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We study ring-theoretic (in)finiteness properties -- such as emph{Dedekind-finiteness} and emph{proper infiniteness} -- of ultraproducts (and more generally, reduced products) of Banach algebras. Whilst 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 {L}oss 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 emph{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 textit{stable rank one} is also studied for ultraproducts.
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For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${mathcal B}$, its {em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]in pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${mathcal B}$. The pre-image of $p(A)$ in ${cc}^{n+1}$ is denoted by $P(A)$. When ${mathcal B}$ is the $ktimes k$ matrix algebra $M_k(cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${mathcal B}$ is reflexive or is a $C^*$-algebra, the {em projective resolvent set} $P^c(A):=cc^{n+1}setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${mathcal B}$ is a $C^*$-algebra with a trace $phi$, then $phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), cc)$.
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We examine the condition that a complex Banach algebra $A$ have dense invertible group. We show that, for commutative algebras, this property is preserved by integral extensions. We also investigate the connections with an old problem in the theory of uniform algebras.
A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $varphicolon Atimes Ato X$, where $X$ is an arbitrary Banach space, which satisfies $varphi(a,b)=0$ whenever $a$, $bin A$ are such that $ab+ba=0$, is of the form $varphi(a,b)=sigma(ab+ba)$ for some continuous linear map $sigma$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications.
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