pairwise orthogonal projections in an inseparable $C^* $ algebra

Question

If $A$ is a separable $C^*$ algebra,then there are at most countable pairwise orthogonal projections.If $A$ is inseparable,how many pairwise orthogonal projections in $A$? If it has, is it uncountable?

Answer 1

It may have zero. For instance, take $$ A=\prod_{t\in[0,1]} C_0(\mathbb R). $$ With the norm $\|a\|=\sup\{\|a_t\|:\ t\in[0,1]\}$, this is a non-separable C$^*$-algebra. And it has no projections other that $0$, since any projection has to be a product of projections and $C_0(\mathbb R)$ has no nonzero projections.

As s.harp mentions, any family of pairwise orthogonal projections will have cardinality less than or equal that of $A$.

Answer 2

If $H$ is a separable infinite dimensional Hilbert space, then the C-algebra $B(H)$ of all operators on $H$ is not separable, but any orthogonal family of nonzero projections must be countable. However, if $K(H)$ is the ideal of all compact operators on $H$, then there is an uncountable orthogonal family of projections in $B(H)/K(H)$. If $A$ is a separable C-algebra then $A$ can be embedded in $B(H)$, so any orthogonal family of nonzero projections must be countable.