There is a theory that fractional reserve banking is a fraud, allowing banks to create money "out of thin air". The theory implies that honest banking must be full reserve.

This theory hinges on the definition of the word "bank". Rothbard makes the above argument in "Man, Economy, and State", but explicitly limits his definition of a bank to that of a "warehouse" for money:

When a man deposits goods at a warehouse, he is given a receipt and pays the owner of the warehouse a certain sum for the service of storage. He still retains ownership of the property; the owner of the warehouse is simply guarding it for him. When the warehouse receipt is presented, the owner is obligated to restore the good deposited. A warehouse specializing in money is known as a "bank."

Murray Rothbard: Man, Economy, and State

Banks do offer this warehousing service, in the name of safe deposit. But banks are not so narrowly defined. They also generally offer interest-bearing accounts such as saving deposit and term deposit. Rothbard uses the expectation of interest to differentiate warehousing money from lending it:

Someone else's property is taken by the warehouse and used for its own money-making purposes. It is not borrowed, since no interest is paid for the use of the money.

In other words, his call for full reserve does not apply to interest-bearing accounts. However he neglects to point out that interest earned on the money represented by deposits can legitimately offset otherwise necessary account fees.

Banks often offer demand deposit (e.g. checking) accounts without interest. The fact of positive yield on the account is not the demarcation between warehousing and lending, even by his own definition. Where a bank account yields 5% at a fee rate of 6%, no distinction from 0% yield with a 1% fee rate exists. The distinction is the contractual agreement between the depositor and the bank.

Since it is convenient to transfer paper in exchange rather than carry gold, money warehouses (or banks) that build up public confidence will find that few people redeem their certificates.

Money certificates representing warehoused money are representative money, a form of money substitute. In the United States, state banks and others formerly issued such certificates. These were eventually replaced by central bank issued gold certificates and silver certificates.

The banks will be particularly subject to the temptation to commit fraud and issue pseudo money certificates to circulate side by side with genuine money certificates as acceptable money-substitutes. The fact that money is a homogeneous good means that people do not care whether the money they redeem is the original money they deposited. This makes bank frauds easier to accomplish.

To the extent that central bank certificates ever represented all of the warehoused money (e.g. gold and silver), they eventually followed the course described by Rothbard.

As the sum of certificates became too large to support redeemability, they were abrogated and people were compelled to convert them to fiat. These large scale frauds occurred in the lifetimes of both Rothbard and his precursor von Mises, and were perpetrated by state and central banks under the protection of statute (i.e. the state).

The theory does not limit its condemnation of banking to warehousing (safe deposit) fraud, it extends to honest lending of deposits by banks generally, including demand deposit, saving deposit and often term deposit. As such the theory is invalid. Furthermore it implies a condemnation of lending and investing generally. And as Rothbard himself points out, lending is indistinct from investing:

Whether saved capital is channeled into investments via stocks or via loans is unimportant. The only difference is in the legal technicalities. Indeed, even the legal difference between the creditor and the owner is a negligible one.

All lending originates from a person's accumulated capital, whether deposited in bank or otherwise. There is no source for lending other than savings deposited. There is a related theory that people are too stupid to understand contractual terms of deposit.

Huerta de Soto considers the possibility "that a certain group of bank customers (or for the sake of argument, all of them) enter into a deposit contract aware and fully accepting that banks will invest (or loan, etc.) a large portion of the money they deposit". In this case, argues Huerta de Soto, "the supposed authorization from the depositors lacks legal validity" because few lay-persons understand the instability inherent in fractional-reserve banking: they believe their deposit is guaranteed, which Huerta de Soto considers a (near universal) misconception.

Wikipedia: Jesús Huerta de Soto

Yet those who make this argument believe themselves able to understand it. As such the theory is invalid. Given the moral distinction of nonaggression, it is the right of every individual to contract with another voluntarily. Taking this right away would be the crime. References to the "unbanked" generally assume that vast numbers of people do not have "access" to banking services. This is generally not the case, banking is widely available all over the world. These are the people who understand the risks and chose not to take them.

A related theory is that money substitutes trade at the same value as the money, representing a fraud. To the extent that money substitutes (e.g. deposit accounts) are insured by the taxpayer, the discount against the money they substitute is lower. However, even given full insurance, it is an error to assume these trade at par with the money. Money substitutes manifest as deposit accounts and are generally transacted electronically. Settling money accounts incurs time, money and risk costs. Credit card and check fraud is rampant, and this cost is surfaced in all transaction and account fees. Settling can take days if not months. Merchants necessarily discount money substitutes against money. Even electronic transfer directly between banks incurs a material settlement cost:

Banks are charged a gross transfer fee of $0.82 for every transaction, however there is a three-tiered discount schedule, which results in actual transaction fees costing between $0.034 and $0.82 per transaction depending on transaction volume.

Wikipedia: Fedwire

This is why many business are "cash only", others do not accept checks, others charge a premium to offset the discount, and why there are ATM fees, etc. As such the observation that money substitutes are not discounted is refuted by a mountain of evidence to the contrary. More importantly, this discount is provably necessary, invalidating the theory.

A related theory is that bank lending creates price inflation as a consequence of credit expansion. Given that lending and money have necessarily evolved together, there is never a time where credit expansion itself changes the level of money substitutes. This requires either an expansion of the money supply, or a reduction in time preference, reflected as the economic rate of interest. Credit expansion is strictly a function of these two factors, not lending itself. As such the theory is invalid.

A related theory is that banks may legitimately lend only "their own" money. All capital lent is someone's savings. If anyone can run a bank (i.e. borrow against their own savings and lend it to others) then this is a distinction without a difference. Aggregating savings with other people (i.e. through bank deposits) does not create any meaningful distinction. As such the theory is invalid.

A related theory is that banks may legitimately lend only against time deposits. There is no economic distinction between a time deposit and a demand deposit, as both imply fractional reserve. The nature of deposit, even safe deposit, implies that time and other constraints (e.g. identification) are required for withdrawal. Even taxpayer-insured checking and savings accounts are effectively time deposits:

For all savings accounts and all personal interest-bearing checking accounts, we reserve the right to require seven days' prior written notice of withdrawal.

Chase Bank: Deposit Agreement

Default risk and credit expansion remain despite maturity matching. As such the theory is invalid. The only true demand deposit is no deposit at all (money), and of course people have this option and that of time deposit to the extent they prefer it.

A related theory is that banks may legitimately lend only against fully-insured deposits. However the only true risk free return is no return. This is why only taxpayers insure loans (i.e. through compulsion). Full insurance is economically equivalent to no lending whatsoever, making the theory a contradiction, and therefore invalid.

A related theory is that even free banking has an inherent ability to create money out of thin air. Yet if this is true then anyone can do so, since free banking confers no special powers on people who refer to themselves as banks. If money can be created at no cost, it cannot be property. As such the theory is invalid. Even state fiat incurs a production cost, a cost to maintain its monopoly on production, and a political cost of monetary inflation. Free banking, such as with Gold or Bitcoin, enjoys no seigniorage privilege, due to the nature of competition.

Finally, it is often the case that people advocating for full reserve lending are the same people advocating for lower time preferences. This is a direct contradiction, as the former implies infinite time preference.