Reports of the Academy of Sciences of the USSR
1960. Volume 134, No. 2

MATHEMATICS

M. A. NAIMARK

ON FACTOR-REPRESENTATIONS OF A LOCALLY COMPACT GROUP

(Presented by Academician A. N. Kolmogorov on 6 V 1960)

As is known, every unitary representation (g \to U_g) of a locally compact(^) group decomposes into a (generally speaking, continuous) direct sum of irreducible representations (see, for example, ({}^{(1)}) or ({}^{(2)}), Ch. II). In this decomposition there may occur representations that are equivalent to one another. It is natural to pose the question of such a decomposition in which all mutually equivalent representations are collected together, so that the continuous sum is already taken over classes of mutually equivalent representations({}^{*}).

In the present paper decompositions of a representation into factor-representations are studied; from the results obtained there follows, in particular, a positive solution of the question posed above for groups of type I({}^{***}).

We note that all the results set forth remain valid for (*)-representations of separable normed rings with involution.

1. Canonical decomposition of a representation into factor-representations. In what follows (G) will denote a locally compact group satisfying the second axiom of countability, and the term “representation (U) of the group (G)” will mean a continuous unitary representation (g \to U_g) of the group (G) in a separable Hilbert space (\mathfrak H). A representation (U) is called a factor-representation if the weakly closed ring (M), generated by all (U_g,\ g \in G), is a factor in the sense of Murray and von Neumann. As is known, if the space (\mathfrak H) of the representation (U) is decomposed into a continuous sum

[
\mathfrak H = \int_{\Lambda} \mathfrak H(\lambda)\, d\mu(\lambda)
]

with respect to the center (Z) of the ring (M)({}^{*}), then the representation (U) decomposes into representations (U(\lambda)) in (\mathfrak H(\lambda)), which will be factor-representations for almost all (\lambda \in \Lambda). This decomposition is called the *canonical decomposition of the representation into factor-representations.

Theorem 1({}^{*}). Let

[
\mathfrak H = \int_{\Lambda} \mathfrak H(\lambda)\, d\mu(\lambda)
]

and (U={U(\lambda)}) be the canonical decomposition of the representation (U). Then there exists a set (\Lambda_0 \subset \Lambda) of (\mu)-measure zero such that for any (\lambda,\lambda' \in \Lambda-\Lambda_0,\ \lambda \ne \lambda'), the representations (U(\lambda)) and (U(\lambda')) are disjoint({}^{**}).

If (G) is a group of type I, then (U(\lambda)) is a factor-representation of type I and therefore is a multiple of an irreducible representation; the disjointness of (U(\lambda)) and

(^*) By compactness here and below is meant bicompactness in the sense of P. S. Aleksandrov.

(^ {**}) This question also arises in connection with certain problems in probability theory, to which A. M. Yaglom drew the author’s attention.

(^ {***}) For the definition of a ring of type I and a group of type I see ({}^{(2,3)}); see also ({}^{(4)}), Ch. I.

(^ {****}) That is, so that (Z) consists of all operators (A={c(\lambda)1}), where (c(\lambda)) is a numerical function in (L^\infty_\mu(\Lambda)).

(^ {*}) A special case of this theorem was recently obtained by Guichardet ({}^{(5)}).

(^ {**}) Two representations (U,V) are called disjoint** (see ({}^{(3)}), p. 1) if no part of the representation (U) is equivalent to any part of the representation (V).

$U(\lambda')$ means then that these representations are multiples of nonequivalent irreducible representations.

The application of Theorem 1 to the case under consideration thus leads to the following theorem, which answers the question posed at the beginning of the article.

Theorem 2. Let

[
\mathfrak H=\int_{\Lambda}\mathfrak H(\lambda)\,d\mu(\lambda)
]

and (U={U(\lambda)}) be the canonical decomposition of a representation (U) of a group (G) of type I.

Then there exists a set (\Lambda_0\subset \Lambda) of (\mu)-measure zero and measurable families (\mathfrak H_k(\lambda)), (k=1,2,\ldots), such that:

1) for (\lambda,\lambda'\in\Lambda-\Lambda_0,\ \lambda\ne\lambda'), the representations (U(\lambda)) and (U(\lambda')) are multiples of nonequivalent irreducible representations;

2) (\mathfrak H(\lambda)=\sum_k \mathfrak H_k(\lambda)) for (\lambda\in\Lambda-\Lambda_0)*;

3) (\mathfrak H_k(\lambda)) is invariant with respect to (U(\lambda)) for (\lambda\in\Lambda_0);

4) if (\lambda\in\Lambda-\Lambda_0) and (\mathfrak H_k(\lambda)\ne(0)), then the restriction of (U(\lambda)) to (\mathfrak H_k(\lambda)) is irreducible.

