Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling, stir in the linguine, and return to a boil. Cook the pasta uncovered, stirring occasionally, until the pasta has cooked through, but is still firm to the bite, about 11 minutes. Drain well in a colander set in the sink.
Slice abalone into 1/2 inch slices. Place the abalone between two sheets of heavy plastic (resealable freezer bags work well) on a solid, level surface. Firmly pound the abalone with the smooth side of a meat mallet to a thickness of 1/4 inch. Mix lime juice and garlic in a large bowl. Toss abalone slices and bell pepper in the lime garlic mixture to coat.
Heat olive oil in a large skillet over high heat and stir in the abalone and bell pepper, including the juice. Cook and stir until the cooked through, 5 to 10 minutes. Remove the abalone slices and place on a plate and set aside. Continue cooking the peppers, until tender. Meanwhile, pour Alfredo sauce into a saucepan and heat over medium-low heat until hot, stirring occasionally. Serve abalone over a bed of linguine and Alfredo sauce.