Broadcast screen over wireless
I'll go ahead and post this as an answer. Android 4.2, including Sony's package on the Z1, supports Miracast - a wireless display standard that's sorta becoming a thing now. Miracast is what Sony uses to connect you to their own TV's, so it's not a proprietary Sony thing (surprisingly).

In order to use Miracast, you need either a Smart TV with the feature built in. Or you can use a Miracast Receiver - they range from $50-$80. I personally use this one - it's worked great for me from Android and Windows without any hiccups.

Note that there is real latency between the phone and the TV's displays. You're looking at a 500ms lag at best, so this won't work well (or at all) for games. It's really good for watching videos, and things that don't require much interaction.