C*-algebras and the C*-identity

An involution sends a to a-star, is conjugate-linear, reverses multiplication order, and squares to the identity operation. In complex matrices it is conjugate transpose, not plain transpose. In continuous complex-valued functions it is pointwise complex conjugation. An element is self-adjoint when it equals its adjoint, normal when it commutes with its adjoint, and unitary when multiplying by its adjoint on either side gives the unit. These are different conditions.

The C*-identity says the norm of a-star times a equals the square of the norm of a. Along with the Banach-algebra structure, it makes the involution isometric. For bounded operators on a complex Hilbert space, the identity follows from the definition of the adjoint and the operator norm. Using another matrix norm can preserve submultiplicativity yet destroy this identity, so calling an algebra a C*-algebra requires checking the compatible norm rather than only listing its operations.

The algebra B(H) of all bounded operators on a complex Hilbert space is unital and complete in operator norm. A norm-closed *-subalgebra is again a C*-algebra, though it may not contain B(H)'s identity. On an infinite-dimensional Hilbert space, the compact operators form a nonunital C*-algebra. The finite-rank operators form a *-algebra but are not norm closed; their completion is the compact-operator algebra. Thus finite-dimensional approximability does not itself supply a unit or completeness.

For a locally compact Hausdorff space X, continuous functions vanishing at infinity form C₀(X), with supremum norm and pointwise operations. If X is compact, this becomes C(X), containing the constant function one. If X is noncompact, that constant does not vanish at infinity. This provides a commutative nonunital example without using matrices. Whenever we later write one or normalize a state by its value at one, we will explicitly be in the unital setting or use the appropriate nonunital definition.

For a normal element of a C*-algebra, the spectral radius equals its norm. Applying this to the positive element a-star a links the norm to spectral information intrinsic to the *-algebra. In particular, there is at most one C*-norm making a fixed unital *-algebra complete. This rigidity is stronger than the flexibility of norms on a general vector space. It does not say that every abstract *-algebra admits such a completion or such a norm.

A *-homomorphism between C*-algebras is automatically contractive, and an injective one is isometric. In the unital case, one proof first compares spectra of self-adjoint elements and then uses the C*-identity for general elements. These conclusions concern maps preserving the adjoint; an arbitrary algebraic linear map need not share them. The representation theorem that every C*-algebra admits a faithful Hilbert-space representation will be reached through states, rather than assumed as the definition of every abstract algebra.