Accept or deny center pawn trade?

In general, I find myself declining the trade, and hope that the space pressure forces my opponent into sub-optimal moves.  It stops a N from getting to c6 (or f6).  I also know that this setup is key as lead-in for the Greek Gift sacrifice.

Of course there are situations where the pawn trade is advisable:

I think that leaving the tension in the center as long as possible without making positional concessions is the best way to go about it. Of course you should break that rule if you can calculate something better.

 However for your second example, that position is very drawish at higher levels. You did deny black to right to castle, but with queens off the board and no pieces developed, it matters very little. Black will look to trade off some minor pieces and then the fact that he had not castled becomes an asset, since his king is closer to the center for the endgame. It should be noted that black is in no way playing for a win in that position, but its very tricky for white to come up ahead, which is why White typically avoids the trade.

I do accept pawn trades when I know that it will be strategically or tactically useful for me in the future. If I can claim the newly opened files and use them to my advantage, I don't hesitate to trade.
On the other hand, if the trading of pawns doesn't make my position any better, I usually advance, in hope of a long-term space advantage. However, I can also simply keep the tension in the center and ignore the pawns, developing my pieces, if that results in a good position.

Thanks guys, great advice so far.

On "keeping the tension": this is a concept I'm not very comfortable with yet.  It's always bewildering when I watch high level players go at it, and they leave stuff hanging out all the time.  I don't feel like I have enough brain-space to hold all those little "to-do" items, in addition to my own plans, and figuring out what my opponent is trying to do.

Is there somewhere I can find out more about that idea?  I think it's something that could benefit my play.

Word of advice in general when it comes to trades, and this isn't limited to pawn trades.

Only trade when it's absolutely necessary (for some defensive reason due to a threat) or when it gains you something (i.e. The win of material, or some poisitonal factor, like control of the only open file, or an eternal knight, which is a knight on a very strong post where no pawn can ever attack it, and it's likely that your opponent's only minor piece is the Bishop of the color opposite the color square the Knight sits on, just to name a couple).

Otherwise, this is an adage you ought to always keep in mind: 

"The initiator of any trade always loses a tempo!"

The best way to illustrate this is imagine the d-file as the only open file.  Black has Rooks on a8 and d8.  White has Rooks on d1 and f1.  It's White to move.  Both sides have 1 rook that controls the open file, and one that doesn't.  If White plays 1.Rxd8, Black can recapture the Rook with his other Rook, and now Black's only Rook controls the d-file, White's doesn't.  To contest the d-file, White would have to play 2.Rd1, and now instead of it being White to move, it's Black to move, and so White cost himself a tempo to eliminate the a8 and f1 Rooks (in essence).  Instead, If White makes another move, and Black initiates the trade, then White recaptures on d1, Black must spend time to reload with ...Rd8, and Now it's White's move again, and White already made a productive move with another piece!

In essence, this is the whole basic concept of "Tension", and the object is to play moves that will force your opponent to break the tension as an inopportune time.  For example, if you can get your Bishop to hit d8 without it being captured and without the ability for Black to block it, you force Black to take on d1 (or abandon the d-file, or move forward, like to say, d6, where he'll have to recapture with something other than the Rook, then a reload of Rd1 after trading rooks might be to your benefit, like if now instead of contesting an open d-file, you are attacking a backwards pawn, for example.

So keep that in mind the next time you think about initiating trades.  I only initiate trades in the following conditions:

1) It leads to a winning combination, like it deflects a defender, and I win material afterwards.

2) It's necessary to prevent the opponent from gaining a strong position.  For example, after 1.e4 c5 2.Nf3 Nc6 3.d4, it's necessary to eliminate the d-pawn with 3...cxd4.  The consequences are worse than initiating the trade, even though it brings White's Knight to a central square.

3) I'm already up material, and the loss of a tempo is worth it to eliminate potential defenders or potential attackers that may cause problems via attacking my King.  Just because you are up a pawn or two doesn't mean you are winning.  You have to consider every circumstance individually.

4) The opponent has wasted time as well to force me to do it.  In this case, you really aren't losing a tempo because the opponent's move is also a time waster.  A prime example of this is in the Saemisch Variation of the Nimzo-Indian Defense.  After 1.d4 Nf6 2.c4 e6 3.Nc3 Bb4, White's move of 4.a3 virtually forces Black to take on c3 and allow White to get an additional pawn that controls a central square (i.e. after 4...Bxc3+ 5.bxc3, the extra c-pawn controls d4).  However, a3 is not exactly the most productive move for White.  It's a legitimate line, but you can't say that a3 was a tempo-gainer.  White would rather be able to play a move like 4.e3, or 4.Nf3, or 4.Qc2, and watch Black voluntarily obligue with 4...Bxc3+, which Black should avoid until White forces him to do so with the exception of the Hubner Variation, which has other positional reasons to take on c3 without White advancing a3 first.

Nobody ever said chess was an easy game, and generalities outside of the very basic ones, like control the center, won't get you very far.

Keeping the tension can be particularly useful in the opening when saving time is so important.  Notice if you spend a move on capturing, and your opponent recaptures, the net result for the pair of moves may be he develops a piece with the recapture while the piece you moved is gone from the board, so he gains time.  This is generally why you often seen strong players just keep bringing out new pieces in the opening.

As for the pawn structure, you've gotten some good advice already... you just have to know what each structure means for the middlegame.  Usually it's simply a matter of preference.  But if you do want some general guidelines, what you should be working for in the opening is to establish and maintain a pawn in the center, and to bring your pieces into the game as quickly as possible.  Capture, advance, or ignore, is not as important as meeting those goals.  If you do, then it's a good move whatever it is.  As you learn more about the middlegame plans associated with different openings you can being to develop a preference.

It depends on the situation.

Generally I prefer to have 2 pawns in the center, but if there is a good reason to trade them, then it is OK.

If there is no reason at all, then NO.

You should play your game, not your opponent's game.

Black mar-Diemer Gambit into an Advance French or exchange variation.Check it out!!!

GL!
 

It's important to remember that in the French, it's the Black Pawn on e6, not the White Pawn on e5, that confines Black's c8-Bishop.

All other things being equal... which they rarely are... I prefer to keep the tension in the center.