Loop gain less than one is stable
I think, some clarification is necessary:

When A is the open-loop gain of the amplifier (without feedback) and beta is the feedback factor, it is important if the feedback network is connected to the inverting (negative feedback) or the non-inverting input of the opamp.

In most cases (amplification) we have negative feedback and the closed-loop gain is 

Gcl=A/(1+A*beta)

However, with respect to the correct sign, it is important to realize that the quantity we call LOOP GAIN includes the negative sign at the summing node, hence 

Loop Gain LG=-A*beta

It is important to include the negative sign at the summing node in the loop gain definition because that is the way we are measuring/simulating the loop gain function: We open the loop at a suitable point and measure the total gain (and phase) of the whole loop - of course including the minus sign).

Therefore:

Gcl=A/(1-LG)

Note that Barkhausen`s oscillation condition (there is no "stability criterion" from Barkhausen) requires LG=1 for a circuit to oscillate (unity loop gain with a phase shift of zero deg).

That means: All feedback circuits are stable for LG&lt;1 (examples: pos. feedback with LG=0.5 or negative feedback with LG&lt;0).

I hope this clarifies some points.