Abstract
Full Text
UDC 517
MATHEMATICS
A. I. LOGINOV
BANACH COMMUTATIVE SYMMETRIC ALGEBRAS OF OPERATORS IN THE PONTRYAGIN SPACE \(\Pi_1\)
(Presented by Academician I. M. Vinogradov, 15 VI 1967)
Let \(R\) be a commutative symmetric algebra of bounded operators in the Pontryagin space \(\Pi_1\). In the works of M. A. Naimark \((^1,^2)\) a classification of such algebras was given. It was shown that, under the assumptions of separability of \(R\) and \(\Pi_1\), all such algebras (with the exception of certain simple cases) are determined, up to equivalence, by means of a certain model. There the conditions for equivalence of algebras defined by such a model were also found. In fact, two distinct types of inequivalent models were indicated: a) and b) (see \((^1)\)); the algebras corresponding to them will be called, respectively, singular and nonsingular.
The operators of a singular algebra \(R\) are given by the formulas
\[ A\xi_0=\lambda(A)\xi_0, \]
\[ Ap(t)=(p,p_{A^*})\xi_0+A_1p= \tag{1} \]
\[ =-\int_{T_1}(A_1(t)-\lambda(A))(p(t),\xi(t))\,d\sigma\cdot \xi_0+A_1(t)p(t), \]
\[ Aq=(q,q_{A^*})\xi_0+\lambda(A)q, \]
\[ A\eta_0=\gamma(A)\xi_0+\lambda(A)\eta_0+p_A+q_A= \]
\[ =\gamma(A)\xi_0+\lambda(A)\eta_0+(A_2(t)-\lambda(A))\xi(t)+q_A, \]
where \(T\) is a bicompact space, \(\sigma\) is a regular Borel measure on \(T\), and \(\lambda(A)=A_1(t_0)\) and \(T_1=T-\{t_0\}\), \(t_0\in T\). The point \(t_0\in T\) will be called a singular point of the singular algebra \(R\).
For the operators of a nonsingular algebra \(R\) we have formulas of the form
\[ A\xi_0=\lambda(A)\xi_0, \]
\[ Ap(t)=(p,p_{A^*})\xi_0+A_1p= \]
\[ =-\int_T(A_1(t)-\lambda(A))(p(t),\xi(t))\,d\sigma\cdot \xi_0+A_1(t)p(t), \tag{2} \]
\[ A\eta_0=\gamma(A)\xi_0+\lambda(A)\eta_0+p_A= \]
\[ =\gamma(A)\xi_0+\lambda(A)\eta_0+(A_1(t)-\lambda(A))\xi(t), \]
with now \(\lambda(A)\ne A_1(t_0)\) for any point \(t_0\in T\). (For a detailed explanation of these formulas and of the notation connected with them, see \((^2)\).)
Thus, each operator \(A\in R\) is defined by the system of parameters \(\{A_1(t), q_A, \gamma(A), \lambda(A)\}\). When \(A\) runs through the algebra \(R\), \(A_1(t)\) runs through a certain symmetric algebra of functions \(R_1\subset C(T)\), uniformly dense in \(C(T)\); the vectors \(q_A\) form a certain linear manifold \(\mathcal L\), and \(\gamma(A)\) and \(\lambda(A)\) are linear functionals on \(R\) (see \((^2)\)). The totality \(\mathcal B\) of all systems \(\{A_1(t), q_A, \gamma(A), \lambda(A)\}\) corresponding to a certain algebra \(R\) forms a linear manifold in the totality \(G\) of all systems \(\{A_1(t), q_A, \gamma(A), \lambda(A)\}\), when all parameters independently run through their ranges of values. This manifold is called the defining manifold of the algebra \(R\). The problem
of the present work is to describe the determining manifolds of complete singular and nonsingular algebras with identity.
- Denote
\[ M_0=\{A:\ A\in R,\ \lambda(A)=0\}. \]
Then, since \(\lambda(A)=\overline{\lambda(A^*)}\), \(M_0\) is a symmetric maximal ideal of the algebra \(R\). Moreover, \(R=1+M_0\), since for \(B=A-\lambda(A)\cdot 1\) we have \(\lambda(B)=0\) and \(B\in M_0\).
Denote \(M_1=\{A:\ A\in R,\ A_1(t)\equiv 0\}\). Then, since the homomorphism \(A\to A_1(t)\) is continuous and symmetric (see (2)), \(M_1\) is a closed symmetric ideal in \(R\).
Denote also by \(\widetilde M_0\) the image of the ideal \(M_0\) under the homomorphism \(A\to A_1(t)\). \(\widetilde M_0\) is a symmetric maximal ideal in \(R_1\), proper or improper. The following assertions are valid with respect to \(\widetilde M_0\).
I. If \(R\) is nonsingular, then the ideal \(\widetilde M_0\) is uniformly dense in \(C(T)\).
II. If \(R\) is singular, then the ideal \(\widetilde M_0\) is uniformly dense in \(C_{t_0}(T)\).
- Every operator of the algebra given by formulas (1) or (2) is bounded. The operator norm in the algebra can be expressed in terms of the parameters of \(R\) via the kernel of the homomorphism \(R\to R_1\) isomorphic to the algebra \(R_1\), i.e. the operator norm up to equivalence has the form
\[ \|A\|=|A_1|+|\gamma(A)|+|p_A|+|q_A| =\sup_{t\in T}|A_1(t)|+|\gamma(A)|+|q_A|+ \]
\[ +\left[\int_{T_1}|A_1(t)-A_1(t_0)|^2\,d\mu\right]^{1/2}, \tag{3} \]
where \(d\mu=(\xi(t),\xi(t))\,d\sigma\).
