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$C^{\ast }$-ALGEBRA
We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a 
$C^{\ast }$-algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all 
$wg$-Drazin invertible elements of a 
$C^{\ast }$-algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.