Remark. The assertion of Theorem 2 will not be valid for arbitrary representations of a group not of type I. Indeed, the assertion of Theorem 2 means that in the canonical decomposition of the representation (U) almost all the (U(\lambda))-factors are of type I; hence it follows (see, for example, ((^2)), exercise on p. 125) that (U) is a representation of type I. Therefore, if (G) is a group not of type I, then there exist representations of the group (G) for which the assertion of Theorem 2 will not be valid.

2. Continuous sum of quasi-equivalent factor representations

Theorem 3. Let the representation (U) in the space (\mathfrak H) be a continuous sum of representations (U(\lambda)) in the spaces (\mathfrak H(\lambda)), so that

[
\mathfrak H=\int_{\Lambda}\mathfrak H(\lambda)\,d\mu(\lambda)
]

and (U={U(\lambda)}), and let there exist a set (\Lambda'\subset\Lambda) of (\mu)-measure zero such that all the representations (U(\lambda)), for (\lambda\in\Lambda-\Lambda'), are pairwise quasi-equivalent** factor representations. Then (U) is also a factor representation.

If, in addition, all (U(\lambda)), for (\lambda\in\Lambda-\Lambda'), are factor representations of type I, then (U) is also a factor representation of type I and therefore is a finite or countable discrete sum of mutually equivalent irreducible representations***.

Corollary. Let representations (U_1) and (U_2) in the spaces (\mathfrak H_1) and (\mathfrak H_2) be continuous sums of representations (U_1(\lambda_1)), (U_2(\lambda_2)) in the spaces (\mathfrak H_1(\lambda_1)), (\mathfrak H_2(\lambda_2)), so that

[
\mathfrak H_1=\int_{\Lambda_1}\mathfrak H_1(\lambda_1)\,d\mu_1(\lambda_1),\quad
U_1={U_1(\lambda_1)},
]

[
\mathfrak H_2=\int_{\Lambda_2}\mathfrak H_2(\lambda_2)\,d\mu_2(\lambda_2),\quad
U_2={U_2(\lambda_2)},
]

and let there exist sets (\Lambda_1'\subset\Lambda_1,\ \Lambda_2'\subset\Lambda_2) of respectively (\nu_1)- and (\nu_2)-measure zero such that all (U_1(\lambda_1)), (\lambda_1\in\Lambda_1-\Lambda_1'), and (U_2(\lambda_2)), (\lambda_2\in\Lambda_2-\Lambda_2'), are factor representations quasi-equivalent to one another. Then (U_1) and (U_2) are quasi-equivalent factor representations.

3. Application to positive-definite functions

Applying Theorem 2 of § 1 to the representation defined by the given—

* For some (\lambda\in\Lambda) it may be that (\mathfrak H_k(\lambda)=(0)).

* Two representations (U,V) are called quasi-equivalent* (see ((^3)), § 1) if no part of (U) is disjoint from (V) and no part of (V) is disjoint from (U).

*** The first assertion of the theorem is a continuous analogue of a proposition of Mackey (see ((^3)), Lemma 1.2); the second assertion generalizes results of Mautner ((^6)) and Pukanszky ((^7)).

positive-definite function, we arrive at the following result.

Theorem 4. Every continuous positive-definite function (\varphi(g)) on a group (G) of type I can be represented in the form

[
\varphi(g)=\int_{\Lambda}\left[\sum_k \varphi_k(g,\lambda)\right]\,d\mu(\lambda),
]

where (\varphi_k(g,\lambda)) are elementary continuous positive-definite functions of (g) and measurable functions of (\lambda) such that:

1) (\varphi_k(g,\lambda)) and (\varphi_\ell(g,\lambda)) define equivalent irreducible representations;

2) (\varphi_k(g,\lambda)) and (\varphi_\ell(g,\lambda')), for (\lambda\ne\lambda'), define inequivalent irreducible representations.

Moscow
Institute of Physics and Technology

Received
7 IV 1960

CITED LITERATURE

(^{1}) M. A. Naimark, S. V. Fomin, Uspekhi Mat. Nauk, 10, 2, 64, 111 (1955).
(^{2}) J. Dixmier, Les algèbres d’opérateurs dans l’espace Hilbert’en, Paris, 1957.
(^{3}) G. W. Mackey, Ann. Math., 58, 2, 193 (1953).
(^{4}) I. Kaplansky, Functional Analysis, Surveys in Applied Mathematics, 4, N. Y., 1958; Russian transl. Matematika, IL, 3, 5, 1959, p. 91.
(^{5}) A. Guichardet, C. R., 250, 962 (1960).
(^{6}) F. I. Mautner, Ann. Math., 52, 528 (1950).
(^{7}) L. Pukanszky, Acta Szeged, 15, 2, 145 (1954).