It is well known that a Lie groupoid G canonically defines both a C *-algebra C * (G) and a Poisson manifold A * (G). We show that the maps G → C * (G) and G → A * (G) are functorial with respect to suitably defined categories. The arrows (Hom-spaces) between Lie groupoids are taken to be isomorphism classes of regular bibundles (Hilsum–Skandalis maps), composed by a canonical bibundle tensor p...

It is well known that a measured groupoid G defines a von Neumann algebra W ∗(G), and that a Lie groupoid G canonically defines both a C∗-algebra C∗(G) and a Poisson manifold A∗(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C∗-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of obj...

(a) isomorphism ∼=, elementary equivalence ≡, elementary equivalence ≡α for Lω1,ω sentences of rank < α. (b) Isomorphism of countable graphs, linear orders, countable Boolean algebras is ≤B complete for orbit equivalence relations of continuous S∞ actions (≤B is Borel reducibility, S∞ is the Polish group of permutations of ω). (c) for ∼= is partially answered in computable model theory, with no...

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

The article establishes a necessary and sufficient condition for the isomorphism of log-algebras constructed on different von Neumann algebras by faithful normal finite trace.

A differential Azumaya algebra, and in particular a differential matrix algebra, over a differential field K with constants C is trivialized by a Picard–Vessiot (differential Galois) extension E. This yields a bijection between isomorphism classes of differential algebras and Picard–Vessiot cocycles Z(G(E/K), PGLn(C)) which cobound in Z (G(E/K), PGLn(E)).

We introduce a method that produces bijection between the posets silt − A $\textrm {{{\bf silt}}-}{A}$ and B silt}}-}{B}$ formed by isomorphism classes of basic silting complexes over finite-dimensional k $k$ -algebras $A$ $B$ , lifting to two [ X ] $k[\![X]\!]$ -orders which are isomorphic as rings. apply this class algebras generalising Brauer graph weighted surface algebras, showing their mu...