Integration in Banach algebra Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a bounded measurable function on the real line $\mathbb{R}$ taking values in a Banach algebra $A$. Is there a general theory that makes sense of integrals of the form
$$
\int_{\mathbb{R}} f(x)\mu(dx) \in A.
$$
I would be also interested in special cases: $A$ a sub-algebra of $B(H)$, where $H$ -- a Hilbert space or $A$ a commutative algebra.
 A: I believe OP's setting is an instance of the more general so called bilinear integral, defined for the following data:

*

*a measurable space $(S,\mathfrak{S})$;

*a function $f: S \to X$, where $X$ is a normed linear space;

*an additive measure $\mu: \mathfrak{S} \to Y$, where $Y$ is another normed linear space;

*a bilinear map $\beta: X \times Y \to Z$, where $Z$ is a Banach space;

Then, for $E \in \mathfrak{S}$, one can consider
$$
\int_E \beta(f(s),\mu(ds))
$$
In OP's setting, the bilinear mapping is given by the multiplication in the Banach algebra.
The canonical reference is Bartle - A General Bilinear Vector Integral (1955), but some further references, that de facto establish it as a theory, are:

*

*Dunford - A Bilinear Vector Integral (1975)

*Freniche, Garcia-Vazquez - The Bartle Bilinear Integration and Carleman Operators (1998)

*Jefferies - Bilinear Integrals and Radon-Nikodym Derivatives

*Jefferies - Some Recent Applications of Bilinear Integration
A generalization of Bartle's integral is obtained and substantially expanded upon in Dobrakov's series of 13 papers titled "Integration in Banach Spaces". There he considers $f: S \to X$, where $X$ is a Banach space, and $\mu: \mathfrak{S} \to BL(X,Y)$, where $Y$ is another Banach space, but this reduces to OP's setting via the adjoint action of the Banach algebra.
EDIT: Apologies, forgot to mention that Panchapagesan has a kind of a nice survey/summary on Dobrakov's integral, titled "On the Distinguishing Features of the Dobrakov Integral (1991)", which should help navigate Dobrakov's series of papers. Moreover, Panchapagesan apparently has a whole book "The Bartle-Dunford-Schwartz Integral" that is relatively recent (from 2000 or so), but I am not sure to what extent it addresses OP's setting, but it might be worth looking into.
