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K-theory classification of graded ultramatricial algebras with involution

  • Roozbeh Hazrat ORCID logo und Lia Vaš ORCID logo EMAIL logo
Veröffentlicht/Copyright: 13. November 2018
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 31 Heft 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.

References

[1] G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293 (2005), no. 2, 319–334. 10.1016/j.jalgebra.2005.07.028Suche in Google Scholar

[2] G. Abrams, G. Aranda Pino, F. Perera and M. Siles Molina, Chain conditions for Leavitt path algebras, Forum Math. 22 (2010), no. 1, 95–114. 10.1515/forum.2010.005Suche in Google Scholar

[3] G. Abrams and M. Tomforde, Isomorphism and Morita equivalence of graph algebras, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3733–3767. 10.1090/S0002-9947-2011-05264-5Suche in Google Scholar

[4] P. Ara, Matrix rings over *-regular rings and pseudorank functions, Pacific J. Math. 129 (1987), no. 2, 209–241. 10.2140/pjm.1987.129.209Suche in Google Scholar

[5] P. Ara, M. A. Moreno and E. Pardo, Nonstable K-theory for graph algebras, Algebr. Represent. Theory 10 (2007), no. 2, 157–178. 10.1007/s10468-006-9044-zSuche in Google Scholar

[6] G. Aranda Pino, K. Rangaswamy and L. Vaš, *-regular Leavitt path algebras of arbitrary graphs, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 5, 957–968. 10.1007/s10114-011-0106-8Suche in Google Scholar

[7] G. Aranda Pino and L. Vaš, Noetherian Leavitt path algebras and their regular algebras, Mediterr. J. Math. 10 (2013), no. 4, 1633–1656. 10.1007/s00009-013-0320-ySuche in Google Scholar

[8] A. A. Baranov, Classification of the direct limits of involution simple associative algebras and the corresponding dimension groups, J. Algebra 381 (2013), 73–95. 10.1016/j.jalgebra.2013.02.004Suche in Google Scholar

[9] S. K. Berberian, Baer *-rings, Grundlehren Math. Wiss. 195, Springer, New York, 1972. 10.1007/978-3-642-15071-5Suche in Google Scholar

[10] S. K. Berberian, Baer rings and Baer *-rings, preprint (1988), https://www.ma.utexas.edu/mp_arc/c/03/03-181.pdf. Suche in Google Scholar

[11] K. R. Davidson, C*-algebras by Example, Fields Inst. Monogr. 6, American Mathematical Society, Providence, 1996. 10.1090/fim/006Suche in Google Scholar

[12] G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. 10.1016/0021-8693(76)90242-8Suche in Google Scholar

[13] J. Gabe, E. Ruiz, M. Tomforde and T. Whalen, K-theory for Leavitt path algebras: Computation and classification, J. Algebra 433 (2015), 35–72. 10.1016/j.jalgebra.2015.03.009Suche in Google Scholar

[14] K. R. Goodearl, Von Neumann Regular Rings, 2nd ed., Robert E. Krieger, Malabar, 1991. Suche in Google Scholar

[15] K. R. Goodearl and D. E. Handelman, Classification of ring and C*-algebra direct limits of finite-dimensional semisimple real algebras, Mem. Amer. Math. Soc. 69 (1987), no. 372, 1–147. 10.1090/memo/0372Suche in Google Scholar

[16] D. Handelman, Coordinatization applied to finite Baer *-rings, Trans. Amer. Math. Soc. 235 (1978), 1–34. 10.1090/S0002-9947-1978-0463230-1Suche in Google Scholar

[17] R. Hazrat, The graded Grothendieck group and the classification of Leavitt path algebras, Math. Ann. 355 (2013), no. 1, 273–325. 10.1007/s00208-012-0791-3Suche in Google Scholar

[18] R. Hazrat, The graded structure of Leavitt path algebras, Israel J. Math. 195 (2013), no. 2, 833–895. 10.1007/s11856-012-0138-5Suche in Google Scholar

[19] R. Hazrat, Graded Rings and Graded Grothendieck Groups, London Math. Soc. Lecture Note Ser. 435, Cambridge University Press, Cambridge, 2016. 10.1017/CBO9781316717134Suche in Google Scholar

[20] Z. Mesyan and L. Vaš, Traces on semigroup rings and Leavitt path algebras, Glasg. Math. J. 58 (2016), no. 1, 97–118. 10.1017/S0017089515000087Suche in Google Scholar

[21] N. Prijatelj and I. Vidav, On special *-regular rings, Michigan Math. J. 18 (1971), 213–221. 10.1307/mmj/1029000680Suche in Google Scholar

[22] D. Quillen, Higher algebraic K-theory. I, Algebraic K-Theory. I: Higher K-Theories (Seattle 1972), Lecture Notes in Math. 341, Springer, Berlin (1973), 85–147. 10.1007/BFb0067053Suche in Google Scholar

[23] E. Ruiz and M. Tomforde, Classification of unital simple Leavitt path algebras of infinite graphs, J. Algebra 384 (2013), 45–83. 10.1016/j.jalgebra.2013.03.004Suche in Google Scholar

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

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Graded Grothendieck group; classification; graded ring; involutive ring; ultramatricial algebras; Leavitt path algebra; isomorphism conjecture

Artikel in diesem Heft

  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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