K-theory Classification of Graded Ultramatricial Algebras with Involution

Abstract

We consider a generalization $K_0^{\operatorname{gr}}(R)$ of the standard Grothendieck group $K_0(R)$ of a graded ring $R$ with involution. If $\Gamma$ is an abelian group, we show that $K_0^{\operatorname{gr}}$ completely classifies graded ultramatricial $*$-algebras over a $\Gamma$-graded $*$-field $A$ such that (1) each nontrivial graded component of $A$ has a unitary element in which case we say that $A$ has enough unitaries, and (2) the zero-component $A_0$ is 2-proper (for any $a,b\in A_0,$ $aa^*+bb^*=0$ implies $a=b=0$) and $*$-pythagorean (for any $a,b\in A_0,$ $aa^*+bb^*=cc^*$ for some $c\in A_0$). If the involutive structure is not considered, our result implies that $K_0^{\operatorname{gr}}$ completely classifies graded ultramatricial algebras over any graded field $A.$ If the grading is trivial and the involutive structure is not considered, we obtain some well known results as corollaries. If $R$ and $S$ are graded matricial $*$-algebras over a $\Gamma$-graded $*$-field $A$ with enough unitaries and $f: K_0^{\operatorname{gr}}(R)\to K_0^{\operatorname{gr}}(S)$ is a contractive $\mathbb Z[\Gamma]$-module homomorphism, we present a specific formula for a graded $*$-homomorphism $\phi: R\to S$ with $K_0^{\operatorname{gr}}(\phi) = f.$ If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. As an application of our results, we show that the graded version of the Isomorphism Conjecture holds for a class of Leavitt path algebras: if $E$ and $F$ are countable, row-finite, no-exit graphs in which every path ends in a sink or a cycle and $K$ is a 2-proper and $*$-pythagorean field, then the Leavitt path algebras $L_K(E)$ and $L_K(F)$ are isomorphic as graded rings if any only if they are isomorphic as graded $*$-algebras.