Look up brown rot of stone fruits.

Injured fruit ripens first- sometimes from coddling moth which can leave quite a bit of edible flesh but some rot in the center. There's always going to be some wounded fruit from something.

If most of the fruit remains sound I wouldn't worry about it but pick off any fruit that shows rot so it doesn't spread- although in S. CA that may not be much of a threat.

Jean, I think it may be brown rot. The tree was not taken care of when we moved in. It had terrible pruning of large branches so I'm grateful for any fruit at all, really. As I said, the fruit tastes good so I'm going to pick some and see if they will ripen a bit in a brown bag. Thanks for both the responses. And yes, harvestman, there seems to be always something attacking the weak!