We classify all essential extensions of the form $$ 0 \rightarrow \mathcal{B} \mathcal{D} C(X) where $\mathcal{B}$ is a nonunital, simple, separable, finite, real rank zero, $\mathcal{Z}$-stable $C^\*$-algebra with continuous scale, and $X$ finite CW complex. In fact, we prove that there group isomorphism \mathbf{Ext}(C(X), \mathcal{B}) KK(C(X), \mathcal{M}(\mathcal{B})/\mathcal{B}).

Let $f\colon S'\longrightarrow S$ be a cyclic branched covering of smooth projective surfaces over $\mathbb{C}$ whose branch locus $\Delta\subset is ample divisor. Pick very complete linear system $|\mathcal{H}|$ on $S$, such that the polarized surface $(S, |\mathcal{H}|)$ not scroll nor has rational hyperplane sections. For general member $[C]\in|\mathcal{H}|$ consider $\mu_{n}$-equivariant is...

Let $P$ be a submonoid of group $G$ and let $\mathcal{E}=(\mathcal{E}_p)_{p\in P}$ product system over with coefficient C*-algebra $A$. We show that the following C*-algebras are canonically isomorphic: C*-envelope tensor algebra $\mathcal{T}_\lambda(\mathcal{E})^+$ $\mathcal{E}$; reduced cross sectional Fell bundle associated to canonical coaction on covariance $A\times_{\mathcal{E}}P$ cosyste...

‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

‎Let $mathcal A$ and $mathcal B$ be unital rings‎, ‎and $mathcal M$‎ ‎be an $(mathcal A‎, ‎mathcal B)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal A$-module and also as a right $mathcal B$-module‎. ‎Let ${mathcal U}=mbox{rm Tri}(mathcal A‎, ‎mathcal M‎, ‎mathcal‎ ‎B)$ be the triangular ring and ${mathcal Z}({mathcal U})$ its‎ ‎center‎. ‎Assume that $f:{mathcal U}rightarrow{mathcal U}$ is...