From Peter F. Basile, John Landon-Lane, and Hugh Rockoff's "Money and Interest Rates in the United States during the Great Depression" (2010):
This paper reexamines the debate over whether the United States fell into a liquidity trap in the 1930s. We first review the literature on the liquidity trap focusing on Keynes's discussion of “absolute liquidity preference” and the division that soon emerged between Keynes, who believed that a liquidity trap had not been reached, and the American Keynesians who believed that the United States had fallen into a liquidity trap. We then explore several interest rates that have been neglected in previous analyses: yields on corporate debt (from Aaa to junk), bank lending rates, and mortgage rates. In general, our results strengthen the case for believing that there was no liquidity trap in the 1930s in the sense of one that covered the full spectrum of interest rates. The small segment of time in which a liquidity trap might have been present, however, makes drawing firm conclusions risky.
At least from my perspective, I think Basile, Landon-Lane, and Rockoff are either confused or confusing. I have always understood "liquidity trap" to mean a situation in which cash is effectively a perfect substitute for Treasury bills and in which as a result open-market operations in their standard form have no effect on anything.
However, that does not mean that central banks are powerless in a liquidity trap:
By taking duration, default, and systemic tail risk onto their balance sheets, they can diminish default and risk premia on debt instruments other than short-term Treasury bills.
By changing expectations of future inflation rates, they can alter business decisions without taking any action to change the current levels of nominal interest rates.
The problem is that these attempts to influence aggregate demand are likely to be less effective and certainly to have less predictable effects than normal open-market operations in normal times. Thus there is here an analytical distinction worth preserving. And the central bank is not necessarily the most effective organization for the task of altering duration, default, and systemic tail risk premia.
Basile et al. seem to me to be redefining "liquidity trap" so that an economy is only in a liquidity trap when central banks can do nothing to affect the structure of nominal interest rates anywhere along the yield curve. But if that is the definition, then I don't believe a liquidity trap can ever exist.