Ooops... "nonassociative" would imply "octonion Hilbert spaces".
On Sun, Jan 31, 2021 at 9:03 AM James Bowery <[email protected]> wrote: > What you guys are discussing is above my head but this seems pertinent: > > On quaternionic functional analysis > <https://arxiv.org/pdf/math/0609160.pdf> > > In this article, we will show that the category of quaternion vector > spaces, the category of (both one-sided and two sided) quaternion Hilbert > spaces and the category of quaternion B∗-algebras are equivalent to the > category of real vector spaces, the category of real Hilbert spaces and the > category of real C∗-algebras respectively. We will also give a Riesz > representation theorem for quaternion Hilbert spaces and will extend the > main results in [12] (namely, we will give the full versions of the > Gelfand-Naimark theorem and the Gelfand theorem for quaternion > B∗-algebras). On our way to these results, we compare, clarify and unify > the term “quaternion Hilbert spaces” in the literatures. > > On Sun, Jan 31, 2021 at 4:46 AM YKY (Yan King Yin, 甄景贤) < > [email protected]> wrote: > >> On 1/30/21, Ben Goertzel <[email protected]> wrote: >> > Unless I remember wrong (which is possible), function application in a >> > Scott domain is not associative, e.g. >> > >> > (f(g) ) (h) >> > >> > is not in general equal to >> > >> > f( g(h) ) >> > >> > However function composition is associative, and the standard products >> > on vectors in Hilbert space are associative >> > >> > So it seems what you're doing may not be quite right, and you need to >> > be looking at some sort of fairly general nonassociative algebras over >> > Hilbert space instead ... or something... >> > >> > Or am I misunderstanding something? >> >> >> Damn... you're right 😆 >> >> I have no idea how to make one function "apply" to another, except by >> function composition, so that was what I did. But I forgot about >> associativity... >> >> I don't know if I should pursue along this any further.... it seems >> computer-implementable but it's very complicated... and I currently >> don't know how to make non-associative algebras.... >> >> Thanks for pointing out my mistake, it's helpful 😅 >> YKY ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T54594b98b5b98f83-M18cf18e69bcaeb167665ae31 Delivery options: https://agi.topicbox.com/groups/agi/subscription
