I think it's worth a try. I have some Labrador violets, Viola labradorica purpurea; Australian violets, Viola hederacea; and Viola odorata "Duchesse de Parme," Parma violets, all growing successfully in lots of shade. Viola tricolor, Johnny Jump Ups (sometimes listed as Viola "Helen Mount"), are very adaptable and would be worth a try too. They won't do well in the kind of deep shade you get underneath the deck. But brighter shade is another matter. Some of mine are growing on the north side of the house where they don't get direct sun except in the height of summer. Others are beneath a tree in a spot with dappled shade for part of the day, and just plain shade during the rest of the day. So it can't hurt to try!

In general, violas prefer cool conditions, so you will see the most bloom during spring and fall. Even though you often see them listed as annuals, they are in fact perennial. So even if they don't look so good during summer, don't pull them out. They'll make a nice recovery come fall. And if by chance your winter is too cold and they don't make it, they will probably reseed themselves, especially the Johnny Jump Ups.