  We extend some aspects of vacuum string field theory to superstring field
theory in Berkovits' formulation, and we study the star algebra in the
fermionic matter sector. After clarifying the structure of the interaction
vertex in the operator formalism of Gross and Jevicki, we provide an algebraic
construction of the supersliver state in terms of infinite-dimensional
matrices. This state is an idempotent string field and solves the matter part
of the equation of motion of superstring field theory with a pure ghost BRST
operator. We determine the spectrum of eigenvalues and eigenvectors of the
infinite-dimensional matrices of Neumann coefficients in the fermionic matter
sector. We then analyze coherent states based on the supersliver and use them
in order to construct higher-rank projector solutions, as well as to construct
closed subalgebras of the star algebra in the fermionic matter sector. Finally,
we show that the geometric supersliver is a solution to the superstring field
theory equations of motion, including the (super)ghost sector, with the
canonical choice of vacuum BRST operator recently proposed by Gaiotto,
Rastelli, Sen and Zwiebach.
