Reports of the Academy of Sciences of the USSR

1. Volume 191, No. 6

MATHEMATICS

EDWIN HEWITT, DUSA McDUFF

SOME PATHOLOGICAL MAXIMAL IDEALS IN THE ALGEBRA OF MEASURES ON A COMPACT GROUP

(Presented by Academician P. S. Aleksandrov on 26 I 1970)

§ 1. In this note we consider the algebra \(\mathbf{M}(G)\) of all regular complex Borel measures on a compact group \(G\). With respect to the operations of addition, scalar multiplication, multiplication (convolution), given by the formula

\[ \mu * \nu(E)=\int_G \mu(Et^{-1})\,d\nu(t), \]

and the norm \(\|\mu\|=|\mu|(E)\), \(\mathbf{M}(G)\) is a Banach algebra. It has as identity the measure \(\varepsilon_e\) of mass 1 concentrated at the identity \(e\) of the group \(G\), and is commutative if and only if the group \(G\) is commutative. The mapping \(\mu\mapsto\mu^{\sim}\), where \(\mu^{\sim}(E)=\mu(E^{-1})\), is an involution in \(\mathbf{M}(G)\) (the elementary theory of the algebra \(\mathbf{M}(G)\) is developed in \((^1)\)). The algebra \(\mathbf{M}(G)\) contains the algebra of measures absolutely continuous with respect to Haar measure \(\lambda\) as a closed two-sided ideal. This algebra can be identified with the algebra \(\mathfrak{L}_1(G)\) of all \(\lambda\)-integrable functions on \(G\). For the algebra \(\mathfrak{L}_1(G)\) it is well known (see, for example, \((^1)\), § 38) that all its closed ideals, both one- and two-sided, can be described in terms of representations of the group \(G\). In particular, every maximal two-sided proper ideal in \(\mathfrak{L}_1(G)\) has the form

\[ \left\{ f\in\mathfrak{L}_1(G): \int_G U_x f(x)\,d\lambda(x)=0 \right\}=\mathfrak{I}_U, \]

where \(x\mapsto U_x\) is some continuous irreducible unitary representation of the group \(G\). The quotient algebra \(\mathfrak{L}_1(G)/\mathfrak{I}_U\) is isomorphic to the algebra of all linear operators on the representation space of \(U\), and therefore is finite-dimensional.

For several years the question remained open whether the algebra \(\mathbf{M}(G)\) also has the latter property: is it true that a simple homomorphic image of \(\mathbf{M}(G)\) is necessarily finite-dimensional? In this note a negative answer is given to this question: there exist compact groups \(G\) for which \(\mathbf{M}(G)\) contains such a maximal two-sided ideal \(J\) that \(\mathbf{M}(G)/J\) is not only infinite-dimensional, but also contains a “scattered” set of any prescribed cardinality.

§ 2. We begin the proof by constructing maximal ideals of some algebras of operators. Let \(H\) be a finite-dimensional Hilbert space of dimension \(d\), let \(\mathfrak{B}(H)\) be the algebra of all linear operators on \(H\), and let \(\mathfrak{U}(H)\) be the compact group of unitary operators in \(\mathfrak{B}(H)\). For any \(A\in\mathfrak{B}(H)\) we shall write \(AA^*\) in the form \(\sum_{k=1}^n a_k P_k\), where \(P_k\) are the projections onto pairwise orthogonal one-dimensional subspaces of \(H\), and \(a_k\) are—

nonnegative real numbers. For \(1\le p<\infty\) let
\[ \|A\|_{\Phi_p}=\left(\sum_{k=1}^{d} a_k^{p/2}\right)^{1/p} \]
and
\[ \|A\|_{\Phi_\infty}=\max\{a_1^{1/2},a_2^{1/2},\ldots,a_d^{1/2}\}. \]
von Neumann [2] showed that all the functions \(\|\cdot\|_{\Phi_p}\) are norms on \(\mathfrak B(H)\). It is easy to show that \(\|\cdot\|_{\Phi_\infty}\) is also the operator norm on \(\mathfrak B(H)\).

