arXiv subjects
Publications and source records attributed to Michael Skeide.
We give conditions when not necessarily adjointable operators between Hilbert modules allow for a polar decomposition involving not necessarily adjointable partial isometries. While the latter have been introduced and discussed by Shalit and Skeide [SS23,Ske25], here we are led, as a basic new ingredient, to the notion of not necessarily adjointable operators $a$ that admit a modulus $|a|$, so-called modular operators.
Motivated by questions raised in the preprint [AL20] by Accardi and Lu (private communication), we examine criteria for when the product of two partial isometries between Hilbert spaces is again a partial isometry and we use this to define a new composition operation that always yields again a partial isometry. Then, we aim at promoting these results to (not necessarily adjointable) partial isometries between Hilbert modules as proposed by Shalit and Skeide [SS23]. The case of Hilbert spaces is elementary and rather simple, though not trivial, but -- we expect -- folkloric. The case of Hilbert modules suffers substantially from the fact that bounded right linear maps need not possess necessarily an adjoint. In fact, we show that the new composition law for partial isometries between Hilbert spaces can in no way be promoted directly to partial isometries between Hilbert modules, but that we have to pass to the more flexible class of partially defined isometries.
We provide a Hunt type formula for the infinitesimal generators of Lévy process on the quantum groups $SU_q(N)$ and $U_q(N)$. In particular, we obtain a decomposition of such generators into a gaussian part and a `jump type' part determined by a linear functional that resembles the functional induced by the Lévy measure. The jump part on $SU_q(N)$ decomposes further into parts that live on the quantum subgroups $SU_q(n)$, $n\le N$. Like in the classical Hunt formula for locally compact Lie groups, the ingredients become unique once a certain projection is chosen. There are analogous result for $U_q(N)$.
In these notes we prove two main results: 1) It is well-known that two strongly continuous $E_0$-semigroups on $B(H)$ can be paired if and only if they have anti-isomorphic Arveson systems. For a new notion of pairing (which coincides only for $B(H)$ with the existing one), we show: For a von Neumann algebra $B$, a strongly continuous $E_0$-semigroup on $B$ and a strongly continuous $E_0$-semigroup on $B'$ can be paired if and only if their product systems are commutants of each other. 2) On the way to prove the former, en passant we have to fill in a long standing important gap in the theory of intertwiner product systems \`a la Arveson (known, so far, only for $B(H)$ in the separable case): Intertwiner product systems of faithful strongly continuous $E_0$-semigroups on von Neumann algebras have sufficiently many strongly continuous sections. We explain why both results are entirely out of reach for Arveson's methods [Arv89,Arv90] and depend essentially on the alternative approach from Skeide [Ske16].
We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on $\mathscr{B}(G)$ ($G$ a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of $E_0$-semigroups on $\mathscr{B}(H)$ by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general $C^*$-algebras) have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here in order to make the result available without any appeal to Hilbert modules.)
Bringing forward the concept of convergence in moments from classical random variables to quantum random variables is what leads to what can be called algebraic central limit theorem for (classical and) quantum random variables. I reflect in a very personal way how such an idea is typical for the spirit of doing research in mathematics as I learned it in Wilhelm von Waldenfels's research group in Heidelberg.
Apart from presenting some new insights and results, one of our main purposes is to put some records in the development of von Neumann modules straight. The von Neumann or $W^*$-objects among the Hilbert ($C^*$-)modules are around since the first papers by Paschke (1973) and Rieffel (1974) that lift Kaplansky's setting (1953) to modules over noncommutative $C^*$-algebras. While the formal definition of $W^*$-modules} is due to Baillet, Denizeau, and Havet (1988), the one of von Neumann modules as strongly closed operator spaces started with Skeide (2000). It has been paired with the definition of concrete von Neumann modules in Skeide (2006). It is well-known that (pre-)Hilbert modules can be viewed as ternary rings of operators and in their definition of Hilbert-von Neumann modules, Bikram, Mukherjee, Srinivasan, and Sunder (2012) take that point of view. We illustrate that a (suitably nondegenerate) Hilbert-von Neumann module is the same thing as a strongly full concrete von Neumann module. We also use this occasion to put some things in earlier papers right. We show that the tensor product of (concrete, or not) von Neumann correspondences is, indeed, (a generalization of) the tensor product of Connes correspondences (claimed in Skeide (2008)), viewed in a way quite different from Bikram et al.. We also present some new arguments that are useful even for (pre-)Hilbert modules.
Answering a long standing question, we give an example of a Hilbert module and a nonzero bounded right linear map having a kernel with trivial orthogonal complement. In particular, this kernel is different from its own double orthogonal complement.
