Are there partial isometries, isometries?

Question

Given a unital $C^*$-algebra A and partial isometries $w_1, \cdots, w_n$ such that $\sum_{i = 1}^{n}w_iw_i^* = 1$ and $w_i^*w_j = 0$ if $i \neq j$ then is it true that $w_i$ is an isometry?

Answer 1

If you take your C* Algebra to be the set of operators on a two dimensional Hilbert space you can consider the following counter example:

Let $w_1$ be the projection on a 1 dimensional subspace and $w_2$ the projection on the orthogonal complement. As these are projections the partial isometry conditions are satisfied. As they are projections on orthogonal subspaces that span the entire space: $1=w_1 + w_2=w_1^*w_1 + w_2^*w_2$ and $w_1^* w_2 = 0 = w_2^* w_1$. However the $w_i$ themselves are not isometries.