Contractor did work that was not authorized be me - am I liable?

[Howie W.]

BLUF - a maintenance company my property manager uses approved the installation of an A/C unit to a third party vendor without our authorization. 

The installation was completed last September 2021 and I received the information from the PM last week (April 2022). When the request came last September, I declined the first vendor and wanted a second opinion/quote. The second vendor mentioned the A/C was working, but a part could fail and it was no longer in stock. I talked with the second vendor directly and told them directly that when the part fails, we'll begin to explore a new A/C system. 

Shortly after, the first vendor approved the work order and sent it to a third party. Fast forward to April of this year and my PM informs me of the invoice. I declined the full amount and asked for an itemized invoice to determine the cost of parts and to perhaps pay a portion of that. This is after discussing the matter with other owners, property managers, and A/C Companies. I have been getting several different numbers from my PM and the Maintenance company on the actual cost - from $7450 to $4750, but not an itemized list of parts - at least one I can feel comfortable with (cost of "parts", which was only one line and broken down specifically was approx. $4200 and not from the third party vendor who performed the installation. Note - the maintenance company is not a certified HVAC company. 

The PM came down to a $4750 cost total and would pay $1000 of the cost if I pay the remainder. My offer was $1500, which would be part of the parts, but I was never given a firm number that I am conformable with. This was sent via email, but during our phone conversation the PM mentioned a potential lawsuit/collection - apparently from the Maintenance Company. My reply was I never authorized a repair and would also inform the BBB and FTC. 

Thoughts? Advice? Am I liable for any maintenance cost that I did not authorize? My offer of $1500, IMO, was more than fair to help compensate the third party vendor. This is the short story, as there are other details I can fill in.