Timeline for Which $\ast$-algebras are $C^\ast$-algebras?
Current License: CC BY-SA 4.0
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| Jan 28, 2022 at 4:44 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Sep 29, 2019 at 10:15 | comment | added | Dave Penneys | @TimCampion - I added more discussion to hopefully clarify what is going on. The proof is: $A$ is a C* algebra if and only if $A$ has a C*-norm (which is determined by spectral radius) if and only if the spectral radius defines a C*-norm on $A$. As for whether some of these properties hold automatically, that's a very interesting question for which I don't know the answer and would need to think more. | |
| Sep 29, 2019 at 9:59 | history | edited | Dave Penneys | CC BY-SA 4.0 |
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| Sep 28, 2019 at 14:51 | comment | added | Tim Campion | @YemonChoi Thanks! Palmer looks great. I guess I'm just kind of surprised that there doesn't seem to be a crisp axiomatization of which $\ast$-algebras are $C^\ast$-algebras when it's known to be possible in principle. | |
| Sep 28, 2019 at 0:02 | comment | added | Yemon Choi | @TimCampion My suggestion is to either learn about Banach algebras (I know, I know, but some of us in functional analysis have tried to learn about category theory) or consult Palmer as I suggested above. Your hopes for things to hold automatically read to me as if I, as a functional analyst, assumed every monoidal category is symmetric, closed, and compact | |
| Sep 27, 2019 at 20:07 | history | edited | Dave Penneys | CC BY-SA 4.0 |
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| Sep 27, 2019 at 19:51 | comment | added | Nik Weaver | @DavePenneys: the OP said this in his first sentence: "It's well-known that the norm on a Cā-algebra is uniquely determined by the underlying ā-algebra by the spectral radius formula." | |
| Sep 27, 2019 at 19:34 | comment | added | Dave Penneys | @NikWeaver - Thanks for your comment. The point is that a being a C*-algebra is a property of a unital complex *-algebra, and not extra structure. This property is expressed in terms of the spectral radius function as the OP points out. | |
| Sep 27, 2019 at 18:24 | comment | added | Nik Weaver | You've just listed the axioms of a C*-algebra and added that the norm comes from spectral radius. | |
| Sep 27, 2019 at 18:22 | comment | added | Nik Weaver | How does spectral radius enter into this characterization? Your three bullet points are the usual axioms for C*-algebras. | |
| Sep 27, 2019 at 18:01 | comment | added | Tim Campion | I suppose Yemon Choi's second comment above shows that the implication $\|a\| = 0 \Rightarrow a = 0$ is false in general. | |
| Sep 27, 2019 at 17:58 | comment | added | Tim Campion | Thanks! I think I misunderstood the definition of the spectral radius -- I thought this modification where one looks at $a^\ast a$ was built into the definition. Now I suppose what I'm wondering is "how close" these conditions are to holding automatically. For example, when $\|-\|$ is defined as above on a $\ast$-algebra, and assuming that $\|a\|$ is always finite, do any of the (in)equalities $\|a\| = 0 \Rightarrow a = 0$, $\|a+b\| \leq \|a\|+\|b\|$, $\|ab \| \leq \|a\|\|b\|$, $\|a^\ast a\| = \|a\|^2$ hold autormatically? Also, it would be nice to see a proof/reference... Or is it trivial? | |
| Sep 27, 2019 at 17:49 | history | answered | Dave Penneys | CC BY-SA 4.0 |