We present an analysis of the diffuse X-ray emission in 19 compact groups of galaxies (CGs) observed with Chandra. The hottest, most X-ray luminous CGs agree well with the galaxy cluster X-ray scaling relations in $L_X-T$ and $L_X-sigma$, even in CGs where the hot gas is associated with only the brightest galaxy. Using Spitzer photometry, we compute stellar masses and classify HCGs 19, 22, 40, and 42 and RSCGs 32, 44, and 86 as fossil groups using a new definition for fossil systems that includes a broader range of masses. We find that CGs with total stellar and HI masses $gtrsim10^{11.3}$ M$_odot$ are often X-ray luminous, while lower-mass CGs only sometimes exhibit faint, localized X-ray emission. Additionally, we compare the diffuse X-ray luminosity against both the total UV and 24 $mu$m star formation rates of each CG and optical colors of the most massive galaxy in each of the CGs. The most X-ray luminous CGs have the lowest star formation rates, likely because there is no cold gas available for star formation, either because the majority of the baryons in these CGs are in stars or the X-ray halo, or due to gas stripping from the galaxies in CGs with hot halos. Finally, the optical colors that trace recent star formation histories of the most massive group galaxies do not correlate with the X-ray luminosities of the CGs, indicating that perhaps the current state of the X-ray halos is independent of the recent history of stellar mass assembly in the most massive galaxies.
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