Erratum to: Acta Appl. Math. (2014) DOI
10.1007/s10440-014-9976-y
There is an error in the proof of Theorem 2.15 in the article Generalizing the Kantorovich Metric to Projection Valued Measures [1]. The result is still true. The theorem is restated below:
$$E(\cdot) \mapsto\sum_{i=0}^{N-1} S_i E\bigl(\sigma_i^{-1}(\cdot) \bigr)S_i^* $$
The following is a corrected version of the part of the proof which contains the error. The author wants to acknowledge Krystal Taylor (University of Minnesota) for identifying the error.
The following argument replaces the argument which begins on the line after Claim 2.16. Let E,F∈P(X). Recall that r=max0≤i≤N−1{r
i
}, where r
i
is the Lipschitz constant associated to σ
i
, and note that 0<r<1. Choose ϕ∈Lip1(X), and \(h \in\mathcal{H}\) with ∥h∥=1. Then
$$\begin{aligned} & \biggl\vert \biggl\langle \biggl(\int\phi d\varPhi(E) - \int\phi d\varPhi (F) \biggr)h, h \biggr\rangle \biggr\vert \\ &\quad = \biggl\vert \biggl\langle \biggl(\int\phi d\varPhi(E) \biggr)h,h \biggr\rangle - \biggl\langle \biggl( \int\phi d\varPhi(F) \biggr)h, h \biggr\rangle \biggr\vert = \biggl\vert \int_X \phi d\varPhi(E)_{h,h} - \int _X \phi d\varPhi (F)_{h,h} \biggr\vert \\ &\quad = \Biggl\vert \sum_{i=0}^{N-1} \int _X \phi dE_{S_i^*h, S_i^*h}\bigl(\sigma _i^{-1}( \cdot)\bigr) - \sum_{i=0}^{N-1} \int _X \phi dF_{S_i^*h, S_i^*h}\bigl(\sigma_i^{-1}( \cdot)\bigr) \Biggr\vert \\ &\quad = \Biggl\vert \sum_{i=0}^{N-1} \int _X (\phi\circ\sigma_ i)dE_{S_i^*h, S_i^*h} - \sum _{i=0}^{N-1} \int_X (\phi \circ\sigma_i) dF_{S_i^*h, S_i^*h} \Biggr\vert \\ &\quad = \Biggl\vert \sum_{i=0}^{N-1} \biggl( \int _X (\phi\circ\sigma_ i)dE_{S_i^*h, S_i^*h} - \int _X (\phi\circ\sigma_i) dF_{S_i^*h, S_i^*h} \biggr) \Biggr\vert \\ &\quad = \Biggl\vert \sum_{i=0}^{N-1} r \biggl( \int _X \biggl( \frac{\phi\circ\sigma_ i}{r} \biggr) dE_{S_i^*h, S_i^*h} - \int_X \biggl(\frac{\phi\circ\sigma _i}{r} \biggr) dF_{S_i^*h, S_i^*h} \biggr) \Biggr\vert \\ &\quad \leq r \Biggl( \sum_{i=0}^{N-1} \biggl\vert \int _X \biggl( \frac{\phi\circ\sigma_ i}{r} \biggr) dE_{S_i^*h, S_i^*h} - \int_X \biggl( \frac{\phi\circ \sigma_i}{r} \biggr) dF_{S_i^*h, S_i^*h} \biggr\vert \Biggr) \\ &\quad =r \Biggl( \sum_{i=0}^{N-1} \biggl\vert \biggl\langle \biggl( \int \biggl(\frac {\phi\circ\sigma_i}{r} \biggr) dE - \int \biggl( \frac{\phi\circ\sigma _i}{r} \biggr) dF \biggr)S_i^*h, S_i^*h \biggr\rangle \biggr\vert \Biggr) \\ &\quad \leq r \Biggl( \sum_{i=0}^{N-1} \biggl\Vert \int \biggl( \frac{\phi\circ \sigma_i}{r} \biggr) dE - \int \biggl( \frac{\phi\circ\sigma_i}{r} \biggr) dF \biggr\Vert \bigl\Vert S_i^* h\bigr\Vert ^2 \Biggr). \end{aligned}$$
Note that the function \(\frac{\phi\circ\sigma _{i}}{r} \in\text{Lip}_{1}(X)\) for all 0≤i≤N−1. Hence
$$\begin{aligned} & r \Biggl( \sum_{i=0}^{N-1} \biggl\Vert \int \biggl( \frac{\phi\circ \sigma_i}{r} \biggr) dE - \int \biggl( \frac{\phi\circ\sigma_i}{r} \biggr) dF \biggr\Vert \bigl\Vert S_i^* h\bigr\Vert ^2 \Biggr) \\ &\quad \leq r \rho(E,F) \Biggl( \sum_{i=0}^{N-1} \bigl\langle S_i^* h, S_i^* h \bigr\rangle \Biggr) = r \rho(E,F) \Biggl( \sum_{i=0}^{N-1} \bigl\langle S_iS_i^* h, h \bigr\rangle \Biggr) \\ &\quad = r \rho(E,F) \Biggl\langle \Biggl( \sum_{i=0}^{N-1} S_iS_i^* \Biggr) h, h \Biggr\rangle = r \rho(E,F) \langle h, h \rangle = r \rho(E,F). \end{aligned}$$
Therefore
$$\biggl\Vert \int\phi d\varPhi(E) - \int\phi d\varPhi(F) \biggr\Vert \leq r \rho(E,F). $$
Since ϕ is an arbitrary element of Lip1(X),
$$\rho\bigl(\varPhi(E), \varPhi(F)\bigr) \leq r \rho(E,F). $$
This proves that Φ is a Lipschitz contraction in the ρ metric on P(X).
