One of the key ingredients of A. Connes’ noncommutative geometry is a generalized Dirac operator which induces a metric(Connes’ distance) on the state space. We generalize such a Dirac operator devised by A. Dimakis et al , whose Connes’ distance recovers the linear distance on a 1D lattice, into 2D lattice. This Dirac operator being “naturally” defined has the “local eigenvalue property” and i...

Abstract. We reformulate the Baum-Connes conjecture with coe cients by introducing a new crossed product functor for C⇤-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to...

let $a$ be an arbitrary banach algebra and $varphi$ a homomorphism from $a$ onto $bbb c$. our first purpose in this paper is to give some equivalent conditions under which guarantees a $varphi$-mean of norm one. then we find some conditions under which there exists a $varphi$-mean in the weak$^*$ cluster of ${ain a; |a|=varphi(a)=1}$ in $a^{**}$.

For horocyclic products of percolation subtrees of regular trees, we show almost sure amenability. Under a symmetry condition concerning the growth of the two percolation trees, we show the existence of an increasing Følner sequence (which we call strong amenability).