Let $A$ be a commutative separable $C^*$ algebra and $X$ a spectrum of $A$. We can identify $A$ with $C(X)$. Then every separable non-degenerate representation $\pi$ of A is unitarily equivalent to $$\pi = \pi_1 \oplus \pi_2 \oplus \dots \oplus \pi_\infty,$$ where $\pi_i$ is a representation of $A$ on $L_2(X,\mu_i,H_i)$ such that $\pi_i(f)$ are multiplication operators $M_f$, $i$ is the dimension of $H_i$ and $\mu_i$ is a finite Borel measure and $\mu_i\perp\mu_j,\ i\neq j$.
This is a summary from Arveson's An Invitation to $C^*$-algebras, Section 2.2.
It is based on two results:
- Theorem 2.1.8 and Corollary 2.1.9 which are decomposition theory for separable type I representations (note that every representation of a commutative $C^*$ algebra is type I)
- Theorem 2.2.4 which is correspondence between equivalence classes of non-degenerate separable multiplicty-free representations of commutative separable $C^*$-algebras and equivalence classes of finite Borel measures on $X$.
I have the following questions:
Is it implicitly assumed that $A$ is unital (hence $A=C(X)$) and if so, is it true that in the case of $C_0(X)$ measures in the classification can be non-finite? Or it's just that it can be chosen to be finite, like for example representation of $C_0(\mathbb{R})$ as multiplication operators on $L_2(\mathbb{R})$ is equivalent to a representation on some $L_2(\mathbb{R},\mu)$ where $\mu$ is a finite and absolutely continuous measure equivalent to Lebesgue measure?
After each of the two results above there are intriguing paragraphs. After the first one:
We remark that with only minor variations the entire analysis of this section may be carried out for representations on Hilbert spaces which are inseparable.
And after the second:
We remark that the above results used separability of A in only a superficial way, and in fact the results are valid, in only slightly modified form, for separable multiplicity-free representations of arbitrary commutative C*-algebras. In place of Borel measures on X, for example, one deals with finite regular Borel measures $\mu$ which are "separable" in the sense that $L_2(X,\mu)$ is separable. One may also extend the results to representations on inseparable Hilbert spaces, at the expense of finiteness of the measure $\mu$. For the somewhat fussy details, see ([7], Chapter I, Section 7).
What is the correct analog of the result above for the representations on possibly non-separable Hilbert spaces of non-separable commutative $C^*$-algebras?
I assume that this is an old result but I can't find a proper reference explicitly. Reference [7] in Arveson is Les Algebres d’Opérateurs dans l’Espace Hilbertien. (Algebres de von Neumann.) By J. Dixmier, 1957 which is indeed somewhat fussy.
In other words, if the "minor variations" in Arveson's note after the first result refers to the fact that the direct sum should include terms of higher cardinalities and the note after the second result is true, then every non-degenerate representation $\pi$ of A on a Hilbert space $H$ is unitarily equivalent to $$\pi = \pi_1 \oplus \pi_2 \oplus \dots \oplus \pi_\kappa,$$ where $\kappa$ is the cardinal number corresponding to dimension of $H$ (cardinality of orthogonal basis) and $\pi_i(A)$ acts on $L_2(X,\mu_i,H_i)$ as multiplication operators, $i$ is the dimension of $H_i$ and $\mu_i$ is regular(?) Borel measure such that $\mu_i\perp\mu_j$, $i\neq j$.
Is it correct? For example, the counting measure on $\mathbb{R}$ is not regular, so what is non-separable representation of $C_0(\mathbb{R})$ on $l_2(\mathbb{R})$ equivalent to?
P.S. This is a natural continuation of another question.
