Let $A$ be a unital C*-algebra.
Let me define its "$K$-theory space" to be the image of its $K$-theory spectrum under the functor $\Omega^\infty:$ Spectra $\to$ Spaces. I denote the $K$-theory space of $A$ by $K(A)$.
Let $\mathcal K$ be the C*-algebra of compact operators on an infinite dimensional Hilbert space (usually taken to be separable).
Write $Pr(A \otimes \mathcal K)$ for the space of projections in $A \otimes \mathcal K$. This is an $E_\infty$-space with respect to the operation of direct sum (the latter involves reshuffling the coordinates on $\mathcal K$)
Question: Is $K(A)$ equivalent to the group completion of $Pr(A \otimes \mathcal K)$?
I'm being told by Ulrich Pennig that the above statement is false for non-unital algebras, as there exist non-unital C*-algebras which are stably projectionless. They contain no projections, even after forming the tensor product with $\mathcal K$. ($C_0(\mathbb R)$ is such an algebra.)