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Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has a unique conditional expectation, and when it has a unique pseudo-expectation (in the sense of Pitts). Likewise for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. As an application, we (slightly) strengthen results of Kishimoto and Archbold-Spielberg concerning $C^*$-simplicity of $\mathcal{A} \rtimes_r G$.

Starting from an arbitrary endomorphism \alpha of a unital C*-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C*-dynamical system (A,\alpha) but also on the choice of an ideal orthogonal to kernel of \alpha. The article gives an explicit description of the internal structure of this crossed product and, in particular, discusses the interrelation between relative Cuntz-Pimsner algebras and partial isometric crossed products. We present a canonical procedure that reduces any given C*-correspondence to the 'smallest' C*-correspondence yielding the same relative Cuntz-Pimsner algebra as the initial one. In the context of crossed products this reduction procedure corresponds to the reduction of C*-dynamical systems and allow us to establish a coincidence between relative Cuntz-Pimsner algebras and crossed products introduced., Comment: The article is based on papers arXiv:math.OA/0703801 and arXiv:math.OA/0704.3811, and in essence forms their unification, refinement and development

If X is a compact Hausdorff space, supplied with a homeomorphism, then a crossed product involutive Banach algebra is naturally associated with these data. If X consists of one point, then this algebra is the group algebra of the integers. In this paper, we study spectral synthesis for the closed ideals of this associated algebra in two versions, one modeled after C(X), and one modeled after the group algebra of the integers. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when the homeomorphism has no periodic points., Comment: 28 pages

Given a closed ideal I in a C*-algebra A, an ideal J (not necessarily closed) in I, a *-homomorphism \al:A --> M(I) and a map L:J --> A with some properties, based on [3] and [9] we define a C*-algebra O(A,\al,L) which we call the "Crossed Product by a Partial Endomorphism." In the second section we introduce the Crossed Product by a Partial Endomorphism O(X,\al,L) induced by a local homeomorphism \sigma:U --> X where X is a compact Hausdorff space and U is an open subset of X.The main result of this section is that every nonzero gauge invariant ideal of O(X,\al,L) has nonzero intersection with C(X). We present the example which motivated this work, the Cuntz-Krieger algebra for infinite matrices. We show in the third section a bijection between the gauge invariant ideals of O(X,\al,L) and the \sigma,\sigma^{-1} -invariant open subsets of X. The last section is dedicated to the study of O(X,\al,L) in the case where the pair (X,\sigma) has an extra property, wich we call topological freeness. We prove that in this case every nonzero ideal of O(X,\al,L) has nonzero intersection with C(X). If moreover (X,\sigma) has the property that (X',\sigma_X') is topologically free for each closed sigma,\sigma^{-1}-invariant subset X' of X then we obtain a bijection between the ideals of O(X,\al,L) and the open \sigma,\sigma^{-1}-invariant subsets of X. We conclude this section by showing a simplicity criteria for the Cuntz-Krieger algebras for infinite matrices., Comment: 31 pages, no figures

Let $M$ be a von Neumann algebra, let $\alpha$ be a $*$ -automorphism of $M$ , and let $M\mathrm{x}_{\alpha}Z$ be the crossed product determined by $M$ and $\alpha$ . In this paper, considering the Cholesky decomposition for a positive operator in $M\mathrm{x}_{\alpha}Z$ , we give a factorization theorem for positive operators in $M\mathrm{x}_{\alpha}Z$ with respect to analytic crossed product $M\mathrm{x}_{\alpha}Z_{+}$ determined by $M$ and $\alpha$ . And we give a necessary and sufficient condition that every positive operator in $M\mathrm{x}_{\alpha}Z$ can be factored by the form $A$ $A$ , where $A$ belongs to $M\mathrm{x}_{\alpha}Z_{+}\cap(M\mathrm{x}_{\alpha}Z_{+})^{-1}$ .