*-semigroup endomorphisms of B(H)

The composition series of ideals of the partial-isometric crossed product by semigroup of endomorphisms

Let $Γ^{+}$ be the positive cone in a totally ordered abelian group $Γ$, and $α$ an action of $Γ^{+}$ by extendible endomorphisms of a $C^{\ast}$-algebra $A$. Suppose $I$ is an extendible $α$-invariant ideal of $A$. We prove that the partial-isometric crossed product $\mathcal{I}:=I\times_α^{\textrm{piso}}Γ^{+}$ embeds naturally as an ideal of $A\times_α^{\textrm{piso}}Γ^{+}$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\mathcal{I}$ together with the kernel of a natural homomorphism $ϕ: A\times_α^{\textrm{piso}}Γ^{+}\righta…

The Partial-Isometric Crossed Products by Semigroups of Endomorphisms as Full Corners

Suppose $Γ^{+}$ is the positive cone of a totally ordered abelian group $Γ$, and $(A,Γ^{+},α)$ is a system consisting of a $C^*$-algebra $A$, an action $α$ of $Γ^{+}$ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\times_α^{\piso}Γ^{+}$ is a full corner in the subalgebra of $Ł(\ell^{2}(Γ^{+},A))$, and that if $α$ is an action by automorphisms of $A$, then it is the isometric-crossed product $(B_{Γ^{+}}\otimes A)\times^{\iso}Γ^{+}$, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realiza…

Crossed product of a C*-algebra by a semigroup of bounded positive linear maps. Interactions

The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev's crossed product by an endomorphism, and is related to Exel's interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms.

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Let $V$ be a standard subspace in the complex Hilbert space $H$ and $G$ be a finite dimensional Lie group of unitary and antiunitary operators on $H$ containing the modular group $(Δ_V^{it})_{t \in R}$ of $V$ and the corresponding modular conjugation~$J_V$. We study the semigroup \[ S_V = \{ g\in G \cap U(H) : gV \subseteq V\} \] and determine its Lie wedge $L(S_V) = \{ x \in L(G) : exp(R_+ x) \subseteq S_V\}$, i.e., the generators of its one-parameter subsemigroups in the Lie algebra $L(G)$ of~$G$. The semigroup $S_V$ is analyzed in terms of antiunitary representations and their analytic exte…

Crossed products by abelian semigroups via transfer operators

We propose a generalisation of Exel's crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler's theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems.

Cocycle conjugacy of quasifree endomorphisms semigroups on the CAR algebra

W. Arveson has described a cocycle conjugacy class $U(α)$ of $E_0$-semigroup $α$ on B(H) which is a factor of type $\rm I$. Under some conditions on $α$, there is a $E_0$-semigroup $β\in U(α)$ being a flow of shifts in the sence of R.T. Powers. We study quasifree endomorphisms semigroups $α$ on the hyperfinite factor $M=π(A(K))''$ generated by the representations $π$ of the algebra of canonical anticommutation relations A(K) over a separable Hilbert space K. The type of M can be $\rm I$, $\rm II$ or $\rm III$ depending on $π$. The cocycle conjugacy class $U(α)$ is described in the terms of ini…

Endomorphism Semigroups and Lightlike Translations

Borchers and Wiesbrock have demonstrated certain results concerning the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables. These results are abstracted and analyzed as essentially operator-theoretic. Criteria are then demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms. All this is combined to establish a von Neumann-algebraic converse to the Borchers-Wiesbrock results. This analysis is then applied to questions of isotony and co…

Two families of Exel-Larsen crossed products

Larsen has recently extended Exel's construction of crossed products from single endomorphisms to abelian semigroups of endomorphisms, and here we study two families of her crossed products. First, we look at the natural action of the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group $Γ$, and the induced action on $C(Γ)$. We prove a uniqueness theorem for the crossed product, and we find a class of connected compact abelian groups $Γ$ for which the crossed product is purely infinite simple. Second, we consider some natural actions of the additive semigroup $\mathbb{N}^2$ …

A new look at the Crossed-Product of a C*-Algebra by a Semigroup of Endomorphisms

Let G be a group and let P be a subsemigroup of G. In order to describe the crossed product of a C*-algebra A by an action of P by unital endomorphisms we find that we must extend the action to the whole group G. This extension fits into a broader notion of "interaction groups" which consists of an assignment of a positive operator V_g on A for each g in G, obeying a partial group law, and such that (V_g,V_{g^{-1}}) is an interaction for every g, as defined in a previous paper by the author. We then develop a theory of crossed products by interaction groups and compare it to other endomorphism…

Type II$_1$ factors with arbitrary countable endomorphism group

In \cite{Ioana:vNsuperrigidity}, Ioana introduced three new invariants of type II$_1$ factors: the one-sided fundamental group, the endomorphism semigroup and the set of right-finite bimodules. In \cite{Ioana:vNsuperrigidity}, he does not provide many computations of these invariants. In particular, the question whether these invariants can be trivial is left open. We give an explicit example of a type II$_1$ factor for which all three invariants are trivial. More generally, for any countable left-cancellative semigroup $G$, we construct a type II$_1$ factor $M$ whose endomorphism semigroup is…

