We study ring-theoretic (in)finiteness properties—such as Dedekind-finiteness and proper infiniteness—of ultraproducts (and more generally, reduced products) of Banach algebras. While we characterise when an ultraproduct has these ring-theoretic properties in terms of its underlying sequence of algebras, we find that, contrary to the $C^*$ -algebraic setting, it is not true in general that an ultraproduct has a ring-theoretic finiteness property if and only if “ultrafilter many” of the underlying sequence of algebras have the same property. This might appear to violate the continuous model theoretic counterpart of Łoś’s Theorem; the reason it does not is that for a general Banach algebra, the ring theoretic properties we consider cannot be verified by considering a bounded subset of the algebra of fixed bound. For Banach algebras, we construct counter-examples to show, for example, that each component Banach algebra can fail to be Dedekind-finite while the ultraproduct is Dedekind-finite, and we explain why such a counter-example is not possible for $C^*$ -algebras. Finally, the related notion of having stable rank one is also studied for ultraproducts.

We examine the phenomenon when surjective algebra homomorphisms between algebras of operators on Banach spaces are automatically injective. In the first part of the paper we shall show that for certain Banach spaces X the following property holds: For every non-zero Banach space Y every surjective algebra homomorphism ψ : B(X) → B(Y ) is automatically injective. In the second part of the paper we consider the question in the opposite direction: Building on the work of Kania, Koszmider, and Laustsen (Trans. London Math. Soc., 2014) we show that for every separable, reflexive Banach space X there is a Banach space Y X and a surjective but not injective algebra homomorphism ψ : B(Y X ) → B(X).1 2 B. HORVÁTH Theorem 1.3. These simple observations ensure that the following definition is not vacuous.The purpose of this paper is to initiate the study of this property. The paper is structured as follows.In the second part of Section 1 we establish our notations and introduce the necessary background. We begin Section 2 by giving a list of examples of Banach spaces which lack the SHAI property, see Example 2.4. We continue by extending our list of examples of Banach spaces with the SHAI property. Since ℓ 2 possesses this property, it is therefore natural to ask the same question for other classical sequence spaces. We obtain the following result:Proposition 1.2. Suppose X is one of the Banach spaces c 0 or ℓ p for 1 ≤ p ≤ ∞. Then X has the SHAI property.Another way of generalising the ℓ 2 -case is to ask whether all, not necessarily separable Hilbert spaces have the SHAI property. As we will demonstrate, the answer is affirmative: Theorem 1.3. A Hilbert space of arbitrary density character has the SHAI property.We shall also provide more "exotic" examples of Banach spaces with the SHAI property, including Schlumprecht's arbitrarily distortable Banach space S, constructed in [42]:Theorem 1.4. Let X be a complementably minimal Banach space such that it has a complemented subspace isomorphic to X ⊕X. Then X has the SHAI property. In particular, Schlumprecht's arbitrarily distortable Banach space S has the SHAI property.When studying the SHAI property of a Banach space X, understanding the complemented subspaces of X and the lattice of closed two-sided ideals of B(X) appears to be immensely helpful. For the Banach spacewhere Y is c 0 or ℓ 1 , the complemented subspace structure was studied by Bourgain, Casazza, Lindenstrauss, and Tzafriri in [6] and the ideal lattice of B(X) by Laustsen, Loy, and Read in [27] and later by Laustsen, Schlumprecht, and Zsák in [28]. Their results allow us to show the following:

We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on X, to $\mathscr{B}(Y)$, where X is one of the following long sequence spaces: c0(λ), $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1 \leqslant p \lt \infty$) and Y is arbitrary. En route to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the ‘sequential strong operator topology’.

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A remarkable result of Molnár [Proc. Amer. Math. Soc., 126 (1998), 853-861] states that automorphisms of the algebra of operators acting on a separable Hilbert space are stable under "small" perturbations. More precisely, if φ, ψ are endomorphisms of B(H) such that φ(A) − ψ(A) < A and ψ is surjective, then so is φ. The aim of this paper is to extend this result to a larger class of Banach spaces including ℓp and Lp spaces, where 1 < p < ∞.En route to the proof we show that for any Banach space X from the above class all faithful, unital, separable, reflexive representations of B(X) which preserve rank one operators are in fact isomorphisms.