Let now \(I\) be a nonempty set of indices, and for each \(i\in I\) let \(H_i\) be a finite-dimensional Hilbert space of dimension \(d_i\). Finally, let \(\mathfrak C_\infty(I)\) be the set of all elements \(A=(A_i)_{i\in I}\) of the Cartesian product \(\prod_{i\in I}\mathfrak B(H_i)\) such that
\[ \|A\|=\sup_{i\in I}\|A_i\|_{\Phi_\infty}. \]
With respect to coordinatewise operations and the norm defined above, \(\mathfrak C_\infty(I)\) is, obviously, a \(C^*\)-algebra with identity, commutative if and only if all \(d_i=1\).

§ 3. Following Wright [3], we construct maximal two-sided ideals in \(\mathfrak C_\infty(I)\).

3.1. Theorem 1. Let \(\mathcal U\) be any ultrafilter on the set \(I\), and let \(\mathfrak I_{\mathcal U}\) be the set of all such \(A\in\mathfrak C_\infty(I)\) that
\[ \lim_{\mathcal U}\frac1{d_i}\|A\|_{\Phi_1}=0. \]
Then \(\mathfrak I_{\mathcal U}\) is a maximal two-sided ideal in \(\mathfrak C_\infty(I)\), and, conversely, every maximal two-sided ideal in \(\mathfrak C_\infty(I)\) has this form for a suitable choice of an ultrafilter \(\mathcal U\) on the set of indices \(I\).

3.2. Theorem 2. Suppose that for some positive integer \(m\) the set \(\{i\in I:d_i\le m\}\) belongs to the ultrafilter \(\mathcal U\). Then the algebra \(\mathfrak C_\infty(I)/\mathfrak I_{\mathcal U}\) is isomorphic to the algebra \(\mathfrak B(L)\) for some Hilbert space \(L\) of dimension \(m\).

Let \(X\) be a metric space and let \(\rho\) be its metric. A subset \(S\subset X\) is called scattered if there exists a positive number \(\alpha\) such that \(\rho(x,y)\ge\alpha\) for any pair \(x,y\) of distinct elements of \(S\).

3.3. Theorem 3. Suppose that for every positive integer \(m\) the set \(\{i\in I:d_i>m\}\) belongs to \(\mathcal U\). Then the quotient algebra \(\mathfrak C_\infty(I)/\mathfrak I_{\mathcal U}\) contains a scattered set \(S\) of cardinality \(2^{\aleph_0}\) such that every element of \(S\) has norm \(1\). In particular, the quotient algebra is infinite-dimensional.

3.4. Theorem 4. Let \(\mathfrak m\) be an infinite cardinal number. There exist sets of indices \(I\) of cardinality \(\mathfrak m\), families of finite-dimensional Hilbert spaces \(\{H_i\}_{i\in I}\), and an ultrafilter \(\mathcal V\) on the set \(I\), such that the quotient algebra \(\mathfrak C_\infty(I)/\mathfrak I_{\mathcal V}\) contains a scattered set of cardinality \(2^{\mathfrak m}\), all elements of which have norm \(1\).

