  The identity of Type Ia supernova progenitors remains a mystery, with various
lines of evidence pointing towards either accretion from a non-degenerate
companion, or the rapid merger of two degenerate stars leading to the
thermonuclear destruction of a white dwarf. In this paper we spectroscopically
scrutinize 24 of the brightest stars residing in the central 38" x 38" of the
SN 1604 (Kepler) supernova remnant to search for a possible surviving companion
star. We can rule out, with high certainty, a red giant companion star - a
progenitor indicated by some models of the supernova remnant. Furthermore, we
find no star that exhibits properties uniquely consistent with those expected
of a donor star down to L>10Lsun. While the distribution of star properties
towards the remnant are consistent with unrelated stars, we identify the most
promising candidates for further astrometric and spectroscopic follow-up. Such
a program would either discover the donor star, or place strong limits on
progenitor systems to luminosities with L<<Lsun.