Ergodic extensions and Hilbert modules associated to endomorphisms of MASAS

We show that a class of ergodic transformations on a probability measure space $(X,μ)$ extends to a representation of $\mathcal{B}(L^2(X,μ))$ that is both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in this work. During the analysis of the existence and uniqueness of such a Cuntz family we give several results of individual interest. Most notably we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal basis of Hilbert modules. We remark that the trivial Hilbert module of the Cuntz alge…

On the existence of $E_0$-semigroups

Product systems are the classifying structures for semigroups of endomorphisms of B(H), in that two $E_0$-semigroups are cocycle conjugate iff their product systems are isomorphic. Thus it is important to know that every abstract product system is associated with an $E_0$-semigrouop. This was first proved more than fifteen years ago by rather indirect methods. Recently, Skeide has given a more direct proof. In this note we give yet another proof by an elementary construction.

Partial-isometric crossed products of dynamical systems by left LCM semigroups

Let P be a left LCM semigroup, and $α$ an action of $P$ by endomorphisms of a $C^{*}$-algebra $A$. We study a semigroup crossed product $C^{*}$-algebra in which the action $α$ is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which we provide a structure theorem as it behaves well under short exact sequences and tensor products.

Classification of q-pure q-weight maps over finite dimensional Hilbert spaces

An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup of $*$-endomorphisms of $B(H)$ that preserve the identity. Every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an $E_0$-semigroup induced by the Bhat induction of a $CP$-flow over a separable Hilbert space $K$. We say an $E_0$-semigroup $α$ is $q$-pure if the $CP$-subordinates $β$ of norm one (i.e. $\Vertβ_t(I)\Vert = 1$ and $α_t-β_t$ is completely positive for all $t \geq 0$) are totally ordered in the sense that if $β$ and $γ$ are two $CP$-subordin…

$E_{0}^{P}$-semigroups and product systems

Let $P$ be a closed convex cone in $\mathbb{R}^{n}$. Assume that $P$ is spanning i.e. $P-P=\mathbb{R}^{n}$ and pointed i.e. $P \cap -P=\{0\}$. Let $α:=\{α_{x}:x \in P\}$ be a $σ$-weakly continuous family of unital normal endomorphisms on $B(H)$. Denote the "product system" associated to $α$ by $\mathcal{E}_α$. We show that $\mathcal{E}_α$ is a concrete product system and $α$, up to cocycle conjugacy, can be recovered completely from $\mathcal{E}_α$

Unit Vectors, Morita Equivalence and Endomorphisms

We solve two problems in the theory of correspondences that have important implications in the theory of product systems. The first problem is the question whether every correspondence is the correspondence associated (by the representation theory) with a unital endomorphism of the algebra of all adjointable operators on a Hilbert module. The second problem is the question whether every correspondence allows for a nondegenerate faithful representation on a Hilbert space. We also resolve an extension problem for representations of correspondences and we provide new efficient proofs of several w…

Gauge-equivariant Hilbert bimodules and crossed products by endomorphisms

C*-endomorphisms arising from superselection structures with non-trivial centre define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz-Pimsner algebra of a vector bundle having the above-mentioned rank and first Chern class, and can be used to construct a duality for abstract (nonsymmetric) tensor categories vs. group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e., the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with …

Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces

Let V be a standard subspace in the complex Hilbert space H and U : G \to U(H) be a unitary representation of a finite dimensional Lie group. We assume the existence of an element h in the Lie algebra of G such that U(exp th) is the modular group of V and that the modular involution J_V normalizes U(G). We want to determine the semigroup $S_V = \{ g\in G : U(g)V \subseteq V\}.$ In previous work we have seen that its infinitesimal generators span a Lie algebra on which ad h defines a 3-grading, and here we completely determine the semigroup S_V under the assumption that ad h defines a 3-grading…

CP-H-Extendable Maps between Hilbert modules and CPH-Semigroups

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 ar…

Semicrossed Products of Operator Algebras by Semigroups

We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope fo…

Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem

Let A be a C*-algebra and I a closed two-sided ideal of A. We use the Hilbert C*-modules picture of the Cuntz semigroup to investigate the relations between the Cuntz semigroups of I, A and A/I. We obtain a relation on two elements of the Cuntz semigroup of A that characterizes when they are equal in the Cuntz semigroup of A/I. As a corollary, we show that the Cuntz semigroup functor is exact. Replacing the Cuntz equivalence relation of Hilbert modules by their isomorphism, we obtain a generalization of Kasparov's Stabilization theorem.

On the existence of $E_{0}$-semigroups -- the multiparameter case

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $Ω$. Let $E$ be a product system over $Ω$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $α:=\{α_x\}_{x \in P}$ of unital normal $*$-endomorphisms of $B(\mathcal{H})$ such that $E$ is isomorphic to the product system associated to $α$.

Pure Semigroups of Isometries on Hilbert C*-Modules

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.

