Truncation and spectral variation in Banach algebras

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• 作者：C. Touré ; ^{cheickkader89@hotmail.com" class="auth_mail" title="E-mail the corresponding author} ; F. Schulz ^{francoiss@uj.ac.za" class="auth_mail" title="E-mail the corresponding author} ; R. Brits ; ^{rbrits@uj.ac.za" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Spectrum ; Truncation ; Spectral radius ; Subharmonic ; C⋆C⋆-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：23-31
• 全文大小：309 K

文摘

Let a and b be elements of a semisimple, complex and unital Banach algebra A  . Using subharmonic methods, we show that if the spectral containment class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303146&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303146&_rdoc=1&_issn=0022247X&md5=ac379e070068392974463c5ca11b9806" title="Click to view the MathML source">σ(ax)⊆σ(bx)class="mathContainer hidden">class="mathCode">$\sigma (ax)\subseteq \sigma (bx)$ holds for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303146&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303146&_rdoc=1&_issn=0022247X&md5=c39cf3bfe3b9dd6d5ff20550e0c0c6a6" title="Click to view the MathML source">x∈Aclass="mathContainer hidden">class="mathCode">$x\in A$, then ax belongs to the bicommutant of bx   for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303146&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303146&_rdoc=1&_issn=0022247X&md5=c39cf3bfe3b9dd6d5ff20550e0c0c6a6" title="Click to view the MathML source">x∈Aclass="mathContainer hidden">class="mathCode">$x\in A$. Given the aforementioned spectral containment, the strong commutation property then allows one to derive, for a variety of scenarios, a precise connection between a and b. The current paper gives another perspective on the implications of the above spectral containment which was also studied, not long ago, by J. Alaminos, M. Brešar et al.