Projection in a C*-algebra

Let $A$ be a $C^*$ -algebra. A projection in $A$ is an element $p\in A$ satisfying

p=p^*=p^2.

Thus a projection is a self-adjoint idempotent. The definition makes sense whether or not $A$ is unital; the zero element is always a projection, while a multiplicative identity is a projection when it exists. If $A$ is represented faithfully on a Hilbert space, $p$ acts as the orthogonal projection onto its closed range. This abstract definition does not require a particular representation or a subspace chosen in advance.

Spectral and geometric properties

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.

Equivalence and stabilization

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.

Role in 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.

References

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