486 research outputs found
Purely infinite crossed products by endomorphisms
We study the crossed product -algebra associated to injective
endomorphisms, which turns out to be equivalent to study the crossed product by
the dilated autormorphism. We prove that the dilation of the Bernoulli
-shift endomorphism is topologically free. As a consequence, we have a way
to twist any endomorphism of a \D-absorbing -algebra into one whose
dilated automorphism is essentially free and have the same -theory map than
the original one. This allows us to construct purely infinite crossed products
-algebras with diverse ideal structures.Comment: Corrected misprints. Clarified some points in Section 3. Added a
reference. Re-submitted to JMA
Algebraic Cuntz-Pimsner rings
From a system consisting of a right non-degenerate ring , a pair of
-bimodules and and an -bimodule homomorphism we construct a -graded ring
called the Toeplitz ring and (for certain systems) a -graded quotient
of called the
Cuntz-Pimsner ring. These rings are the algebraic analogs of the Toeplitz
-algebra and the Cuntz-Pimsner -algebra associated to a
-correspondence (also called a Hilbert bimodule).
This new construction generalizes for example the algebraic crossed product
by a single automorphism, corner skew Laurent polynomial ring by a single
corner automorphism and Leavitt path algebras. We also describe the structure
of the graded ideals of our graded rings in terms of pairs of ideals of the
coefficient ring.Comment: 55 pages. Version 3 is a complete rewrite of version 2. In version 4
Def. 3.14, Def. 4.6, Def. 4.8 and Remark 4.9 have been added and some minor
adjustments have been mad
The Maximal C*-Algebra of Quotients as an Operator Bimodule
We establish a description of the maximal C*-algebra of quotients of a unital
C*-algebra as a direct limit of spaces of completely bounded bimodule
homomorphisms from certain operator submodules of the Haagerup tensor product
labelled by the essential closed right ideals of into .
In addition the invariance of the construction of the maximal C*-algebra of
quotients under strong Morita equivalence is proved.Comment: 8 pages; submitte
The Cuntz Semigroup and Comparison of Open Projections
We show that a number of naturally occurring comparison relations on positive
elements in a C*-algebra are equivalent to natural comparison properties of
their corresponding open projections in the bidual of the C*-algebra. In
particular we show that Cuntz comparison of positive elements corresponds to a
comparison relation on open projections, that we call Cuntz comparison, and
which is defined in terms of-and is weaker than-a comparison notion defined by
Peligrad and Zsid\'o. The latter corresponds to a well-known comparison
relation on positive elements defined by Blackadar. We show that Murray-von
Neumann comparison of open projections corresponds to tracial comparison of the
corresponding positive elements of the C*-algebra. We use these findings to
give a new picture of the Cuntz semigroup
Topological Full Groups of Ample Groupoids with Applications to Graph Algebras
We study the topological full group of ample groupoids over locally compact
spaces. We extend Matui's definition of the topological full group from the
compact, to the locally compact case. We provide two general classes of
groupoids for which the topological full group, as an abstract group, is a
complete isomorphism invariant. Hereby extending Matui's Isomorphism Theorem.
As an application, we study graph groupoids and their topological full groups,
and obtain sharper results for this class. The machinery developed in this
process is used to prove an embedding theorem for ample groupoids, akin to
Kirchberg's Embedding Theorem for -algebras. Consequences for graph
-algebras and Leavitt path algebras are also spelled out. In particular,
we improve on a recent embedding theorem of Brownlowe and S{\o}rensen for
Leavitt path algebras.Comment: 58 pages. V4: Deleted an obsolete reference. Updated Journal
reference. Minor typographical changes. (V3: Rewritten the "Our results" and
"Pr\'ecis" part of the Introduction. Added Remark 6.20 and Definition 7.8.
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