Quaternionic Hilbert Spaces

Question

what do we mean by two sided quaternionic hilbert spaces?How do we define an inner product on a right quaternionic hilbert space which is left quaternionic hilbert space as well?

Answer 1

A quaternionic Hilber space ought to be a complete inner product space which is a quaternionic vector space, and in which the inner product is with respect to the quaternionic multiplication. If it is a left quaternionic vector space (i.e. a vector space over $\mathbb{H}$ in which we muliply vectors by scalars on the left, e.g. $\lambda v$) then I expect the inner product to be conjugate-linear in the second argument and linear in the first argument, or in other words $\langle \alpha u,\beta v\rangle=\alpha\langle u,v\rangle\overline{\beta}$, and if it is a right quaternionic vector space (multiply vectors by scalars on the right, e.g. $v\lambda$) then I expect it is the other way around.

A two-sided quaternionic vector space admits quaternionic multiplication from both sides, with the additional condition of associativity, i.e. $(\alpha v)\beta=\alpha(v\beta)$. An inner product on such a quaternionic vector space I expect is with respect to either the left or right multiplication, but not both, in which case the other scalar multiplication doesn't play nice with the inner product. For instance, if the inner product is with respect to the right scalar multiplication, then $\langle \alpha u,\beta v\rangle$ can't be determined from knowing $\langle u,v\rangle$ and the scalars $\alpha,\beta$ (unless both scalars are real of course).