Bible Question

Someone help me out. I came across this verse in Acts and trying to figure out how it does not mean what it obviously appears to mean.

Context, first. The event takes place just after Pentecost and the arrival of the Holy Spirit (Acts 2). Peter and John are going to the Temple to pray (Acts 3:1). They come across a man, lame from birth, begging at the gate. Peter gives his famous, "Silver and gold have I none; but such as I have give I thee" (Acts 3:6) line (Can you tell I was raised on King James?) and commands him to stand and is healed. Wow! (Apparently this man didn't bring faith to the equation.) Well, of course, the crowd is all impressed so Peter launches into his next big sermon (Acts 3:12-26). Okay, that's the context. Now, in his explanation of the event they had all witnesses, Peter says this.

"And His name--by faith in His name--has made this man strong whom you see and know, and the faith that is by Jesus has given the man this perfect health in the presence of you all." (Acts 3:16)

Now, I know this is a touchy subject and not a few are unhappy about it, so I'm asking the question. What did Peter mean? Most of it is abundantly clear. The name of Christ made this man strong. Got it. "By faith in His name." Clear. But about this faith Peter says that "the faith that is by Jesus" made this man whole. It looks like Peter is saying that, indeed, this lame man didn't bring his own faith to the equation. It looks like Peter is saying that the "faith in His name" required to heal the man was provided by Jesus. Now, of course, he may have been referring not to any faith on the part of the man. Peter may have been referring to Peter and John's faith in the name of Jesus that healed the man. Fine with me. But it still looks like Peter names the source of that faith as Jesus Himself.

What am I missing?