That's not the same problem as we are having, I think. When a program is running, its icon should appear in the dock if it's not there already and there should be an indicator underneath -or next to- the icon to indicate that the program it represents is really running. This is not the case. When I tried to launch App Store, it icon briefly appeared in my dock before it disappeared. Nothing indicated that App Store was running, not even the activity monitor.
As of now, my problems have been fixed by re-installing Mavericks through the recovery partition. I hope that the problem doesn't show up again. I re-installed a couple of days ago already and the native apps started to malfunction in less that 24 hours. I re-installed again and the MacBook, including the native apps, worked fine until I decided to spill 7UP on the trackpad:( I'll check if my apps still work as soon as my MacBook is done drying up using the ancient "put it in a box with rice"-method.