I would like to Know if it is possible to open a Personal Geodatabase of Esri with Openlayer with FileGDB_API_VS2010_1_2
Thank in advance
You can create an MXD from that personal geodatabase and publish it via ArcGIS Server. This will allow you to expose the data in a standardized OGC format (WMS, WFS) so that OpenLayers can read it.
It is possible to consume data from an ESRI Personal GDB using OpenLayers, but not by using the File GDB API. The reason is that the Personal GDB and the File GDB are two completely different storage formats, and thus have different programmatic access methods.
The Personal GDB is built inside a Microsoft Access database.
The File GDB is a completely file-based structure somewhat similar to, but far more complex than, an ArcINFO coverage.
Here is ESRI's description of the different types: Types of geodatabases
In order to access the Personal GDB in Openlayers, you may need to make it a multi-step process. It is important to note that this process would be the same when using a File GDB, as the API you are referring to has now been incorporated into the GDAL/OGR tools as well. There are significant limitations to be aware of as mentioned here: ESRI File GDB API
Please note that this is one method, using open-source tools, of accomplishing your task. There are undoubtedly other open-source mthods, as well as ways that utilize ESRI products. Using ESRI products would remove the need for the File GDB API, but would include additional licensing costs.
If you are not necessirly after a client-server setup, but just looking for a software tool to inspect and manipulate a personal gdb, check out QGIS desktop GIS. It makes use of the OGR library mentioned before to open the ESRI file format.