I absolutely love the terrazzo. Being from Palm Springs, I think your very lucky! Ceiling has to be painted white, it reflects off the floor, and makes the space feel lighter. The supports need some weight, potted plants would help. Furnishings need to be cohesive, and entry door has to be defined. Move dining to the right, runner for front door. Good luck!

Wow! What a distinctive entryway! Completely different than anything in my part of the world. Here's a few thoughts on how to add some character and some green...

You could build square planters at the base of the posts, with a vacant (sort of "hollowed out") center to prevent problems with the posts, painted white. Shrubs/flowering vines to run up at the post-side, annuals at the outer edge. Paint the ceiling either pale blue or white, and the purple portion white. It looks like there's a GIANT fan out there. Would it be helpful to wire for a couple of outdoor ceiling fans in white at the two beams just off of the center one? And it also looks like you have a mobile (?) which you might move more towards center of the space. A few large hanging ferns along the outer edges of the very outside beams would be great. A runner and some potted plants to define the front entry, as Marc suggested. (Don't want to sound like a nit-picker, but I noticed the white tags on the outdoor dining room set which you could pull off for a cleaner look.) Move the curved back bench over by the stone wall. I'd visit this terrific outdoor space regardless, for sure! :)

I love your wall also, it is pretty different than the house. What if you beefed up the supports of the porch with similar materials to the rock wall? Paint the built up posts (above the rock) a darker olivie-ish green and painted the ceiling a white with green tint. The greenish color will help balance out all the reds. Add large pots and plants...a huge outdoor rug. With the size of that space you could add a rock wall and/or a fireplace. I love your covered space it is uber cool. Marc Russell has great ideas!

Some ideas: Windows in the soffit (or at least the same color as the 'ceiling'). Bigger pillars and maybe some woodwork in the gable. Dark trims (windows and doors) and offwhite ceiling..
And change the lighting, a little more indirect lighting would give a more romantic feel. One smaller fixture in the middle should do. Also change the layout of the furniture. A stonegrey wicker lounge set and a sleek steel black/grey (or white with a bright color) dining set will update the look even more. Place the teak bench somewhere further out on the patio/garden.

Painting the doors, window trims and soffit one bold color will look great too.
Make sure to choose a color that goes well with the brick.

I LoVE your ideas! I'm definitely going to add a fire pit, and I really like the thought of adding stone to the base of the posts.

The light fixture is actually pretty cool. It is a light from a surgical suite, circa 1950's. the original owner was a physician!

Concrete and steel gray might be a good option and easy to combine other colors with.

A fire pit next to the roof overhang I assume?
Otherwise the ceiling will turn black..