Answer: Basically, the answer is no. The bulb lilies such as "Star Gazer" need their foliage growing along that main stem for as long as possible in order to rebuild the bulbs for the best flowers next year. You should, however, "deadhead" by removing the wiry little individual flower stems and any developing seed pods from the site of each flower (or remove just the tip of the main stem) to keep the plant from directing energy to seed production. Once the large main stems are frosted and browned in the fall they should be carefully removed.
The liatris also need a similar replenishing period. They are difficult to deadhead bloom by bloom, so you might remove the tips of the stalks if you wish but it isn't necessary. Similarly, since the leaves are on the stems, if you use the blooms as cut flowers, you should leave as much stem (and leaves) on the plant as possible.
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