We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we show that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs, instead, are motivated from a third equivalent notion, linking ideals (Section 2), and a Theorem (Section 3) that all extends nicely to (reduced) linking algebras. As an application, in Section 4, we introduce ternary extensions of Hilbert modules and prove most of the basic properties (some new even for the known notion of extensions of Hilbert modules), by reducing their proof to the well-known analogue theorems about extensions of $C^*$-algebras. Finally, in Section 5, we propose several open problems that our method naturally suggests.
We point out that a sequence of natural numbers is the dimension sequence of a subproduct system if and only if it is the cardinality sequence of a word system (or factorial language). Determining such sequences is, therefore, reduced to a purely combinatorial problem in the combinatorics of words. A corresponding (and equivalent) result for graded algebras has been known in abstract algebra, but this connection with pure combinatorics has not yet been noticed by the product systems community. We also introduce Cartesian systems, which can be seen either as a set theoretic version of subproduct systems or an abstract version of word systems. Applying this, we provide several new results on the cardinality sequences of word systems and the dimension sequences of subproduct systems.
We prove many new results about interacting Fock spaces. We pose many open problems; for most of them we prove that their solutions have no choice but being nontrivial. We ask the kind reader to consult the extended abstract in the paper.
These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d-parameter semigroups - and beyond. While exisiting work on the two- and d-parameter case is based on the approach via the Arveson-Stinespring correspondence of a CP-map by Muhly and Solel (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (A comparison is postponed to Appendix A(iv).) The generalizations are multi-fold, the difficulties often enormous. In fact, our only true if-and-only-if theorem, is the following: A Markov semigroup over (the opposite of) an Ore monoid admits a full (strict or normal) dilation if and only if its GNS-subproduct system embeds into a product system. Already earlier, it has been observed that the GNS- (respectively, the Arveson-Stinespring) correspondences form a subproduct system, and that the main difficulty is to embed that into a product system. Here we add, that every dilation comes along with a superproduct system (a product system if the dilation is full). The latter may or may not contain the GNS-subproduct system; it does, if the dilation is strong - but not only. Apart from the many positive results pushing forward the theory to large extent, we provide plenty of counter examples for almost every desirable statement we could not prove. Still, a small number of open problems remains. The most prominent: Does there exist a CP-semigroup that admits a dilation, but no strong dilation? Another one: Does there exist a Markov semigroup that admits a (necessarily strong) dilation, but no full dilation?
We show that pure strongly continuous semigroups of adjointable isometries on a Hilbert C*-module are standard right shifts. By counter examples, we illustrate that the analogy of this result with the classical result on Hilbert spaces by Sz.-Nagy, cannot be improved further to understand arbitrary isometry semigroups of isometries in the classical way.
This paper presents the complete classification of E_0-semigroups by product systems in the case of von Neumann correspondences, and under countability assumptions in the case of C*-correspondences.
One may ask which maps between Hilbert modules allow for a completely positive extension to a map acting block-wise between the associated (extended) linking algebras. In these notes we investigate in particular those of such CP-extendable maps whose 22-corner is a homomorphism, the CP-H-extendable maps. We show that they coincide with the maps considered by Asadi [Asa09], by Bhat, Ramesh, and Sumesh [BRS12], and by Skeide [Ske10]. We also give an intrinsic characterization that generalizes the characterization by Abbaspour and Skeide [AbSk07] of homomorphicly extendable maps as those which are ternary homomorphisms. For general strictly CP-extendable maps we give a factorization theorem that generalizes those of Asadi, of Bhat, Ramesh, and Sumesh, and of Skeide for CP-H-extendable maps. As an application, we examine semigroups of CP-H-extendable maps, so-called CPH-semigroups, and illustrate their relation with a sort of generalized dilation of CP-semigroups, CPH-dilations.
We present a far reaching generalization of a factorization theorem by Bhat, Ramesh, and Sumesh (stated first by Asadi) and furnish a very quick proof.
We show that the Markov semigroup obtained by Floricel in [Flo08] compressing the $E_0$-semigroup of Skeide [Ske06], does not consist of endomorphisms. It, therefore, cannot be the tail flow of an $E_0$-semigroup. As a corollary of our result, Floricel's construction will allow to get examples of proper type III Markov semigroups that are not tensor products of simpler ones, provided we find type III Arveson systems that do not factor into tensor products.
In this note we prove that there exist at least two examples of three commuting, unital, completely positive maps that have no dilation on a type I factor, and no minimal dilation on any von Neumann algebra.