On Sun, Jan 31, 2021 at 11:04 PM 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. > I looked at some books briefly, and found that the Cayley-Dickson process gives rise to some "alternative" non-associative algebras. These are not "arbitrary" non-associative algebras, but they satisfy some "alternative" associative laws, such as: (aa)b = a(ab), ie, the associative law limited to 2 elements. >From my limited uderstanding, the quaternion / octonion algebras would not be directly usable as a Turing-complete language (which requires an arbitrary non-associative algebra...) But thanks for the information :) YKY ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/T54594b98b5b98f83-Mcdb10fed038a82bdc55bf12d Delivery options: https://agi.topicbox.com/groups/agi/subscription