For a nonsingular algebra we have
\[ \|A\|=|\lambda(A)|+|A_1|+|\gamma(A)|= \]
\[ =|\lambda(A)|+|\gamma(A)|+\sup_{t\in T}|A_1(t)|. \tag{4} \]
- All the preceding assertions are valid in the case of an arbitrary algebra, complete or incomplete. We now impose the condition of completeness on the algebra \(R\). Since the algebra \(R_1\) is homomorphic to the algebra \(R\), the factor algebra of \(R\) by the kernel of the homomorphism \(R\to R_1\) is isomorphic to the algebra \(R_1\), i.e. the algebra \(R/M_1\) is algebraically isomorphic to the algebra \(R_1\). Since the homomorphism \(R\to R_1\) is continuous and, consequently, the ideal \(M_1\) is closed, by a known theorem (see, for example, (3), Ch. I, ยง 4.3) and the completeness of \(R\) in the norm \(\|A\|\), the algebra \(R/M_1\) and, consequently, the isomorphic algebra \(R_1\), will be complete in the natural norm in the quotient space \(R/M_1\). If \(\widetilde A\) is the class of elements from \(R\) congruent modulo \(M_1\) and \(A_1(t)\in R_1\) is the function corresponding to it, then this natural norm is equal to
\[ \|\widetilde A\|=\|A_1\|=\|A_1(t)\|=\inf_{A\in \widetilde A}\|A\|. \]
From formulas (3) and (4) it follows that this norm has the form
\[ \|\widetilde A\|=\sup_{t\in T}|A_1(t)|+\inf_{A\in \widetilde A}\bigl(|\lambda(A)|+|\gamma(A)|\bigr) \tag{5} \]
for a nonsingular algebra \(R\), and
\[ \|\widetilde A\|=\sup_{t\in T}|A_1(t)|+ \]
\[ +\inf_{A\in \widetilde A}\left[|\lambda(A)|+|\gamma(A)|+|q_A|+ \left(\int_{T_1}|A_1(t)-A_1(t_0)|^2\,d\mu\right)^{1/2}\right] \tag{6} \]
for a singular algebra \(R\).
Thus, in the case of completeness of \(R\) in its norm, the algebra \(R_1\) is not only uniformly dense in \(C(T)\), but also complete in the norm (5) or (6). This fact makes it possible to describe concretely the structure of the algebra \(R_1\) and of the determining manifolds for complete algebras.
- Applying the arguments of Sec. 3 and taking assertion I into account, we obtain that, in the case of a complete nonsingular algebra, \(R_1=C(T)\), and for the defining manifold the following holds.
Theorem 1. Every nonsingular complete algebra \(R\) is equivalent to an algebra given by the formulas:
\[ A\xi_0=\lambda \xi_0, \]
\[ Ap(t)=A_1(t)p(t), \]
\[ A\eta_0=\gamma \xi_0+\lambda \eta_0, \]
where \(\lambda\) and \(\gamma\) are arbitrary complex numbers, and \(A_1(t)\) is an arbitrary function from \(C(T)\).
- In the study of singular algebras it is useful to consider the following collection of functions from \(R_1\). Denote by \(S\) the linear span of all possible products of functions from \(R_1\). \(S\) is a symmetric ideal in \(R_1\). For \(S\) the following holds.
Lemma. The ideal \(S\) is dense, in the norm
\[ \sup_{t\in T}|A_1(t)|+\left(\int_{T_1}|A_1(t)|^2\,d\mu\right)^{1/2} \]
in the ring of all continuous functions on \(T\) that vanish at 0 at the point \(t_0\), for which this norm is finite.
A detailed consideration of the defining manifolds of complete singular algebras, using assertion II, the arguments of Sec. 3, and the lemma, leads to the fact that in this case \(R_1\) consists of all continuous functions on \(T\) for which the norm is finite:
\[ \|A_1\|=\sup_{t\in T}|A_1(t)|+\left(\int_{T_1}|A_1(t)-A_1(t_0)|^2\,d\mu\right)^{1/2}. \tag{7} \]
For defining manifolds of complete singular algebras the following holds.
Theorem 2. Every complete singular algebra is given by a defining manifold of the form
\[ \{A_1(t),q_A,\gamma,\lambda\,(A)=A_1(t_0)\}, \]
where \(\gamma\), \(q_A\), and \(A_1(t)\) independently run through, respectively: the set of complex numbers, a certain complete space \(\mathcal E\), and the ring of all functions for which the norm (7) is finite.
The author expresses gratitude to M. A. Naimark for his attention to the work.
Moscow Institute of Physics and Technology
Received
24 V 1967
REFERENCES
- M. A. Naimark, DAN, 156, 734 (1964).
- M. A. Naimark, Rev. Roumaine Math. pures et appl., 9, 499 (1964).
- M. A. Naimark, Normed Rings, Moscow, 1956.