Hsiang-Ping Huang
For any finite dimensional C^*-algebra A, we give an endomorphism Φof the hyperfinite II_1 factor R of finite Jones index such that: for all k \in \mathbb {N}, Φ^k (R)' \cap R= \otimes^k A. The Jones index [R: Φ(R)]= (rank (A))^2, here rank (A) is the dimension of the maximal abelian subalgebra of A.
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For any finite dimensional C^*-algebra A with a trace vector \vec s whose entries are rational numbers, we give an endomorphism Φof the hyperfinite II_1 factor R such that: for all k \in \mathbb {N}, Φ^k (R)' \cap R= \otimes^k A. The canonical trace τon R extends the trace vector \vec s on A. Therefore the minimal projection is not necessarily equivalent to each other.
We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions. Following the automorphism case, we introduce two notions for endomorphisms of type III factors: modular endomorphisms and Connes-Takesaki modules. Several applications to compact groups of automorphisms and subfactors of type III factors are given from the viewpoint of ergodic theory.
We characterize the condition for two finite index endomorphisms on an AFD factor to be approximately unitarily equivalent. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi--Sutherland--Takesaki and Masuda--Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.
It is a well-known fact that endomorphisms of $B(H)$ are intimately connected with families of mutually orthogonal isometries, i.e. with representations of the so-called Toeplitz $C^*$-algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^*$-algebras associated to graphs and endomorphisms of certain von Neumann subalgebras of $B(H)$. Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.
We introduce the notions of approximate innerness and central triviality for endomorphisms on separable von Neumann factors, and we characterize them for hyperfinite factors by Connes-Takesaki modules of endomorphisms and modular endomorphisms which are introduced by Izumi. Our result is a generalization of the corresponding result obtained by Kawahigashi-Sutherland-Takesaki in automorphism case.
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…
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…
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…
For any finite dimensional C*-algebra A with any trace vector {\vec s} whose components are rational numbers, we give an endomorphism Φ of the hyperfinite II_1 factor R such that: forall k in {\mathbb N} Φ^k (R)' \cap R= \otimes^k A The canonical trace τ on R extends the trace vector {\vec s} on A. As a corollary, we construct a one-parameter family of inclusions of hyperfinite II_1 factors N^λ \subset M^λ with trivial relative commutant (N^λ)' \cap M^λ= {\mathbb C} and with the Jones index [M^λ: N^λ]= λ^{-1} \in (4, \infty) \cap {\mathbb Q} This partially solves the problem of finding all pos…
We compute the index of the inclusions of type $III_{1/2}$ factors arising from endomorphisms of the Cuntz algebra ${\mathcal O}_2$ associated to the rank-two permutation matrices.
Normal endomorphisms of von Neumann algebras need not be extendable to automorphisms of a larger von Neumann algebra, but they always have asymptotic lifts. We describe the structure of endomorphisms and their asymptotic lifts in some detail, and apply those results to complete the identification of asymptotic lifts of unital completely positive linear maps on von Neumann algebras in terms of their minimal dilations to endomorphisms.
We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type $\rm{I}\sb{\infty}.$
In this work we construct a C*-algebra from an injective endomorphisms of some group G, allowing the endomorphism to have infinite cokernel. We generalize results obtained by I. Hirshberg and also by J. Cuntz and A. Vershik. In good cases we show that the C*-algebra that we study is classifiable by Kirchberg's classification theorem, with K-groups equal to K*(C*(G)).
In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras $Ø_θ$of a 2-graph $\Fth$ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of $Ø_θ$ and its unitary pairs with a \textit{twisted property}. We characterize when endomorphisms preserve the fixed point algebra $\fF$ of the gauge automorphisms and its canonical masa $\fD$. Some other properties of endomorphisms are also investigated. As far as the modular theory of $Ø_θ$ is concerned, we show that the algebraic *-algebra generated by the generators of $Ø_…
The problem of existence of standard (i.e. product-type) invariant MASAs for endomorphisms of the Cuntz algebra O_n is studied. In particular endomorphisms which preserve the canonical diagonal MASA D_n are investigated. Conditions on a unitary in O_n equivalent to the fact that the corresponding endomorphism preserves D_n are found, and it is shown that they may be satisfied by unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms which leave all standard MASAs invariant and have identical actions on them are characterized. Finally some properties of examples of finite-…
In our previous articles, we have presented a class of endomorphisms of the Cuntz algebras which are defined by polynomials of canonical generators and their conjugates. We showed the classification of some case under unitary equivalence by help of branching laws of permutative representations. In this article, we construct an automaton which is called the Mealy machine associated with the endomorphism in order to compute its branching law. We show that the branching law is obtained as outputs from the machine for the input of information of a given representation.
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.
We give a new definition for the crossed-product of a C*-algebra A by a *-endomorphism α, which depends not only on the pair (A,α) but also on the choice of a transfer operator (defined in the paper). With this we generalize some of the earlier constructions in the situations in which they behave best (e.g. for monomorphisms with hereditary range), but we get a different and perhaps more natural outcome in other situations. For example, we show that the Cuntz-Krieger algebra O_A arises as the result of our construction when applied to the corresponding Markov subshift and a very natural transf…
We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that the endomorphism itself becomes induced by the bimodule of continuous sections of a vector bundle. Some motivating examples for such a construction are given. Furthermore, we study the C*-algebra of G-invariant elements of the Cuntz-Pimsner algebra associated with a G-vector bundle, where G is a (noncompact, in general) group. In particular, the C*-algebra of invariant elements w.r.t. the action of the group of special unitaries of the given vector bundle is a crossed product in the above sense. We also s…
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