 63 13 Von Neumann case: Spatial Markov semigroups . ................................ 66 Appendix A: Strong type I product systems . ................................... 69 Appendix B: E0–Semigroups and representations for strongly continuous product systems. 79 References ......................................... ....................................... 97 1 Introduction An E –semigroup is a semigroup of unital endomorphisms of a unital –algebra. Our scope in 0 ∗ these notes is to present a complete classification of E –semigroups on Ba(E), the –algebra of 0 ∗ all adjointable operators on some Hilbert module over a unital C∗ or von Neumann algebra , B by product systems of –correspondences. This is a far-reaching generalization of Arveson’s B classification of E0–semigroups on B(H) by product systems of Hilbert spaces (henceforth, Arveson systems). Our motivation to have this generalization, is its (maybe, surprising) im- portance in the study of properly irreversible dynamical systems, that is, of (quantum, or not) Markov semigroups that are not E0–semigroups, and their dilations. (A Markov semigroup is a semigroup of unital completely positive maps on a unital C∗ or von Neumann algebra.) For instance, as an application of our classification, we determine the structure of those Markov semigroups that admit a dilation to a cocycle perturbation of a noise. The theory has many ramifications. (For instance, we always have to distinguish the case of unital C∗–algebras, with C∗–modules and C∗–correspondences, and the case of von Neumann algebras with von Neumann (or W∗) modules and correspondences. Also, the approach that works in the C∗–case and in the von Neumann case, is quite different from Arveson’s approach. Only in the von Neumann case there is a “dual” approach, using commutants of von Neumann correspondences, that resembles Arveson’s original approach.) As we wish, to keep track with these ramifications, we split this introduction into a ‘historical introduction’, a more detailed ‘motivation’, a ‘methodological introduction’, and a brief description of the ‘results’. Many sections have thematic introductions on their own. 2 Historical introduction. One-parameter groups of automorphisms of the algebra B(H) of all adjointable operators on a Hilbert space H are well understood. At least since the work of Wigner [Wig39] (see Sections 1.1, 2.4 and 3.4 of Arveson’s monograph [Arv03]) it is well B known that every σ–weakly continuous automorphism group α = αt t R on (H) is imple- B ∈ mented as αt = ut ut∗ where u = ut t R is a group of unitaries in (H ). Unitary groups, in • ∈ it turn, are understood in terms of their generators by Stone’s theorem: u = e H , where can be t H any (possibly unbounded) self-adjoint operator on H. The generator of α is, therefore, (modulo domain questions) a derivation given as a commutator with the Stone generator. This is also the setting of “standard” quantum mechanics, in which one-parameter groups of automorphisms on (H) are models for the time flow of a quantum mechanical system. B Generalizations in several directions are possible. Firstly, one may allow more general al- gebras than B(H). More advanced examples from quantum physics require this; but also if we wish to include the “classical” theories, then we should allow for commutative algebras. Au- tomorphism groups (one-parameter or not, on C∗ or von Neumann algebras), so-called C∗– or W∗–dynamical systems, are a vast area and far from being understood completely. Secondly, one may relax from groups of automorphisms to E0–semigroups and further to Markov semi- groups. (General contraction CP-semigroups that are non-Markov do occur, too. In particular, n–parameter semigroups have applications to commuting tuples of operators in multivariate operator theory; see, for instance, the introduction of Shalit and Solel [SS09]. Here, we shall completely ignore these ramifications; however, the work Shalit and Skeide [SS14a] is in prepa- ration.) Markov semigroups acting on commutative algebras, indeed, correspond to classical Markovian semigroups of transition probabilities, while Markov semigroups on noncommuta- tive C∗ or von Neumann algebras are models for irreversible quantum dynamics. Powers [Pow88] initiated studying E0–semigroups on B(H) for the sake of classifying gen- eral unbounded derivations of B(H) by classifying the E0–semigroups generated by them; and he wishes to classify E0–semigroups up to cocycle conjugacy. Arveson, in his Memoir [Arv89a], associated with every E0–semigroup on B(H) (H separable and infinite-dimensional) an Arveson system, and he showed that the E0–semigroup is determined by its Arveson system up to cocycle conjugacy. In the three successor papers [Arv90a, Arv89b, Arv90b] he showed that every Arveson system is the Arveson system associated with an E0–semigroup. In con- clusion, Arveson systems form a complete cocycle conjugacy invariant for E0–semigroups on B(H). The construction of product systems from dynamics has been generalized in various ways. In historical order: Bhat [Bha96] associated with a Markov semigroup on B(H) an Arveson system. He did this by, first, constructing a so-called (unique!) minimal dilation of the Markov semigroup to an E0–semigroup and, then, taking the Arveson system of that E0–semigroup. Bhat and Skeide [BS00] took a completely different approach. For each Markov semigroup 3 on a unital C∗ or von Neumann algebra they construct directly a product system of corre- B spondences over . This product system allows to get quite easily the minimal dilation. This B dilation acts on the algebra Ba(E) of adjointable operators on a Hilbert –module E. In Skeide B [Ske02] we presented the first construction of a product system (of –correspondences) from B a E0–semigroups on B (E). The construction from [Ske02] has been refined in various ways, which will be addressed throughout these notes. In this light, Bhat’s Arveson system would generalize to the product system of –correspondences constructed from the E –semigroup B 0 dilating a Markov semigroup that acts on Ba(E), not on . B Arveson systems classify E0–semigroups on B(H) one-to-one up to cocycle conjugacy. However, one should note that this depends on the hidden assumption that H is infinite-dimen-
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