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I'm having a hard time wrapping my head around the following definition taken form the book of Higson & Roe on k-homology.

A Hilbert space is defined to have a $p$-multigrading if $H$ is $\mathbb{Z}_2$-graded and there are $p$ odd unitaries $e_1, ..., e_p$ on $H$ with the properties that $$e_i^\star = -e_i \qquad e_i^2 = -1 \qquad e_ie_j + e_j e_i = 0.$$ The first condition is superfluous, and the third holds for all $i\neq j$. I will not go into the details here, but the definition is such that the Hilbert space $H$ becomes a $\mathbb{Z}_2$-graded module over the universal Clifford algebra of $p$-generators when the Clifford algebra is endowed with the usual grading determined by the length of the basis monomials spanning it.

This definition seems very different from the intuitive definition of a grading as something that decomposes a space into subspaces and a module into subgroups. My question is, does the family $e_1, ..., e_p$ determine a decomposition of $H$ into linear subspaces in some natural way or is this not what a p-multigrading is all about?

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I don't see any nice way in which a $p$-multigrading on a Hilbert space $\mathcal H$ induces a $\mathbb Z_p$-grading on $\mathcal H$. A $\mathbb Z_p$-grading on $\mathcal H$ is given by a unitary $U\in\mathbb B(\mathcal H)$ such that $U^p=1$ (and the subspace in the grading would be given by $\mathcal H_k=\{\xi\in\mathcal H:U\xi=e^{2\pi i k}\xi\})$. I'm not aware of any way to go from the operators $e_1,\ldots,e_p$ to a unitary $U$.

As far as I'm aware, calling this a $p$-multigrading just an unfortunate name, given that the definition already requires a $\mathbb Z_2$-grading.

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