Timeline for Socle of an operator algebra
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 24, 2022 at 13:23 | comment | added | Onur Oktay | Professor Ozawa, thanks again for your kind and helpful reply. | |
| Feb 24, 2022 at 13:11 | comment | added | Narutaka OZAWA | Triviality of the socle of $A$ does not imply the same of the C*-envelope. An example can be derived from Parrott's example. | |
| Feb 23, 2022 at 20:03 | comment | added | Onur Oktay | Professor @NarutakaOZAWA, as opposed to the $C^{\ast}$-algebra generated by the unilateral shift $u$, its $C^{\ast}$-envelope is $C(\mathbb{R}/\mathbb{Z})$ that has trivial socle. Is it generally true that the socle of the $C^{\ast}$-envelope of an operator algebra $A$ is $\{0\}$ if $socle(A)=\{0\}$? If not, the socle of the $C^{\ast}$-envelope is (perhaps?) a boundary ideal. Which property of $A$ suffices the socle of the $C^{\ast}$-envelope to be equal to the Shilov boundary ideal? Thank you for your time in advance. | |
| Feb 23, 2022 at 19:37 | history | edited | Onur Oktay | CC BY-SA 4.0 |
Added Q3
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| Feb 23, 2022 at 14:43 | comment | added | Onur Oktay | @NarutakaOZAWA Thank you for your kind reply and for the precise reference to the section in Pisier's book. It's not so long ago that I started to learn the operator space theory from the basics, starting from the books of Blecher & le Merdy, Effros & Ruan, Pisier, Paulsen, along with next-step books to read on my Jabref shelf. Thanks for bearing with me and your kind response. | |
| Feb 23, 2022 at 13:40 | comment | added | Narutaka OZAWA | Both are false. As to Q1, any nontrivial $A$ such that $A\cap A^*=\mathbb{C}1$ is a counterexample, by considering the conjugation by an invertible operator. It makes more sense to ask if every contractive homomorphism is completely contractive. Pisier calls such operator algebras to be "full" and presents examples and counterexamples in his book [Introduction to Operator Space Theory, Section 26]. As to Q2, the unital operator algebra $A$ generated by the unilateral shift is a counterexample. | |
| Feb 22, 2022 at 23:48 | history | edited | YCor |
edited tags
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| Feb 22, 2022 at 22:27 | history | edited | Onur Oktay | CC BY-SA 4.0 |
edited title
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| Feb 22, 2022 at 22:15 | history | asked | Onur Oktay | CC BY-SA 4.0 |