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Yemon Choi
Yemon Choi
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Bob
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Suppose that $\tilde{X}$ is a compact space. If $C(X)$$C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=C(X)^{**}$$C(\tilde{X})=C_0(X)^{**}$?

P.S. Note that the converse is always true, namely if $X$ is a locally compact space, there is a compact space $\tilde{X}$ such that $C(X)^{**}=C(\tilde{X})$$C_0(X)^{**}=C(\tilde{X})$. The Banach space $C(X)^{**}$$C_0(X)^{**}$ turns into a Banach algebra with the first (second) Arens product. This is a non-trivial product on $C(X)^{**}$$C_0(X)^{**}$.

Suppose that $\tilde{X}$ is a compact space. If $C(X)$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=C(X)^{**}$?

P.S. Note that the converse is always true, namely if $X$ is a locally compact space, there is a compact space $\tilde{X}$ such that $C(X)^{**}=C(\tilde{X})$. The Banach space $C(X)^{**}$ turns into a Banach algebra with the first (second) Arens product. This is a non-trivial product on $C(X)^{**}$.

Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=C_0(X)^{**}$?

P.S. Note that the converse is always true, namely if $X$ is a locally compact space, there is a compact space $\tilde{X}$ such that $C_0(X)^{**}=C(\tilde{X})$. The Banach space $C_0(X)^{**}$ turns into a Banach algebra with the first (second) Arens product. This is a non-trivial product on $C_0(X)^{**}$.

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Bob
Bob
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When is it $C(X)$?

Suppose that $\tilde{X}$ is a compact space. If $C(X)$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=C(X)^{**}$?

P.S. Note that the converse is always true, namely if $X$ is a locally compact space, there is a compact space $\tilde{X}$ such that $C(X)^{**}=C(\tilde{X})$. The Banach space $C(X)^{**}$ turns into a Banach algebra with the first (second) Arens product. This is a non-trivial product on $C(X)^{**}$.