K-theory classification of graded ultramatricial algebras with involution

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 Γ is an abelian group, we show that {K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial {*}-algebras over a Γ-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 ({aa^{*}+bb^{*}=0} implies {a=b=0} for any {a,b\in A_{0}}) and {*}-pythagorean (for any {a,b\in A_{0}} one has {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 Γ-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. If {A_{0}} is 2-proper and {*}-pythagorean, we also show that two graded {*}-homomorphisms {\phi,\psi:R\to S} are such that {K_{0}^{\operatorname{gr}}(\phi)=K_{0}^{\operatorname{gr}}(\psi)} if and only if there is a unitary element u of degree zero in S such that {\phi(r)=u\psi(r)u^{*}} for any {r\in R}. 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 infinite 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. We also present examples which illustrate that {K_{0}^{\operatorname{gr}}} produces a finer invariant than {K_{0}}.