Positive elements and continuous spectral calculus

An element a is positive when it is self-adjoint and its spectrum lies in the nonnegative reals. In a C*-algebra this is equivalent to being b-star b for some b. In a concrete operator algebra it is equivalent to nonnegative quadratic forms against every Hilbert-space vector. These are equivalent tests, not three unrelated definitions. For a finite self-adjoint matrix they say exactly that every eigenvalue is nonnegative; positive entries alone are neither required nor sufficient.

A positive element has a unique positive square root inside its generated C*-algebra. It may have other, nonpositive square roots, so the qualification matters. We write a at most b when b minus a is positive; both a and b are then understood to be self-adjoint. This order is preserved by conjugating with any element: c-star a c is at most c-star b c. That statement follows directly from the adjoint-square characterization and is much safer than arbitrarily multiplying inequalities on one side.

For a normal element a, the unital C*-algebra generated by a is commutative. Continuous functional calculus gives a unique unital *-isomorphism from C(σ(a)) onto that generated algebra, sending the coordinate function z to a. Consequently a continuous scalar function f has an unambiguous value f(a), its norm is the maximum of |f| on the spectrum, and its spectrum is the scalar image f(σ(a)). These statements include complex-valued functions on a possibly nonreal compact spectrum.

For self-adjoint a the spectrum is real, and continuous real-valued functions produce self-adjoint elements. Nonnegative functions produce positive elements. Applying the scalar square-root function to a positive a constructs its positive square root. Nonunital algebras require a unitization or a formulation using functions vanishing at zero, so a constant function cannot silently produce a missing unit. For a nonnormal element, arbitrary continuous functional calculus on its spectrum is unavailable: a nonzero nilpotent with spectrum {0} already contradicts an isometric coordinate-function calculus.

If a and b commute, they can be treated inside one commutative C*-algebra, and pointwise reasoning applies to them together. Without commutation, an increasing scalar function need not preserve their order. Squaring is increasing on nonnegative real numbers, but it is not operator-monotone on all positive matrices. This is not a failure of functional calculus: each square is correctly defined. The failed inference is the attempt to compare two different noncommuting inputs by a scalar rule.

Functional calculus also distinguishes continuous from measurable operations. A spectral projection for a discontinuous indicator function may not belong to the original C*-algebra, although it belongs to a suitable von Neumann closure in a concrete representation. A continuous function can be approximated uniformly by the allowed algebraic combinations on a compact spectrum; a jump cannot generally be approximated that way. This prepares the later distinction between norm-closed algebras and operator-topology-closed algebras.