Projection in a C*-algebra

The spectrum of a projection lies in $\{0,1\}$. Every nonzero projection has norm one, and $1-p$ is a projection when $A$ is unital. In a concrete operator algebra, self-adjointness distinguishes orthogonal projections from general idempotent operators, whose range and kernel need not be orthogonal. These facts follow directly from the continuous functional calculus Murphy, §2.2.

Projections $p$ and $q$ are Murray–von Neumann equivalent when there is a partial isometry $v$ with $v^*v=p$ and $vv^*=q$. This records isomorphism of the corresponding projective modules and is weaker than equality. One also studies projections in matrix algebras $M_n(A)$; block sum provides the stabilization operation used in operator $K$-theory.

For a unital $C^*$-algebra, the group $K_0(A)$ is built from stable equivalence classes of projections in the matrix algebras over $A$. For a nonunital algebra, one passes to a unitization and retains the kernel of the map induced by the scalar quotient. Consequently, “a projection defining a $K_0$-class” may live in a matrix algebra or a unitization rather than in $A$ itself.

1. Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. Publisher record. Relevant: Chapter III, especially §5, on projections and $K_0$.
2. Gerard J. Murphy, $C^*$-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §§2.2–2.3 on projections and functional calculus.