An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center

In Rev. Math. Phys. 4 (1997) 785 we study Hilbert-C* systems {F,G} where the fixed point algebra A has nontrivial center Z and where A'\cap F=Z is satisfied. The corresponding category of all canonical endomorphisms of A contains characteristic mutually isomorphic subcategories of the Doplicher/Roberts-type which are connected with the choice of distinguished G-invariant algebraic Hilbert spaces within the corresponding G-invariant Hilbert Z-modules. We present in this paper the solution of the corresponding inverse problem. More precisely, assuming that the given endomorphism category T of a …

On Bounded Finite Potent Operators on arbitrary Hilbert Spaces

The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate's trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators.

The induced semigroup of Schwarz maps to the space of Hilbert-Schmidt operators

We prove that for every semigroup of Schwarz maps on the von~Neumann algebra of all bounded linear operators on a Hilbert space which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak$^*$ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe th…

Cuntz semigroups of C*-algebras of stable rank one and projective Hilbert modules

Let $A$ be a simple C*-algebra of stable rank one and let $p$ and $q$ be two $σ$-compact open projections. It is proved that there is a continuous path of unitaries in ${\tilde A}$ which connects open sub-projections of $p$ which is compactly contained in $p$ to those in $q.$ It is also shown that every Hilbert module is projective in the category whose morphisms are bounded module maps with adjoints. A discussion of projective Hilbert modules (whose morphisms are bounded module maps) is also given.

Four Lectures on Noncommutative Dynamics

These lectures concern basic aspects of the theory of semigroups of endomorphisms of type $I$ factors that relate to causal dynamics, dilation theory, and the problem of classifying $E_0$-semigroups up to cocycle conjugacy. We give only a few proofs here; full details can be found in the author's upcoming monograph {\em Noncommutative Dynamics and $E$-semigroups}, to be published in the Springer series Monographs in Mathematics.

Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations

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 stoc…

Martingales, endomorphisms, and covariant systems of operators in Hilbert space

We show that a class of dynamical systems induces an associated operator system in Hilbert space. The dynamical systems are defined from a fixed finite-to-one mapping in a compact metric space, and the induced operators form a covariant system in a Hilbert space of L^2-martingales. Our martingale construction depends on a prescribed set of transition probabilities, given by a non-negative function. Our main theorem describes the induced martingale systems completely. The applications of our theorem include wavelets, the dynamics defined by iterations of rational functions, and sub-shifts in sy…

Semigroups of Partial Isometries

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of "generalized weighted composition" operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is irreducible and contains a compact operator then the underlying measure space is p…

Commutants of von Neumann Modules, Representations of B^a(E) and Other Topics Related to Product Systems of Hilbert Modules

We review some of our results from the theory of product systems of Hilbert modules. We explain that the product systems obtained from a CP-semigroup in a paper by Bhat and Skeide and in a paper by Muhly and Solel are commutants of each other. Then we use this new commutant technique to construct product systems from E_0-semigroups on B^a(E) where E is a strongly full von Neumann module. (This improves the construction from a paper by Skeide for Hilbert modules where existence of a unit vector is required.) Finally, we point out that the Arveson system of a CP-semigroup constructed by Powers f…

Lifting fixed points of completely positive semigroups

Let $\{ϕ_s\}_{s\in S}$ be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra $N$. Assume there exists a semigroup $\{α_s\}_{s\in S}$ of weak*-continuous *-endomorphisms of some larger von Neumann algebra $M\supset N$ and a projection $p\in M$ with $N=pMp$ such that $α_s(1-p)\le 1-p$ for every $s\in S$ and $ϕ_s(y)=pα_s(y)p$ for all $y\in N$. If $\inf_{s\in S}α_s(1-p)=0$ then we show that the map $E:M\to N$ defined by $E(x)=pxp$ for $x\in M$ induces a complete isometry between the fixed point spaces of $\{α_s\}_{s\in S}$ …

导子与KMS对称量子马尔可夫半群

Derivations and KMS-Symmetric Quantum Markov Semigroups

本文证明KMS对称量子马尔可夫半群的$L^2$生成元可表示为Hilbert双模值导子的平方，通过引入新的完全正映射推广了迹对称与GNS对称情形的已有结果。

Infinite-dimensional transfer operators, endomorphisms, and measurable partitions

We develop a new duality between endomorphisms of measure spaces, on the one hand, and a certain family of positive operators, called transfer operators, acting in spaces of measurable functions on, on the other. A framework of standard Borel spaces is adopted; and this generality is wide enough to cover a host of applications. While the mathematical structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in a host of settings, both pure and applied, we propose here a unified study. This is the general setting of dynamics in Borel …

The Charge Quantum Numbers of Gauge Invariant Quasi-free Endomorphisms

The representations of a group of gauge automorphisms of the canonical commutation or anticommutation relations which appear on the Hilbert spaces of isometries H_ρimplementing quasi-free endomorphisms ρon Fock space are studied. Such a representation, which characterizes the "charge" of ρin local quantum field theory, is determined by the Fock space structure of H_ρitself: Together with a "basic" representation of the group, all higher symmetric or antisymmetric tensor powers thereof also appear. Hence ρis reducible unless it is an automorphism. It is further shown by the example of the massl…