The pump feeds the exhaust manifold fresh new air during startup. If its dead or dying it makes a whinning noise when you start the car on for a few minutes or seconds then shuts off. I got mine replaced (on e39), I would suggest you replace the hoses, valve, and the bracket thier pretty cheap while changing the air pump.
The link you provided shows the air pump, clamps, hoses, bracket, nuts and bolts. The picture shows the valve but has no # associated with it, click on 11 45 Vacuum control in the subgroup. Hope this helps.