Pure infiniteness and ideal structure of C^⁎-algebras associated to Fell bundles

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• 作者：Bartosz Kosma Kwaśniewski^a ; ^b ; ^{bartoszk@math.uwb.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Wojciech Szymański^a ; ^{szymanski@imada.sdu.dk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fell bundle ; Aperiodicity ; Paradoxicality ; Pure infiniteness ; Ideals ; Cross-sectional algebra
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：898-943
• 全文大小：944 K

文摘

We investigate structural properties of the reduced cross-sectional algebra an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=0f9bbb2c5cc0152c1391e4c3cc57e55e">mg class="imgLazyJSB inlineImage" height="16" width="45" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303687-si1.gif">cript>mg height="16" border="0" style="vertical-align:bottom" width="45" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16303687-si1.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><msubsup><mrow><mi>Cmi>mrow><mrow><mi>rmi>mrow><mrow><mo>⁎mo>mrow>msubsup><mo stretchy="false">(mo><mi mathvariant="script">Bmi><mo stretchy="false">)mo>math>an>an>an> of a Fell bundle an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=ed2251628f151e441e6fdf8c3d0754bf" title="Click to view the MathML source">Ban>an class="mathContainer hidden">an class="mathCode"><math altimg="si2.gif" overflow="scroll"><mi mathvariant="script">Bmi>math>an>an>an> over a discrete group m>G  m>. Conditions allowing one to determine the ideal structure of an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=0f9bbb2c5cc0152c1391e4c3cc57e55e">mg class="imgLazyJSB inlineImage" height="16" width="45" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303687-si1.gif">cript>mg height="16" border="0" style="vertical-align:bottom" width="45" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16303687-si1.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><msubsup><mrow><mi>Cmi>mrow><mrow><mi>rmi>mrow><mrow><mo>⁎mo>mrow>msubsup><mo stretchy="false">(mo><mi mathvariant="script">Bmi><mo stretchy="false">)mo>math>an>an>an> are studied. Notions of aperiodicity, paradoxicality and an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=ed2251628f151e441e6fdf8c3d0754bf" title="Click to view the MathML source">Ban>an class="mathContainer hidden">an class="mathCode"><math altimg="si2.gif" overflow="scroll"><mi mathvariant="script">Bmi>math>an>an>an>-infiniteness for the Fell bundle an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=ed2251628f151e441e6fdf8c3d0754bf" title="Click to view the MathML source">Ban>an class="mathContainer hidden">an class="mathCode"><math altimg="si2.gif" overflow="scroll"><mi mathvariant="script">Bmi>math>an>an>an> are introduced and investigated by themselves and in relation to the partial dynamical system dual to an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=ed2251628f151e441e6fdf8c3d0754bf" title="Click to view the MathML source">Ban>an class="mathContainer hidden">an class="mathCode"><math altimg="si2.gif" overflow="scroll"><mi mathvariant="script">Bmi>math>an>an>an>. Several criteria of pure infiniteness of an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=0f9bbb2c5cc0152c1391e4c3cc57e55e">mg class="imgLazyJSB inlineImage" height="16" width="45" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303687-si1.gif">cript>mg height="16" border="0" style="vertical-align:bottom" width="45" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16303687-si1.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><msubsup><mrow><mi>Cmi>mrow><mrow><mi>rmi>mrow><mrow><mo>⁎mo>mrow>msubsup><mo stretchy="false">(mo><mi mathvariant="script">Bmi><mo stretchy="false">)mo>math>an>an>an> are given. It is shown that they generalize and unify corresponding results obtained in the context of crossed products, by the following duos: Laca, Spielberg an id="bbr0340">[34]a>an>; Jolissaint, Robertson an id="bbr0210">[21]a>an>; Sierakowski, R&oslash;rdam an id="bbr0470">[47]a>an>; Giordano, Sierakowski an id="bbr0180">[18]a>an> and Ortega, Pardo an id="bbr0390">[39]a>an>. For exact, separable Fell bundles satisfying the residual intersection property primitive ideal space of an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=0f9bbb2c5cc0152c1391e4c3cc57e55e">mg class="imgLazyJSB inlineImage" height="16" width="45" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303687-si1.gif">cript>mg height="16" border="0" style="vertical-align:bottom" width="45" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16303687-si1.gif">cript>a>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><msubsup><mrow><mi>Cmi>mrow><mrow><mi>rmi>mrow><mrow><mo>⁎mo>mrow>msubsup><mo stretchy="false">(mo><mi mathvariant="script">Bmi><mo stretchy="false">)mo>math>an>an>an> is determined. The results of the paper are shown to be optimal when applied to graph an id="mmlsi130" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303687&_mathId=si130.gif&_user=111111111&_pii=S0022247X16303687&_rdoc=1&_issn=0022247X&md5=4c011dead91dd035a6cdd6c0ffb12269" title="Click to view the MathML source">C^⁎an>an class="mathContainer hidden">an class="mathCode"><math altimg="si130.gif" overflow="scroll"><msup><mrow><mi>Cmi>mrow><mrow><mo>⁎mo>mrow>msup>math>an>an>an>-algebras. Applications to a class of Exel&ndash;Larsen crossed products are presented.