§ 4. Applications to the algebra \(\mathrm M(G)\). Let again \(G\) be a compact group. The object dual to the group \(G\) is the set \(\Sigma\) of equivalence classes of continuous unitary irreducible representations of the group \(G\). Thus, each \(\sigma\in\Sigma\) is a class of pairwise equivalent \(d_\sigma\)-dimensional continuous unitary irreducible representations of \(G\), where \(d_\sigma\) is some positive integer. In each class \(\sigma\in\Sigma\) choose a representative \(U^{(\sigma)}\in\sigma\). The representation \(U^{(\sigma)}\) acts on the \(d_\sigma\)-dimensional Hilbert space \(H_\sigma\). In \(H_\sigma\) choose an orthonormal basis \(\xi_1,\xi_2,\ldots,\xi_{d_\sigma}\). Let \(D_\sigma\) be the conjugate-linear operator on \(H_\sigma\) such that
\[ D_\sigma\left(\sum_{k=1}^{\sigma} a_k\xi_k\right)=\sum_{k=1}^{\sigma}\overline{a_k}\xi_k. \]
For \(\mu\in \mathrm M(G)\) we define the Fourier–Stieltjes transform \(\hat\mu\) of the element \(\mu\) as the element of \(\mathfrak C_\infty(\Sigma)\) such that
\[ \langle \hat\mu(\sigma)\xi,\eta\rangle =\int_G \langle D_\sigma U_x^{(\sigma)}D_\sigma\xi,\eta\rangle\,d\mu(x) \]
for all \(\xi,\eta\in H_\sigma\) and all \(\sigma\in\Sigma\). The mapping \(\mu\mapsto\hat\mu\) is a norm-nonincreasing involutive isomorphism of \(\mathrm M(G)\) into \(\mathfrak C_\infty(\Sigma)\). For any nonempty

subsets \(P \subset \Sigma\) the mapping \(\mu \mapsto (\mu(\sigma))_{\sigma \in P} = \hat{\mu}|P\) is a norm-nonincreasing involutive homomorphism of \(M(G)\) into \(\mathfrak{C}_{\infty}(G)\).

4.1. Definition. A nonempty subset \(P \subset \Sigma\) is called a Sidon set if the mapping \(\mu \mapsto \hat{\mu}|P\) is a homomorphism of \(M(G)\) onto \(\mathfrak{C}_{\infty}(P)\).

4.2. Example. For any indexed family \(\{H_i\}_{i \in I}\) of finite-dimensional Hilbert spaces, the product
\[ L = \prod_{i \in I} U(H_i) \]
is a compact group, and the mapping \((U_i)_{i \in I} \mapsto U_{i_0}\) is a \(d_{i_0}\)-dimensional continuous unitary irreducible representation of \(L\) for any fixed \(i_0 \in I\). In (1) it was proved (37.5) that the set of all such representations defines a Sidon set in the dual object of \(L\).

4.3. Theorem 5. Let \(G\) be a compact group and let \(P\) be a Sidon set in \(\Sigma\) such that
\[ \sup \{d_{\sigma}:\sigma \in P\}=\infty . \]
Then \(M(G)\) contains a maximal two-sided ideal \(\mathfrak{I}_u\), the quotient algebra \(M(G)/\mathfrak{I}_u\) by which contains a scattered set as in (3.3).

4.4. Theorem 6. Let \(L\) be the group defined in (4.2), and suppose that
\[ \sup\{d_i:i\in I\}=\infty . \]
Then \(M(L)\) contains a maximal two-sided ideal with the properties described in (4.3).

4.5. Theorem 6. Let \(\mathfrak{m}\) be an infinite cardinal number. There exists a product \(L\), as in (4.2), and such a maximal two-sided ideal \(\mathfrak{I}\) in \(M(L)\) that \(M(L)/\mathfrak{I}\) contains a scattered set \(P\) as in (3.4).

Theorems 5 and 6 follow directly, respectively, from Theorems 3 and 4. It is only necessary to note that the Fourier–Stieltjes transform, restricted to \(P\), is a homomorphism “onto.”

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Moscow

Washington University
USA

Faculty of Mechanics and Mathematics
Moscow State University
named after M. V. Lomonosov

Cambridge University
Great Britain

Received
21 I 1969

REFERENCES

1. E. Hewitt, K. A. Ross, Abstract Harmonic Analysis, Heidelberg—Berlin—N. Y. 1, 1963; 2, 1970.
2. I. Neiman, Izv. Research Inst. of Math. and Mech. at Tomsk State Univ. named after V. V. Kuibyshev, 4, 205 (1962).
3. F. B. Wright, Ann. Math. (2) 60, 560 (1954).