We give a new very concrete description of the C*envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid C*algebra. In the non-surjective case, it is a full corner of a such an algebra. We also show that when the space is compact, then the C*-envelope is simple if and only if the system is minimal.2000 Mathematics Subject Classification. 47L55, 47L40, 46L05, 37B20, 37B99.

Abstract. We show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate topological graphs yield completely isometrically isomorphic tensor algebras.

Abstract. We consider the Schatten spaces S p in the framework of operator space theory and for any 1 ≤ p = 2 < ∞, we characterize the completely 1-complemented subspaces of S p . They turn out to be the direct sums of spaces of the form S p (H, K), where H, K are Hilbert spaces. This result is related to some previous work of Arazy and Friedman giving a description of all 1-complemented subspaces of S p in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L p -spaces. Also we show that for any n ≥ 2, there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2n which is not completely isometric, and we investigate L p -versions of such isomorphisms.

We give an operator space characterization of subalgebras of C(Ω, M n ). We also describe injective subspaces of C(Ω, M n ) and then give applications to sub-TROs of C(Ω, M n ). Finally, we prove an 'n-minimal version' of the Christensen-Effros-Sinclair representation theorem.

We prove that von Neumann algebras and separable nuclear C * -algebras are stable for the Banach-Mazur cb-distance. A technical step is to show that unital almost completely isometric maps between C * -algebras are almost multiplicative and almost selfadjoint. Also as an intermediate result, we compare the Banach-Mazur cb-distance and the Kadison-Kastler distance. Finally, we show that if two C * -algebras are close enough for the cb-distance, then they have at most the same length.