  This article takes up the challenge of extending the classical Real
Nullstellensatz of Dubois and Risler to left ideals in a *-algebra A. After
introducing the notions of non-commutative zero sets and real ideals, we
develop three themes related to our basic question: does an element p of A
having zero set containing the intersection of zero sets of elements from a
finite set S of A belong to the smallest real ideal containing S? Firstly, we
construct some general theory which shows that if a canonical topological
closure of certain objects are permitted, then the answer is yes, while at the
purely algebraic level it is no. Secondly for every finite subset S of the free
*-algebra R<x,x*> of polynomials in g indeterminates and their formal adjoints,
we give an implementable algorithm which computes the smallest real ideal
containing S and prove that the algorithm succeeds in a finite number of steps.
Lastly we provide examples of noncommutative real ideals for which a purely
algebraic non-commutative real Nullstellensatz holds. For instance, this
includes the real (left) ideals generated by a finite sets S in the *-algebra
of n by n matrices whose entries are polynomials in one-variable. Further,
explicit sufficient conditions on a left ideal in R<x,x*> are given which cover
all the examples of such ideals of which we are aware and significantly more.
