Pvm integral is a star homomorphism

Unitality: ${\mathbf{1}}_X$ is simple, and ${\mathrm{\Phi }}_E({\mathbf{1}}_X)=E(X)=I$ by the definition of the simple integral and $E(X)=I$.

Linearity on simple functions: presenting $s$ and $t$ over a common refinement of their disjoint normal forms, $\int (as+bt)dE=a\int sdE+b\int tdE$ because both sides are the corresponding coefficient-weighted sum of the same projection values.

Multiplicativity on simple functions: over a common disjoint normal form $s={\sum }_j{c}_j{\mathbf{1}}_{D_j}$, $t={\sum }_j{d}_j{\mathbf{1}}_{D_j}$ one has $st={\sum }_j{c}_j{d}_j{\mathbf{1}}_{D_j}$ and $(\int sdE)(\int tdE)={\sum }_{j,k}{c}_j{d}_{k}E(D_j)E(D_k)={\sum }_j{c}_j{d}_{j}E(D_j)=\int stdE$, because $E(D_j)E(D_k)=E(D_j\cap D_k)$ vanishes for $j\mathrm{\ne }k$ and equals $E(D_j)$ for $j=k$.

Conjugation on simple functions: self-adjointness of the projection values and conjugate-linearity of the adjoint give $(\int sdE{)}^{\ast }=({\sum }_j{c}_{j}E(D_j){)}^{\ast }={\sum }_j\overline{c_j}E(D_j)=\int \bar{s}dE$. The bounded integral agrees with the simple integral by taking a constant approximating sequence.

Linearity, multiplicativity and conjugation pass to uniform limits: if $s_n\to f$ and $t_n\to g$ uniformly then $a{s}_n+b{t}_n\to af+bg$, $s_n{t}_n\to fg$ and $\overline{s_n}\to \bar{f}$ uniformly, and A1 gives convergence of the simple integrals to the integrals of each of these limits, so ${\mathrm{\Phi }}_E(af+bg)=a{\mathrm{\Phi }}_E(f)+b{\mathrm{\Phi }}_E(g)$, ${\mathrm{\Phi }}_E(fg)={\mathrm{\Phi }}_E(f){\mathrm{\Phi }}_E(g)$ and ${\mathrm{\Phi }}_E(\bar{f})={\mathrm{\Phi }}_E(f{)}^{\ast }$ by taking norm limits and using norm continuity of the adjoint.

Strong convergence: if ${\sup }_n\mathrm{\parallel }{f}_n{\mathrm{\parallel }}_{\mathrm{\infty }}\le C<\mathrm{\infty }$ and $f_n\to f$ $E$-almost everywhere, then for each $x$ the functions $\mathrm{\mid }{f}_n-f{\mathrm{\mid }}^2$ converge to $0$ $E_x$-almost everywhere and are dominated by the constant $(C+\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}{)}^2$, which is integrable for the finite measure $E_x$; hence $\mathrm{\parallel }{\mathrm{\Phi }}_E(f_n)x-{\mathrm{\Phi }}_E(f)x{\mathrm{\parallel }}^2=\int \mathrm{\mid }{f}_n-f{\mathrm{\mid }}^{2}d{E}_x\to 0$.

${\mathrm{\Phi }}_E$ is a unital star homomorphism on the bounded measurable functions, and bounded pointwise $E$-almost everywhere convergence with a uniform bound implies strong convergence of the operators.