I am still having problems getting the air pump to come on when it is supposed to. Here is an update on my findings so far:
1. The pump runs fine and will come on if I remove the relay and jumper the power terminals.
2. The SAS controller workes fine. I connected a vacuum gauge to the SAS vacuum supply line and started the engine (engine cold). Vacuum was present for about two minutes and then it was cut off.
3. During the warm up period, I had a voltmeter connected to the power supply lead for the air pump, reading was 0 volts.
4. With the ignition in position II, I read 12.5 volts at the air pump supply terminals in the relay box. I read 9 volts at the solenoid terminals in the relay box.
5. With the engine running and warmed up, I read 14 volts at the power terminals and 10.8 volts at the solenoid terminals.
6. I replaced the airpump relay and still no luck.
NOTE: If the airpump relay is removed, the CHECK ENGINE light will come on soon after the engine is started. With the CHECK ENGINE light on there is no voltage present at the air pump terminals in the relay box.

It seems to me that if the ECU controls the air pump relay, the problem may be in the ECU. All other components appear to be functioning properly. Is there any test which would confirm a faulty ECU? The ECU is covered under the extended warranty for certain emissions control components and it would be nice to have Volvo foot the bill for this problem.

My dealer diagnosed this as the typical SAS valve/ air pump failure and wanted to replace those components to the tune of $550.

I would greatly appreciate any additional assistance on this situation as the dealer has not been receptive to my telling them they misdiagnosed the problem.
(Thanks Rich K. at Bay13 for your help so far.)

The dealer uses their VST scan tool to run it through its diagnostic test mode and by doing that, they can properly determine if the pump's running, not electrically inop or siezed by doing so.
The pump doesn't always come on everytime the engine's started cold, there's a few conditions that need to be met and it's determined by the fuel ECU.

I've never tryed activating the pump by the means that you're listing as I use the VST and it's never let me down, nor have I seen where the air pump circuit was anything more involved that a bad pump and/or relay.

In your case, if you're sure that your relay isn't getting the proper voltages, then your next step is to 1) get a OEM wiring manual from Volvo (800-25-volvo or online at their site) and 2) using that wiring diagram, trace the circuits to/from the relay and determine what's not getting voltage and why. It's remotely "possible" that you have a ECU problem but I doubt it. It's more likely IMHO that you're not knowing when that air pump is suppose to work and trying to fault check something that might not even be a problem. For example, is the check engine light on and if so, are you sure you've got air pump codes?
If the check engine light's off and you're not getting air pump codes, then I wouldn't even worry about it. BTW, I've seen intermittant air pump codes appear too that commonly occurs when temps go below freezing and I speculate that it's because moisture is in the pump and thus freezing the pump. As the temps increase, the pump later works and will test normally but eventually the moisture becomes excessive and the pump motors get noisy and/or just sieze. It's a very bad design, put into the worst possible location where moisture can't escape.

Herb,

Thanks for the information. I really believe there is some other problem causing the air pump relay not to activate. When the CHECK ENGINE light originally came on, I had the dealer check the codes. They found two air pump related codes and wanted to replace the pump, valve, etc.

I did some checking on my own and found the pump was OK, I removed it and looked inside. There were absolutely no signs of moisture invasion.

Also,I've checked the the system several times when the car is cold and everytime the controller valve opens and activates the check valve. When this happens, I can feel exhuast pulsing back through the air pump inlet hose because the pump is not running. This can't be normal because allowing the check valve to open when the pump is off would eventually cause the same problem as a stuck check valve. It seems that the air pump should run whenever the check valve is open. That's why I'm stumped.

The 2 codes were probably 442 and 446 (flow malfunction). Code 448 occurs if the pump siezes, permanent fault and then one should activate the pump with the VST to be sure it's siezed, not just guess. The first 2 codes are caused by the check valve failing and if that's all that's wrong (the pump works and isn't excessively noisy) then I'd just replace the check valve only. Usually once the check valve goes, the pump's not far behind it as it becomes full of water in short order. Apparently you caught yours in time.
I'd advise replacing your check valve now (only), clearing the code(s) and just drive the car not worrying about how, when or why the pump activates because (like I mentioned before), it doesn't always run cold and I think you might be worrying about it un-necessarily as it's probably fine, functioning normally. If after you replace that valve, clear the codes and drive it for a while you don't see the check engine light again, there's no problems.

I appreciate your additional input. The check valve is working fine, I've tested it several times and it always shuts off. It was replaced some time ago under warranty because the the weld broke on the connecting tube on the original. The whole thing was replaced as an assembly, the dealer told me at that time it was only available that way. So, hopefully, I already have the improved version. I reset the check engine light by disconnecting the battery and I'll see what happens. This is my wife's car and she worries when lights come on. I took your advice and ordered the electrical schematics which seemed reasonable because they will cover both of my cars (I also hsve a 1999 V70, no air pump thank GOD!) Thanks again.

OK, you should then have the latest check valve so hopefully you're check engine light, at least those codes won't return for a long time.