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Let $G$ be a locally compact group and $A$ be a Banach algebra. By $L^1(G,A)$ and $M(G,A)$ we denote the $A$-valued group, and measure algebra.

  1. Is $M(G,A)$ a Banach algebra (with convolution as the product)?
  2. Let $B$ be a Banach algebra contains $A$ as an ideal. Is $L^1(G,A)$ an ideal of $M(G,B)$?
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    $\begingroup$ I'm not quite sure of the exact definitions of $L^1(G,A)$ and (especially) $M(G,A)$. Can you give some references and/or explanation? $\endgroup$
    – Matthew Daws
    Commented Feb 27, 2023 at 8:18
  • $\begingroup$ @MatthewDaws $L^1(G,A)$ is indeed the projective tensor product of $L^1(G)$ and $A$. Another space is defined for example in the book titled "Vector Mesures" written by N. DINCULEANU (sciencedirect.com/book/9781483197623/vector-measures). $\endgroup$
    – MSMalekan
    Commented Feb 27, 2023 at 16:38

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