Using coreApi to publish a component, method execute successfully and the publishing queue in tridion also show that the item published status as success. But the component in tridion which has been published by coreapi does not change the "Publish Status" from Unpublished to Published. The same thing happens to Unpublish method.
By Default if user publish a component which is not dynamically publishable, the Tridion items (where this component is used and are already published) are published not the component.
That is why, the publish status is not changing for the component (independently non publishable item).
You can check this behavior in publish popup by clicking “show items to publish” button.