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Tensor products of Steinberg algebras

Simon Rigby (UGent)
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Abstract
We prove that A(R)(G) circle times(R) A(R)(H) congruent to A(R)(G x H) if G and H are Hausdor ff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between L-2,L-R circle times L-3,L-R and L-2,L- R circle times L-2,L-R. In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every *-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that L-2,L- Z circle times L-3,L- Z not congruent to L-2,L-Z circle times L-2,L- Z (as*-rings).
Keywords
Steinberg algebras, ample groupoids, Leavitt algebras, diagonal-preserving isomorphisms, ETALE GROUPOID ALGEBRAS, LEAVITT PATH ALGEBRAS, INVERSE SEMIGROUP, ASTERISK-ISOMORPHISM, SIMPLICITY

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MLA
Rigby, Simon. “Tensor Products of Steinberg Algebras.” JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, vol. 111, no. 1, 2021, pp. 111–26, doi:10.1017/S1446788719000302.
APA
Rigby, S. (2021). Tensor products of Steinberg algebras. JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 111(1), 111–126. https://doi.org/10.1017/S1446788719000302
Chicago author-date
Rigby, Simon. 2021. “Tensor Products of Steinberg Algebras.” JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY 111 (1): 111–26. https://doi.org/10.1017/S1446788719000302.
Chicago author-date (all authors)
Rigby, Simon. 2021. “Tensor Products of Steinberg Algebras.” JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY 111 (1): 111–126. doi:10.1017/S1446788719000302.
Vancouver
1.
Rigby S. Tensor products of Steinberg algebras. JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY. 2021;111(1):111–26.
IEEE
[1]
S. Rigby, “Tensor products of Steinberg algebras,” JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, vol. 111, no. 1, pp. 111–126, 2021.
@article{8616810,
  abstract     = {{We prove that A(R)(G) circle times(R) A(R)(H) congruent to A(R)(G x H) if G and H are Hausdor ff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between L-2,L-R circle times L-3,L-R and L-2,L- R circle times L-2,L-R. In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every *-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that L-2,L- Z circle times L-3,L- Z not congruent to L-2,L-Z circle times L-2,L- Z (as*-rings).}},
  articleno    = {{PII S1446788719000302}},
  author       = {{Rigby, Simon}},
  issn         = {{1446-7887}},
  journal      = {{JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY}},
  keywords     = {{Steinberg algebras,ample groupoids,Leavitt algebras,diagonal-preserving isomorphisms,ETALE GROUPOID ALGEBRAS,LEAVITT PATH ALGEBRAS,INVERSE SEMIGROUP,ASTERISK-ISOMORPHISM,SIMPLICITY}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{PII S1446788719000302:111--PII S1446788719000302:126}},
  title        = {{Tensor products of Steinberg algebras}},
  url          = {{http://dx.doi.org/10.1017/S1446788719000302}},
  volume       = {{111}},
  year         = {{2021}},
}

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