Pure infiniteness and ideal structure of C<sup>⁎sup>-algebras associated to Fell bundles

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• 作者：Bartosz Kosma Kwaśniewski<sup>asup><sup> ; sup><sup>bsup><sup> ; sup><sup>bartoszk@math.uwb.edu.pl" class="auth_mail" title="E-mail the corresponding authorsup> ; Wojciech Szymański<sup>asup><sup> ; sup><sup>szymanski@imada.sdu.dk" class="auth_mail" title="E-mail the corresponding authorsup>
• 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 <span id="mmlsi1" class="mathmlsrc">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">ss="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">script>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">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>Cr⁎subsup>stretchy="false">(script">Bstretchy="false">)span>span>span> of a Fell bundle <span id="mmlsi2" class="mathmlsrc"><span 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">Bspan><span class="mathContainer hidden"><span class="mathCode">scroll">script">Bspan>span>span> over a discrete group G  . Conditions allowing one to determine the ideal structure of <span id="mmlsi1" class="mathmlsrc">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">ss="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">script>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">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>Cr⁎subsup>stretchy="false">(script">Bstretchy="false">)span>span>span> are studied. Notions of aperiodicity, paradoxicality and <span id="mmlsi2" class="mathmlsrc"><span 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">Bspan><span class="mathContainer hidden"><span class="mathCode">scroll">script">Bspan>span>span>-infiniteness for the Fell bundle <span id="mmlsi2" class="mathmlsrc"><span 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">Bspan><span class="mathContainer hidden"><span class="mathCode">scroll">script">Bspan>span>span> are introduced and investigated by themselves and in relation to the partial dynamical system dual to <span id="mmlsi2" class="mathmlsrc"><span 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">Bspan><span class="mathContainer hidden"><span class="mathCode">scroll">script">Bspan>span>span>. Several criteria of pure infiniteness of <span id="mmlsi1" class="mathmlsrc">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">ss="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">script>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">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>Cr⁎subsup>stretchy="false">(script">Bstretchy="false">)span>span>span> 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 <span id="bbr0340">[34]span>; Jolissaint, Robertson <span id="bbr0210">[21]span>; Sierakowski, R&oslash;rdam <span id="bbr0470">[47]span>; Giordano, Sierakowski <span id="bbr0180">[18]span> and Ortega, Pardo <span id="bbr0390">[39]span>. For exact, separable Fell bundles satisfying the residual intersection property primitive ideal space of <span id="mmlsi1" class="mathmlsrc">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">ss="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">script>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">script><span class="mathContainer hidden"><span class="mathCode">scroll">subsup>Cr⁎subsup>stretchy="false">(script">Bstretchy="false">)span>span>span> is determined. The results of the paper are shown to be optimal when applied to graph <span id="mmlsi130" class="mathmlsrc"><span 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<sup>⁎sup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>C⁎sup>span>span>span>-algebras. Applications to a class of Exel&ndash;Larsen crossed products are presented.