#The notion of conjugate symmetry on an inner product space on an arbitrary field

35 messages · Page 1 of 1 (latest)

lofty rose
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I've been reading into operator theory and linear algebra, and I have come across(more like, just recently noticed) that when trying to find a definition for an inner product, it has the property of conjugate symmetry(pic. 1). This works fine on a complex inner product space, but what happens when considering other fields? I don't know where to look, and I've come up with the idea that a conjugate must be an order-2 automorphism in the space, but in that case why is it not the map z -> -z or the trivial map z -> z?

My main question is what is, if there even is, a generalization of conjugate symmetry to an inner product space acting on an arbitrary field?

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echo hornet
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It’s more interesting to consider this if k is a Galois extension of a field L and f is a galois automorphism of k over L

hollow phoenixBOT
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Exceed

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echo hornet
lofty rose
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Okay, sure. I get what you're saying is that the inner product is conjugate-bilinear in C. But which f, if not conjugation, would be picked for another field?

echo hornet
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If you take a finite field it can be the Frobenius automorphism for example

lofty rose
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Mhmm, interesting

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Do you know where I could look more into this?

echo hornet
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Hm no i don’t have any specific sources sadly

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But if you are concentrating on operator theory, usual inner product spaces and Hilbert spaces (so over C) are really all you need

echo hornet
lofty rose
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and that is what I'm focusing on

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but it was just a thought, because it was framed as being a general field but I only found definitions on complex inner product spaces

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anyways, thanks a lot!!

echo hornet
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If you want a general simple example of such a form take $K$ a field $f \in Aut(K)$ then for $n \geq 1$ define $\mu_{f}((x_i){1 \le i \le n}, (y_i){1 \le i \le n})=\sum_{1 \le i \le n}x_if(y_i)$ then $\mu_{f}$ is $f$-bilinear on $K^{n}$ as I’ve defined

hollow phoenixBOT
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Exceed

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echo hornet
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(for f an involution)

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Else it only satisfies the second property not the first

echo hornet
hollow phoenixBOT
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Exceed

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echo hornet
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(An example other than C)

stone prism
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There are some papers I've seen on p-adic operator theory, but they are few and far between. Most of those papers are just trying to recover the theory in C. I also once attended a talk a friend of mine gave on operators over the quaternions, but I don't remember much of it.

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And if you're something like R, then you don't need to worry about conjugation.

echo hornet
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It probably is but used in specialised domains

stone prism
# stone prism There are some papers I've seen on p-adic operator theory, but they are few and ...

e.g. https://arxiv.org/abs/2403.04046 and https://www.mdpi.com/2073-8994/11/8/953

arXiv.org
Operator algebras over the p-adic integers

We introduce $p$-adic operator algebras, which are nonarchimedean analogues of $C^$-algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) $C^$-algebras - have nonarchimedean counterparts. The category of $p$-adic operator algebras exhibits similar properties to those of the category of real and comp...

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These are more C*-algebra theory admittedly, but close enough.

stone prism
echo hornet
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I should really go back to doing some functional analysis one of these days

stone prism
# echo hornet The only thing I’ve studied on C^*-algebras this year is the equivalence of the ...

That's actually super important. If all commutative unital C*-algebras are exactly C(X) for some compact Hausdorff space X (which is the spectrum of the algebra in fact) then it stands to reason that noncommutative unital C*-algebras (i.e. most of them -- think about matrix algebras as an example) are sort of a non-commutative analogue of compact Hausdorff spaces, and this is the framework that Connes developed and is now called noncommutative geometry (NCG).

echo hornet