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This paper deals with spectral properties of commutative locally holomorphic Banach algebra valued functions. One of the main concepts is that of a spectral set of such a function. This concept, which is due to L. Mittenthal, extends that of a spectral set of a single Banach algebra element. It will be shown that the spectral idempotent associated with a nonvoid spectral set is nonzero. This result is a generalization of a well-known theorem in ordinary spectral theory. It will be used to provedoi:10.2140/pjm.1974.52.321 fatcat:n4shvb6qhrgzxgjn4wdltk7umu