Projection operators nearly orthogonal to their symmetries

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• 作者：Sam Walters ^{walters@unbc.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hilbert space ; Norm approximation ; C*-algebras ; Automorphisms ; Projections ; Orthogonality
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1356-1361
• 全文大小：238 K

文摘

For any order 2 automorphism α of a C*-algebra A (a symmetry of A), we prove that for each projection e   such that class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305169&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305169&_rdoc=1&_issn=0022247X&md5=8d36e4d5d41bf1b4abb231d7b695ab87">class="imgLazyJSB inlineImage" height="23" width="90" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305169-si1.gif">class="mathContainer hidden">class="mathCode">$\Vert e\alpha (e)\Vert \le \frac{9}{20}$, there exists a projection q   with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305169&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305169&_rdoc=1&_issn=0022247X&md5=46f3a29a78113ad75145920a72d9ed2b" title="Click to view the MathML source">qα(q)=0class="mathContainer hidden">class="mathCode">$q\alpha (q)=0$ satisfying the norm estimate In other words, if e   is a projection that is “nearly orthogonal” to its symmetry class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305169&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305169&_rdoc=1&_issn=0022247X&md5=4b74b8b23cebe6e6eb0d7fb263fcee76" title="Click to view the MathML source">α(e)class="mathContainer hidden">class="mathCode">$\alpha (e)$ in the sense that the norm class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305169&_mathId=si5.gif&_user=111111111&_pii=S0022247X16305169&_rdoc=1&_issn=0022247X&md5=5c47069ae87aa3def2f3c5d46d8892b8" title="Click to view the MathML source">‖eα(e)‖class="mathContainer hidden">class="mathCode">$\Vert e\alpha (e)\Vert$ is no more than class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305169&_mathId=si6.gif&_user=111111111&_pii=S0022247X16305169&_rdoc=1&_issn=0022247X&md5=f2294a69524d7b6490e27faafb1dd304">class="imgLazyJSB inlineImage" height="23" width="13" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305169-si6.gif">class="mathContainer hidden">class="mathCode">$\frac{9}{20}$, then e can be approximated by a projection q   that is exactly orthogonal to its symmetry in a fairly optimal fashion. (Optimal in the sense that the first term in the estimate satisfies class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305169&_mathId=si7.gif&_user=111111111&_pii=S0022247X16305169&_rdoc=1&_issn=0022247X&md5=546790978286239e88aff3ac0ceb7ef3">class="imgLazyJSB inlineImage" height="23" width="133" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305169-si7.gif">class="mathContainer hidden">class="mathCode">$\frac{1}{2}\Vert e\alpha (e)\Vert \le \Vert e-q\Vert$ for any such q.)