...if you improve your armor without wearing a shield, are you still below the armor cap? I don't want to hit the cap.

Unless you use a crazy ass fortify smithing potion with crazy fortify smithing gear, you will be below the cap.

Cap is what, 554? Something like that...

567 I believe.

Thanks, that is good to know.

What cap? It's my understanding that you can improve armor to ridiculously high levels with enchanted equipment, potions, and relative perks.

Cap is 567.

It becomes redundant. Once you hit 567 with armour, the amount of damage you sustain or "block" is the same as if you had an armour rating of 1052.

True, but after 567, your armor rating no longer gives you any extra protection. So if you have 567, or 92483783758315387 AR, you receive the same damage.

Cap is actually 667 (x0.12=80.04, cap is 80% mitigation), as there is a 25 point bonus on top of stated value for each slot excluding shield. If you don't wear a helm, you will need to smith to an inventory value of 592 for your three items to hit the cap (592+3x25=667).

667 is the cap if you do not wear armour.

"Overall, anything beyond 567 displayed armor rating when wearing all four pieces of armor or 667 when not wearing any pieces of armor will still be capped at 80% of physical damage reduction, meaning you cannot gain 100% physical damage immunity."

-http://www.uesp.net/wiki/Skyrim:Armor