In 1912 the suburb of Mosman was a quiet place. Taronga Zoological Park was under construction and the tram lines diverted at Spit Junction; with one route continuing down Military Road towards where the new zoo was being built and towards Balmoral Beach, and the other towards The Spit and a wharf which no longer exists. Spit Junction proper was in what is now called Mosman Square and the intersection of Military Road and Spit Road wasn't really anything, except that it had a corner store on what was a proper corner. In July of that year though, the intersection would change dramatically with the construction of two impressive buildings.
On the southwest corner the Bank of New South Wales purchased the land from a resident and then promptly demolished the building and replaced it with a two storey affair which subsequently passed through several hands including the Commonwealth Bank and the Bank of Australasia before eventually being divided in half and becoming offices and a jeweler's shop, which is the current configuration of the building today. Next to that though, on the city side, another building was erected which has a slightly more amusing history.
From what I can gather, the first building on the site was built by James Halstead in 1899. There is some speculation that he was the person who gave the building its name and that this was simply carried over to the new building; however I can't corroborate that with any records that I found in the local history unit of the library. The more likely story is told below.
Warning: Some of this might be wildly untrue. The following contains weapons of mass speculation.
Benjamin Nelson who was a property developer, built his own building which he hoped would be filled with fancy shops. Unfortunately, as the banks decided to locate themselves along Military Road and towards the zoo, the retail traffic that his imagined fancy shops would attract, never materialized. In a bid to bring tenants to his building, Nelson did something which is kind of remarkable. Instead of painting the building or reducing the rent, he ordered that a fancier façade be built on the roof and he decided to put a name on the building. Naming a building is not particularly uncommon but what the name that Nelson had affixed to his new façade wasn't the name of the building: Nelson had chosen to rename the intersection. Moreover; owing to the fact that his name was Nelson, the name of the intersection would be Trafalgar Square. That name is still on the building today.
A lot of names for places are arrived at through historical accident that just happens to settle. Chatswood for example was originally named Chattie's Wood after Charlotte Harnett, wife of then Mayor of Willoughby. That name has been shortened as is often the way in Australia. The Meccano Set is an intersection which is so named because of the steel signage gantry which hangs above it. I'm sure that whoever Thompson's Corner in Pennant Hills and Pearce's Corner in Hornsby are named after are mostly lost to obscurity but the names have lasted longer than their fame.
Trafalgar Square in Mosman sounds like it was named after the maddening gyratory circus in London and maybe it was but at the time of naming, it was a funny shaped t-junction; with a building owner on an ego trip.
It certainly provides a vast contrast to the Trafalgar Square in London which got its name after the great Admiral Lord Horatio Nelson aboard HMS Victory commanded the British fleet to an unmitigated smack down victory (22 French and Spanish ships lost to the British 0) off the coast of Cape Trafalgar.
Even though the name has stood proudly over the intersection for more than a hundred years though, it hasn't stuck. When the tram lines were ripped up by the Cahill Government, what was a sort of mish-mash of an intersection, was turned into a fairly normal t-junction with Military Road towards the zoo becoming a reasonably minor road. Nelson's building with the name Trafalgar Square, found itself in a busier place but still nobody stopped because the road became a main arterial. Worse, the name Spit Junction shifted from the tram junction when it disappeared, to the t-junction which previously hadn't borne the name.
I don't know if Benjamin Nelson was alive by that stage but I can imagine him being petulant when nobody called the t-junction of Military Road and Spit Road the name which he had bestowed upon it: Trafalgar Square. The name still survives as a hundred year old appendix to a story that no-one is really sure about and as evidenced by the fact that everyone calls the intersection Spit Junction, they don't care about it either.