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Let  G =SU(2) and let Ω  G  denote the space of continuous based loops in  G , equipped with the pointwise conjugation action of  G . It is a classical fact in topology that the ordinary cohomology  H *(Ω  G ) is a divided polynomial algebra Γ[ x ]. The algebra Γ[ x ] can be described as an inverse limit as  k →∞ of the symmetric subalgebra in Λ( x  1 , …,  x   k  ), where Λ( x  1 , …,  x   k  ) is the usual exterior algebra in the variables  x  1 , …,  x   k  . We compute the  R ( G )-algebra structure of the  G -equivariant  K -theory  K *  G  (Ω  G ) which naturally generalizes the classical computation of  H *(Ω  G ) as Γ[ x ]. Specifically, we prove that  K *  G  (Ω  G ) is an inverse limit of the symmetric ( S  2 r  -invariant) subalgebra ( K *  G  ((ℙ 1 ) 2 r  ))  S  2 r    of  K *  G  ((ℙ 1 ) 2 r  ), where the symmetric group  S  2 r   acts in the natural way on the factors of the product (ℙ 1 ) 2 r   and  G  acts diagonally via the standard action on each factor.

THE PRODUCT STRUCTURE OF THE EQUIVARIANT K -THEORY OF THE BASED LOOP GROUP OF SU(2)