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Discovery Projects - Grant ID: DP150101598

Title
Discovery Projects - Grant ID: DP150101598
Funding
ARC | Discovery Projects
Contract (GA) number
DP150101598
Start Date
2015/01/01
End Date
2017/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP150101598

 

  • On the geometry of idempotents in von Neumann algebras

    Giordano, Thierry; Sierakowski, Adam (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We consider the general linear group as an invariant of von Neumann factors. We prove that up to complement, a set consisting of all idempotents generating the same right ideal admits a characterisation in terms of properties of the general linear group of a von Neumann factor. We prove that for two Neumann factors, any bijection of their general linear groups induces a bijection of their idempotents with the following additional property: If two idempotents or their two complements generate ...

    Reconstruction of groupoids and C*-rigidity of dynamical systems

    Carlsen, Toke Meier; Ruiz, Efren; Sims, Aidan; Tomforde, Mark (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We show how to construct a graded locally compact Hausdorff \'etale groupoid from a C*-algebra carrying a coaction of a discrete group, together with a suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid. When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees with Renault's Weyl groupoid. We prove that if G is a second-countable locally compact \'etale groupoid carrying a grading of a discrete group, and if the interior of the trivially graded...

    Comparison properties of the Cuntz semigroup and applications to C*-algebras

    Bosa, Joan; Petzka, Henning (2018)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{\o}rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not ha...

    Graded C*-algebras, graded K-theory, and twisted P-graph C*-algebras

    Kumjian, Alex; Pask, David; Sims, Aidan (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner-Voiculescu sequence. We introduce the notion of a twisted P-graph C*-algebra and establish connections with graded C*-algebras. Specifically, we show how a functor from a P-graph into the group of order two determines a grading of the...

    Opposite algebras of groupoid C*-algebras

    Buss, Alcides; Sims, Aidan (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We show that every groupoid C*-algebra is isomorphic to its opposite, and deduce that there exist C*-algebras that are not stably isomorphic to groupoid C*-algebras, though many of them are stably isomorphic to twisted groupoid C*-algebras. We also prove that the opposite algebra of a section algebra of a Fell-bundle over a groupoid is isomorphic to the section algebra of a natural opposite bundle.

    Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

    Clark, Lisa Orloff; Edie-Michell, Cain; Huef, Astrid an; Sims, Aidan (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed subspaces of their unit spaces are all effective. For such a groupoid, we completely describe the ideal lattice of the associated Steinberg algebra over any commutative ring with identity. Our results are new even for the special case of Leavitt path algebras; so we describe explicitly what they sa...

    Reconstruction of graded groupoids from graded Steinberg algebras

    Ara, Pere; Bosa, Joan; Hazrat, Roozbeh; Sims, Aidan (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.

    Unbounded quasitraces, stable finiteness and pure infiniteness

    Pask, David; Sierakowski, Adam; Sims, Aidan (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    We prove that if A is a \sigma-unital exact C*-algebra of real rank zero, then every state on K_0(A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for any k-graph whose C*-algebra is unital and simple, either every twisted C*-algebra associated to that k-graph is stably fin...

    Baer and Baer *-ring characterizations of Leavitt path algebras

    We characterize Leavitt path algebras which are Rickart, Baer, and Baer $*$-rings in terms of the properties of the underlying graph. In order to treat non-unital Leavitt path algebras as well, we generalize these annihilator-related properties to locally unital rings and provide a more general characterizations of Leavitt path algebras which are locally Rickart, locally Baer, and locally Baer $*$-rings. Leavitt path algebras are also graded rings and we formulate the graded versions of these...

    Leavitt path algebras: Graded direct-finiteness and graded $\Sigma$-injective simple modules

    Hazrat, Roozbeh; Rangaswamy, Kulumani M.; Srivastava, Ashish K. (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP160101481 (DP160101481), ARC | Discovery Projects - Grant ID: DP150101598 (DP150101598)
    In this paper, we give a complete characterization of Leavitt path algebras which are graded $\Sigma $-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $\Sigma $-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matr...
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