K-theory classification of graded ultramatricial algebras with involution

Abstract

Abstract We consider a generalization K 0 gr ⁢ ( R ) {K_{0}^{\operatorname{gr}}(R)} of the standard Grothendieck group K 0 ⁢ ( R ) {K_{0}(R)} of a graded ring R with involution. If Γ is an abelian group, we show that K 0 gr {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 {A_{0}} is 2-proper ( a ⁢ a * + b ⁢ b * = 0 {aa^{*}+bb^{*}=0} implies a = b = 0 {a=b=0} for any a , b ∈ A 0 {a,b\in A_{0}} ) and * {*} -pythagorean (for any a , b ∈ A 0 {a,b\in A_{0}} one has a ⁢ a * + b ⁢ b * = c ⁢ c * {aa^{*}+bb^{*}=cc^{*}} for some c ∈ A 0 {c\in A_{0}} ). If the involutive structure is not considered, our result implies that K 0 gr {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 gr ⁢ ( R ) → K 0 gr ⁢ ( S ) {f:K_{0}^{\operatorname{gr}}(R)\to K_{0}^{\operatorname{gr}}(S)} is a contractive ℤ ⁢ [ Γ ] {\mathbb{Z}[\Gamma]} -module homomorphism, we present a specific formula for a graded * {*} -homomorphism ϕ : R → S {\phi:R\to S} with K 0 gr ⁢ ( ϕ ) = f {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 {A_{0}} is 2-proper and * {*} -pythagorean, we also show that two graded * {*} -homomorphisms ϕ , ψ : R → S {\phi,\psi:R\to S} are such that K 0 gr ⁢ ( ϕ ) = K 0 gr ⁢ ( ψ ) {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 ϕ ⁢ ( r ) = u ⁢ ψ ⁢ ( r ) ⁢ u * {\phi(r)=u\psi(r)u^{*}} for any r ∈ R {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 ) {L_{K}(E)} and L K ⁢ ( F ) {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 gr {K_{0}^{\operatorname{gr}}} produces a finer invariant than K 0 {K_{0}} .