Non-commuting projections in non-separable Hilbert space

Question

In the algebra of $2\times 2$ matrices we have continuum many non-commuting projections.

Is it possible to find a (non-separable) Hilbert space $H$ and a family of non-commuting idempotents in $B(H)$ that has cardinality greater than continuum?

Answer 1

Yes. Let $S$ be a set with $\operatorname{card} S > \mathfrak{c}$. Let $H = \ell^2(S)$. Fix $s_0 \in S$, and for $s \in S \setminus \{s_0\}$ let $P_s$ be the orthogonal projection onto the subspace spanned by $e_{s_0} + e_s$. Then $\bigl\{ P_s : s \in S \setminus \{s_0\}\bigr\}$ is a family of non-commuting idempotents in $B(H)$ of cardinality $\operatorname{card} S$.