K-theory classification of graded ultramatricial algebras with involution

Roozbeh Hazrat, Lia Vaš

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

We consider a generalization Kgr0(R) of the standard Grothendieck group K0(R) of a graded ring R with involution. If Γ is an abelian group, we show that Kgr0 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 A0 is 2-proper (aa∗+bb∗=0 implies a=b=0 for any a,b∈A0) and *-pythagorean (for any a,b∈A0 one has aa∗+bb∗=cc∗ for some c∈A0). If the involutive structure is not considered, our result implies that Kgr0 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:Kgr0(R)→Kgr0(S) is a contractive Z[Γ]-module homomorphism, we present a specific formula for a graded *-homomorphism ϕ:R→S with Kgr0(ϕ)=f. If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. If A0 is 2-proper and *-pythagorean, we also show that two graded *-homomorphisms ϕ,ψ:R→S are such that Kgr0(ϕ)=Kgr0(ψ) if and only if there is a unitary element u of degree zero in S such that ϕ(r)=uψ(r)u∗ for any r∈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 LK(E) and LK(F) are isomorphic as graded rings if any only if they are isomorphic as graded *-algebras. We also present examples which illustrate that Kgr0 produces a finer invariant than K0.
Original languageEnglish
Pages (from-to)419-463
Number of pages43
JournalForum Mathematicum
Volume31
Issue number2
DOIs
Publication statusPublished - 2018

Keywords

  • abelian groups
  • algebra
  • graded rings
  • rings with involution

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