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 class="mathmlsrc">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">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">class="mathContainer hidden">class="mathCode">$C_r^{⁎}(\mathcal{B})$ of a Fell bundle class="mathmlsrc">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">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$ over a discrete group G  . Conditions allowing one to determine the ideal structure of class="mathmlsrc">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">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">class="mathContainer hidden">class="mathCode">$C_r^{⁎}(\mathcal{B})$ are studied. Notions of aperiodicity, paradoxicality and class="mathmlsrc">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">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$-infiniteness for the Fell bundle class="mathmlsrc">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">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$ are introduced and investigated by themselves and in relation to the partial dynamical system dual to class="mathmlsrc">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">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$. Several criteria of pure infiniteness of class="mathmlsrc">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">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">class="mathContainer hidden">class="mathCode">$C_r^{⁎}(\mathcal{B})$ 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 [34]; Jolissaint, Robertson [21]; Sierakowski, Rørdam [47]; Giordano, Sierakowski [18] and Ortega, Pardo [39]. For exact, separable Fell bundles satisfying the residual intersection property primitive ideal space of class="mathmlsrc">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">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">class="mathContainer hidden">class="mathCode">$C_r^{⁎}(\mathcal{B})$ is determined. The results of the paper are shown to be optimal when applied to graph class="mathmlsrc">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^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras. Applications to a class of Exel–Larsen crossed products are presented.