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-M6b908724795396033b272d47 Delivery options: https://agi.topicbox.com/groups/agi/subscription
