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K-theory classification of graded ultramatricial algebras with involution

  • Roozbeh Hazrat ORCID logo and Lia Vaš ORCID logo EMAIL logo
Published/Copyright: November 13, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 2

Abstract

We consider a generalization K0gr⁢(R) of the standard Grothendieck group K0⁢(R) of a graded ring R with involution. If Γ is an abelian group, we show that K0gr 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 (a⁢a*+b⁢b*=0 implies a=b=0 for any a,b∈A0) and *-pythagorean (for any a,b∈A0 one has a⁢a*+b⁢b*=c⁢c* for some c∈A0). If the involutive structure is not considered, our result implies that K0gr 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:K0gr⁢(R)→K0gr⁢(S) is a contractive ℤ⁢[Γ]-module homomorphism, we present a specific formula for a graded *-homomorphism ϕ:R→S with K0gr⁢(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 K0gr⁢(ϕ)=K0gr⁢(ψ) if and only if there is a unitary element u of degree zero in S such that ϕ⁢(r)=u⁢ψ⁢(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 K0gr produces a finer invariant than K0.

Keywords: Graded Grothendieck group; classification; graded ring; involutive ring; ultramatricial algebras; Leavitt path algebra; isomorphism conjecture
MSC 2010: 16W50; 16W10; 16W99; 16S99; 16D70

Communicated by Manfred Droste

Funding source: Australian Research Council

Award Identifier / Grant number: DP150101598 and DP160101481

Funding statement: The first author would like to acknowledge Australian Research Council grant DP160101481. A part of this work was done at the University of Bielefeld, where the first author was a Humboldt Fellow.

Acknowledgements

We would like to thank the referee for a careful reading, insightful comments and useful suggestions.

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Received: 2017-12-28
Revised: 2018-08-27
Published Online: 2018-11-13
Published in Print: 2019-03-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
  3. Infinite families of equivariantly formal toric orbifolds
  4. Bounds for GL3L-functions in depth aspect
  5. Nonlinear Dirichlet problems with unilateral growth on the reaction
  6. K-invariant cusp forms for reductive symmetric spaces of split rank one
  7. Cellularity in quotient spaces of topological groups
  8. Shifted convolution sums for higher rank groups
  9. Dimension quotients, Fox subgroups and limits of functors
  10. Large sums of Hecke eigenvalues of holomorphic cusp forms
  11. K-theory classification of graded ultramatricial algebras with involution
  12. Regularity of symbolic powers and arboricity of matroids
  13. On some problems concerning symmetrization operators
  14. An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
  15. Rankin–Selberg gamma factors of level zero representations of GLn
  16. Sobolev’s inequality for double phase functionals with variable exponents
  17. Independence of Artin L-functions
  18. Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
https://doi.org/10.1515/forum-2017-0268
Keywords for this article
Graded Grothendieck group; classification; graded ring; involutive ring; ultramatricial algebras; Leavitt path algebra; isomorphism conjecture

Articles in the same Issue

  1. Frontmatter
  2. Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
  3. Infinite families of equivariantly formal toric orbifolds
  4. Bounds for GL3L-functions in depth aspect
  5. Nonlinear Dirichlet problems with unilateral growth on the reaction
  6. K-invariant cusp forms for reductive symmetric spaces of split rank one
  7. Cellularity in quotient spaces of topological groups
  8. Shifted convolution sums for higher rank groups
  9. Dimension quotients, Fox subgroups and limits of functors
  10. Large sums of Hecke eigenvalues of holomorphic cusp forms
  11. K-theory classification of graded ultramatricial algebras with involution
  12. Regularity of symbolic powers and arboricity of matroids
  13. On some problems concerning symmetrization operators
  14. An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
  15. Rankin–Selberg gamma factors of level zero representations of GLn
  16. Sobolev’s inequality for double phase functionals with variable exponents
  17. Independence of Artin L-functions
  18. Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
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