Given a compact space X and two commuting continuous open surjective maps sigma_1, sigma_2 : X --> X, we construct certain C*-algebras that reflect the dynamics of the N^2-action. When the maps sigma_1, sigma_2 are local homeomorphisms, these are groupoid algebras, but in general, we will use a Cuntz-Pimsner algebra associated to a product system of Hilbert bimodules in the sense of Fowler.
The motivating example for our construction is the dynamical system associated with a rank two graph by Kumjian and Pask. We consider also a two-dimensional subshift of Ledrappier, the case of two covering maps of the circle, and the two-dimensional Bernoulli shift.
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).
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 $\rm{I}\sb{\infty}.$
It is shown that the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism is purely infinite when it is simple, provided the endomorphism does not leave a trace state invariant and maps the unit to a full projection.
It is shown that Voiculescu's toplogical entropy for the canonical endomorphism of a simple Cuntz-Krieger algebra O_A equals the logarithm of the spectral radius of A.
We classify spectrum-preserving endomorphisms of stable continuous-trace C^*-algebras up to inner automorphism by a surjective multiplicative invariant taking values in finite dimensional vector bundles over the spectrum. Specializing to automorphisms, this gives a different approach to results of Phillips and Raeburn.
Let α:G --> G be an endomorphism of a discrete amenable group such that [G:α(G)]<infinity. We study the structure of the C^* algebra generated by the left convolution operators acting on the left regular representation space, along with the isometry of the space induced by the endomorphism.
In this paper we extend the constructions of Boava and Exel to present the C*-algebra associated with an injective endomorphism of a group with finite cokernel as a partial group algebra and consequently as a partial crossed product. With this representation we present another way to study such C*-algebras, only using tools from partial crossed products.
We present a different way to study the C*-algebra associated with an injective endomorphism of a group G of infinite cokernel. We follow the work of Boava and Exel to construct a partial crossed product representation of that C*-algebra and show properties only using such tools.
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.
In this paper we define a Rokhlin property for automorphisms of non-unital C*-algebras and for endomorphisms. We show that the crossed product of a C*-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing C*-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Winter.
We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that it becomes induced by a Hilbert C(X)-bimodule. Furthermore we introduce the notion of C(X)-category, and discuss relationships with crossed products and duality for compact groups.
A study of noncommutative topological entropy of gauge invariant endomorphisms of Cuntz algebras began in our earlier work with Joachim Zacharias is continued and extended to endomorphisms which are not necessarily of permutation type. In particular it is shown that if H is an N-dimensional Hilbert space, V is an irreducible multiplicative unitary on the tensor product of H with itself and F is the tensor flip, then the Voiculescu entropy of the Longo's canonical endomorphism associated with the unitary VF is equal to log N.
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.
A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a uniq…
It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms $λ_u$ of the Cuntz algebra $\mathcal{O}_n$ such that $λ_u (\mathcal{F}_n)\subseteq\mathcal{F}_n$ but $u\not\in\mathcal{F}_n$, and a question was raised if for such a $u$ there must always exist a unitary $v\in\mathcal{F}_n$ with $λ_u|_{\mathcal{F}_n} = λ_v|_{\mathcal{F}_n}$. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms $λ_u$ for which the relative commutant $λ_u(\mathcal{F}_n)'\cap\mathcal{F}_n$ is finite dimensional.
We study the crossed product $C^*$-algebra associated to injective endomorphisms, which turns out to be equivalent to study the crossed product by the dilated autormorphism. We prove that the dilation of the Bernoulli $p$-shift endomorphism is topologically free. As a consequence, we have a way to twist any endomorphism of a $\D$-absorbing $C^*$-algebra into one whose dilated automorphism is essentially free and have the same $K$-theory map than the original one. This allows us to construct purely infinite crossed products $C^*$-algebras with diverse ideal structures.
We consider non-zero endomorphisms of the Dales and Davie algebras of infinitely differentiable functions on intervals in the real line. We discuss necessary and sufficient conditions for a selfmap of the interval to induce a compact endomorphism of these algebras. We also determine the spectra of these compact endomorphisms.
Noncommutative topological entropy estimates are obtained for polynomial gauge invariant endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values for the entropy are computed for a class of permutative endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura.
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)).
We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra $\mathcal{Q}(H)$ is unitarily equivalent to an endomorphism of $\mathcal{Q}(H)$ which is liftable to a unital endomorphism of $\mathcal{B}(H)$. We use this result to classify all unital endomorphisms of $\mathcal{Q}(H)$ up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of $\mathrm{C}^\ast$-algebras that embed into $\mathcal{Q}(H)$ is not closed under tensor product nor countable inductive…
We investigate the relationship between endomorphisms of the Cuntz algebra ${\mathcal O}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$ represented inside the unitary group of ${\mathcal O}_2$. For an endomorphism $λ_u$ of ${\mathcal O}_2$, we show that $λ_u(V)\subseteq V$ if and only if $u\in V$. If $λ_u$ is an automorphism of ${\mathcal O}_2$ then $u\in V$ is equivalent to $λ_u(F)\subseteq V$. Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of ${\mathcal O}_n$, whose properties and examples are investigated.
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 investigate polynomial endomorphisms of graph $C^*$-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the dynamics induced this way on the space of infinite paths (the spectrum of the MASA) is generated by an asynchronous transducer.
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_…
A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra $\mathcal{D}_2$ in the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$ is proved not to be regular. Finally, countably many novel permutative endomorphisms of $\mathcal{Q}_2$ are exhibited with prescribed images of the generator $U$.
Using Poincare duality, we formulate a formula of Lefschetz type which computes the Lefschetz number of an endomorphism of a separable, nuclear C*-algebra satisfying Poincare duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with the complex numbers, as in the classical case.) We then consider endomorphisms of Cuntz-Krieger algebras O_A. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of th…
We consider commuting squares of finite dimensional von Neumann algebras having the algebra of complex numbers in the lower left corner. Examples include the vertex models, the spin models (in the sense of subfactor theory) and the commuting squares associated to finite dimensional Kac algebras. To any such commuting square we associate a compact Kac algebra and we compute the corresponding subfactor and its standard invariant in terms of it.
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-…
We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D_E of C*(E). Our results pertain both automorphisms and proper endomorphisms. Firstly, the Weyl group and the restricted Weyl group of a graph C*-algebra are introduced and investigated. In particular, criteria of outerness for automorphisms in the restricted Weyl group are found. We also show that the…
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…
Previously, we have shown that the CAR algebra for fermions is embedded in the Cuntz algebra ${\cal O}_{2}$ in such a way that the generators are expressed in terms of polynomials in the canonical generators of the latter, and it coincides with the U(1)-fixed point subalgebra ${\cal A}\equiv {\cal O}_{2}^{U(1)}$ of ${\cal O}_{2}$ for the canonical gauge action. Based on this embedding formula, some properties of ${\cal A}$ are studied in detail by restricting those of ${\cal O}_{2}$. Various endomorphisms of ${\cal O}_{2}$, which are defined by polynomials in the canonical generators, are expl…
Based on an embedding formula of the CAR algebra into the Cuntz algebra ${\mathcal O}_{2^p}$, properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various $\ast$-endomorphisms of the Cuntz algebra are explicitly constructed, and transcribed into those of the CAR algebra. In particular, a set of $\ast$-endomorphisms of the CAR algebra into its even subalgebra are constructed. According to branching formulae, which are obtained by composing representations and $\ast$-endomorphisms, it is shown that a KMS state of the CAR algebra is obtained through the a…
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…
