Timeline for Example of a ternary $C^{\ast}$-ring which is not an operator space
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 20, 2022 at 3:52 | vote | accept | Math Lover | ||
| May 19, 2022 at 7:35 | vote | accept | Math Lover | ||
| May 20, 2022 at 3:51 | |||||
| Oct 29, 2021 at 15:52 | comment | added | Math Lover | @MatthewDaws: I was basically looking for an canonical embedding of ternary $C^*$-ring in a $C^*$-algebra like we have for TROs by means of linking $C^*$-algebra. | |
| Oct 29, 2021 at 11:49 | comment | added | Todd Trimble | @YemonChoi In light of Ruy's response, might there be a way of making the question precise to your satisfaction? It sounds like OP is saying: I don't know of any examples of ternary $C^\ast$-rings except the ones of Ruy's form (up to isomorphism). | |
| Oct 29, 2021 at 11:46 | comment | added | Matthew Daws | @MathLover Would it be possible for you to edit your question, and make it a bit clearer as to what is being asked? My personal reading of the question currently is that the example Ruy speaks of is a "ternary ring of operators" which is not all possible examples of an "operator space" (and thus you are asking something more general than Ruy's answer). But I cannot be sure. | |
| Oct 29, 2021 at 0:44 | comment | added | Ruy | @YemonChoi, when the OP refers to an "operator space" in the context of ternary C*-rings, they mean a closed subspace $X\subseteq B(H)$ such that $XX^*X\subseteq X$, and equipped with the ternary product $$[x,y,z]=xy^*z.$$ | |
| Oct 28, 2021 at 23:58 | comment | added | Yemon Choi | @LSpice The point is that an "operator space", in the widely understood sense, does not have any intrinsic multiplication or any ternary operation defined on it. Hence the question that was asked is a bit like asking if there is any semigroup which is not a subset of aleph_2 | |
| Oct 28, 2021 at 23:54 | comment | added | Yemon Choi | @LSpice I don't know off the top of my head, but that was not the question which was asked. As with some of the OP's history of questions, I start to find the lack of precision troubling. | |
| Oct 28, 2021 at 23:26 | answer | added | Ruy | timeline score: 4 | |
| Oct 28, 2021 at 20:49 | comment | added | LSpice | @YemonChoi, can those constructions be made to respect the ternary operation? | |
| Oct 28, 2021 at 14:19 | review | Close votes | |||
| Nov 2, 2021 at 3:01 | |||||
| Oct 28, 2021 at 13:56 | comment | added | Yemon Choi | Your question is not well posed because any Banach space can be realized as an operator space, via the MIN or MAX constructions | |
| Oct 28, 2021 at 11:04 | comment | added | Math Lover | @LSpice: Yes. These are also known as ternary algebras of 2nd kind. In $1$st kind the associativity condition is the natural one. | |
| Oct 28, 2021 at 10:16 | history | edited | LSpice | CC BY-SA 4.0 |
Punctuation
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| Oct 28, 2021 at 10:13 | comment | added | LSpice | Is the middle condition in the definition of associativity really supposed to have $b$ and $d$ switched? | |
| Oct 28, 2021 at 7:39 | history | asked | Math Lover | CC BY-SA 4.0 |