We introduce stabilised property $\Gamma$, a $\mathrm {C}^*$-algebraic variant of $\Gamma$ which is invariant under stable isomorphism. then show that simple separable nuclear {C}^*$-algebras with and {Cu}(A) \cong \mathrm {Cu}(A \otimes \mathcal {Z})$ absorb the Jiang-Su algebra $\mathcal {Z}$ tensorially.

We provide the rigorous foundations for a categorical approach to classification of C*-dynamics up cocycle conjugacy. Given locally compact group $G$, we consider category (twisted) $G$-C*-algebras, where morphisms between two objects are allowed be equivariant maps or exterior equivalences, which leads concept so-called morphisms. An isomorphism in this is precisely conjugacy known sense. show...

We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C∗-algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to class...

Noncommutative multivariable versions of weighted shift operators arise naturally as ‘weighted’ left creation operators acting on the Fock space Hilbert space. We identify a natural notion of periodicity for these N tuples, and then find a family of inductive limit algebras determined by the periodic weighted shifts which can be regarded as noncommutative multivariable generalizations of the Bu...

A C∞-Hopf algebra is a C∞-algebra which is also a convenient Hopf algebra with respect to the structure induced by the evaluations of smooth functions. We characterize those C∞-Hopf algebras which are given by the algebra C∞(G) of smooth functions on some compact Lie group G, thus obtaining an anti-isomorphism of the category of compact Lie groups with a subcategory of convenient Hopf algebras.

we study topological von neumann regularity and principal von neumann regularity of banach algebras. our main objective is comparing these two types of banach algebras and some other known banach algebras with one another. in particular, we show that the class of topologically von neumann regular banach algebras contains all $c^*$-algebras, group algebras of compact abelian groups and cer...

We study automorphic Lie algebras and their applications to integrable systems. Automorphic are a natural generalisation of celebrated Kac-Moody the case when group automorphisms is not cyclic. They infinite dimensional almost graded. formulate concept graded isomorphism classify s l ( 2 , C ) based corresponding all finite reduction groups. show that hierarchies systems, Lax representations ma...