An intrepid developer has created a free Google Music desktop app for OS X, borrowing the Google Music name and logo in the process. You shouldn't have to Google "trademark law" to understand that Google's legal department may be readying a letter to Sajid Anwar, the developer who maintains the app or Jame Fator, its creator. On Hacker News, some have reasoned that such a blatant use of Google's trademarks has to mean the developers seek Google's attention so that their app or they themselves may get acquired by Google.
Whatever the reason behind its development, the Google Music for Mac is available on GitHub at the time of this writing. The app is actually a custom stylesheet that emulates the appearance of a native OS X application. It is unquestionably slick. It mimics the look and feel of the Google Music Web app, complete with the same buttons and controls and settings, and yes, the Google Play name and logo at the top, while adding a useful feature you don't get with the Web app: support for your Mac's media keys.
Using the Google Music for Mac app lets you play/pause, fast-forward, and rewind tracks using the keys at the top of your Mac's keyboard, which means you can control the action without leaving your current window or tab. This feature alone makes the app worth the download. And on my 2011 MacBook Pro laptop, I found that the desktop app was much more responsive than the Web app.
While you get the same music labs settings with the desktop app that you do with the Web app -- the ability to enable desktop notifications, HTML5 audio, ratings, comments, and the Chromecast Fireplace Visualizer -- there are two themes to choose from available in Google Music > Preferences from the menu bar. When the box is checked for "Emulate native application appearance," the top banner makes the app look more like a Mac app. When it's unchecked (and after restarting the app), the top banner of the app looks like what you are accustomed to with the Google Music Web app.
It's too bad you can't choose a theme that gets rid of the Google Play logo and changes the wording to "Guugle Play" or somesuch that may keep Google's lawyers at bay a while longer.
Google Music for Mac requires Mac OS X 10.8 or later and Adobe's Flash Player.