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.
Let H be a complex infinite dimensional Hilbert space. We describe the form of all *-semigroup endomorphisms $ϕ$ of B(H) which are uniformly continuous on every commutative C*-subalgebra. In particular, we obtain that if $ϕ$ satisfies $ϕ(0)=0$, then $ϕ$ is additive.
In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on $C^{*}$-algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes.
Let $Γ^+$ be the positive cone in a totally ordered abelian group $Γ$, and let $α$ be an action of $Γ^+$ by endomorphisms of a $C^*$-algebra $A$. We consider a new kind of crossed-product $C^*$-algebra $A\times_αΓ^+$, which is generated by a faithful copy of $A$ and a representation of $Γ^+$ as partial isometries. We claim that these crossed products provide a rich and tractable family of Toeplitz algebras for product systems of Hilbert bimodules, as recently studied by Fowler, and we illustrate this by proving detailed structure theorems for actions by forward and backward shifts.
We begin this note with a von Neumann algebraic version of the elementary but extremely useful fact about being able to extend inner-product preserving maps from a total set of the domain Hilbert space to an isometry defined on the entire domain. This leads us to the notion of when `good' endomorphisms of a factorial probability space $(M,ϕ)$ (which we call equi-modular) admit a natural extension to endomorphisms of $L^2(M,ϕ)$. We exhibit examples of such extendable endomorphisms. We then pass to $E_0$-semigroups $α= {α_t: t \geq 0}$ of factors, and observe that extendability of this semigroup…
Let $Γ^{+}$ be the positive cone of a totally ordered abelian discrete group $Γ$, and $α$ an action of $Γ^{+}$ by extendible endomorphisms of a $C^*$-algebra $A$. We prove that the partial-isometric crossed product $A\times_α^{\textrm{piso}}Γ^{+}$ is a full corner of a group crossed product $\mathcal{B}\times_βΓ$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(Γ,A)$ generated by a collection of faithful copies of $A$, and the action $β$ on $\mathcal{B}$ is induced by shift on $\ell^{\infty}(Γ,A)$. We then use this realization to show that $A\times_α^{\textrm{piso}}Γ^{+}$ has an essentia…
Laca constructed a minimal automorphic dilation for every semigroup dynamical system arising from an action of an Ore semigroup by injective endomorphisms of a unital $C^*$-algebra. Here we show that the semigroup crossed product with its action by inner endomorphisms given by the implementing isometries has as minimal automorphic dilation the group crossed product of the original dilation. Applications include recent examples studied by Cuntz and the second named author.
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…
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…
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.
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…
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.
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…
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…
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 $\mathbb{N}^2$ …
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…
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…
We consider a family of Hecke C*-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.
When S is a discrete subsemigroup of a discrete group G such that G = S^{-1} S, it is possible to extend circle-valued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilate…
We study completely contractive representations of product systems of $C^*$-correspondences over semigroups. For a product system of $C^*$-correspondences over the semigroup $\mathbb{N}^2$, we prove that every such representation can be dilated to an isometric (or Toeplitz) representation. We use it to prove that every pair of commuting CP maps on a von Neumann algebra $M$ can be dilated to a commuting pair of endomorphisms (on a larger von Neumann algebra).
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.
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.
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 $\mathcal{E}_α$
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…
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…
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 $α$.
Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\Sc{S}_i$, where for each $i\in I$, $\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\B{R}_+$ containing 0. We consider representations of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nica-covariance property: $T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant represe…
We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of $C^*$-correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for $C^*$-precategories. Our results provide an interpretation of conditions identified in work of Fowler and Fowler-Raeburn, and apply also to their crossed product twisted by a product system, in the new context of right LCM semigroups, as well as to a new, Doplicher-Roberts type $C^*$-algebra associated to the Nica-Toeplitz algebra. As a derived construction we develop Nica-Toeplitz cr…
We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the $C^*$-algebra of the graph. We show that the crossed product by this action is stably isomorphic to the $C^*$-algebra of a quotient graph. Our main tool is Laca's dilation theory for endomorphic actions of Ore semigroups on $C^*$-algebras, which embeds such an action in an automorphic action of the enveloping group on a larger $C^*$-algebra.
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.
Since quite a time there were available only two rather difficult and involved proofs, the original one by Arveson and a more recent one by Liebscher, of the fact that for every Arveson system there exists an E_0-semigroup. We put together two recent short proofs, one by Skeide and one by Arveson, to obtain a still simpler one, which unfies the advantages of each proof and discards with their disadvantages.
Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems
Indian Statistical Institute, Delhi Centre（印度统计学院德里中心）；The Institute of Mathematical Sciences, A CI of Homi Bhabha National Institute（数学科学研究所，霍米·巴巴国立大学成员机构）
Cipriani and Sauvageot have shown that for any $L^2$-generator $L^{(2)}$ of a tracially symmetric quantum Markov semigroup on a C*-algebra $\mathcal{A}$ there exists a densely defined derivation $δ$ from $\mathcal{A}$ to a Hilbert bimodule $H$ such that $L^{(2)}=δ^*\circ \overlineδ$. Here we show that this construction of a derivation can in general not be generalised to quantum Markov semigroups that are symmetric with respect to a non-tracial state. In particular we show that all derivations to Hilbert bimodules can be assumed to have a concrete form, and then we use this form to show that i…
A product system E over a semigroup P is a family of Hilbert spaces {E_s:s\in P} together with multiplications E_s \times E_t\to E_{st}. We view E as a unitary- valued cocycle on P, and consider twisted crossed products A \times_{β,E} P involving E and an action βof P by endomorphisms of a C*-algebra A. When P is quasi-lattice ordered in the sense of Nica, we isolate a class of covariant representations of E, and consider a twisted crossed product B_P \times_{τ,E} P which is universal for covariant representations of E when E has finite-dimensional fibres, and in general is slightly larger. In…
We present a procedure to construct families of local, massive and interacting Haag-Kastler nets on the two-dimensional spacetime through an operator-algebraic method. An existence proof of local observable is given without relying on modular nuclearity. By a similar technique, another family of wedge-local nets is constructed using certain endomorphisms of conformal nets recently studied by Longo and Witten.
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 …
