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Primitive graph algebras Abrams, Gene
Description
Let E = (E0,E1,s,r) be an arbitrary directed graph (i.e., no restriction is placed on the cardinality of E0, or of E1, or of s−1(v) for v ∈ E0). Let LK(E) denote the Leavitt path algebra of E with coefficients in a field K, and let C∗(E) denote the graph C∗-algebra of E. (Note: here C∗(E) need not be separable.) We give necessary and sufficient conditions on E so that LK(E) is primitive (joint work with Jason Bell and K.M. Rangaswamy). We then show that these same conditions are precisely the necessary and sufficient conditions on E so that C∗(E) is primitive (joint work with Mark Tomforde). This gives yet another example in a long and growing list of algebraic/analytic properties of the graph algebras LK(E) and C∗(E) for which the graph conditions equivalent to said property are identical, but for which the proof/techniques used are significantly different. In the Leavitt path algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive von Neumann regular algebras (thereby giving a systematic answer to a decades-old question of Kaplansky). In the graph C∗-algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive C∗-algebras (thereby giving a systematic answer to a decades-old question of Dixmier).
Item Metadata
Title |
Primitive graph algebras
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2013-04-22
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Description |
Let E = (E0,E1,s,r) be an arbitrary directed graph (i.e., no restriction is placed on the cardinality of E0, or of E1, or of s−1(v) for v ∈ E0). Let LK(E) denote the Leavitt path algebra of E with coefficients in a field K, and let C∗(E) denote the graph C∗-algebra of E. (Note: here C∗(E) need not be separable.) We give necessary and sufficient conditions on E so that LK(E) is primitive (joint work with Jason Bell and K.M. Rangaswamy). We then show that these same conditions are precisely the necessary and sufficient conditions on E so that C∗(E) is primitive (joint work with Mark Tomforde). This gives yet another example in a long and growing list of algebraic/analytic properties of the graph algebras LK(E) and C∗(E) for which the graph conditions equivalent to said property are identical, but for which the proof/techniques used are significantly different. In the Leavitt path algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive von Neumann regular algebras (thereby giving a systematic answer to a decades-old question of Kaplansky). In the graph C∗-algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive C∗-algebras (thereby giving a systematic answer to a decades-old question of Dixmier).
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Extent |
61 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Colorado at Colorado Springs
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Series | |
Date Available |
2014-08-07
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivs 2.5 Canada
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DOI |
10.14288/1.0043521
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivs 2.5 Canada