数理解析研究所講究録1080 巻 1999 年 137-139

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On a problem of Fell and Doranby W. elazko (Warszawa)$\dot{\mathrm{Z}}$Let $X$ be a real or complex locally convex space andalgebra over the same
field of scalars as $X$ . A representation of $A$ on $X$ is an algebra homomorphism $aarrow T_{a}$of $A$ into the algebra $L(X)$ of all continuous endomorphisms of the space $X$ . We say thatis irreducible if there is no proper closed linear subspace of $X$ which is invariant for$a\in A$ . Thus
all operatorsis irreducible if and only if for each non-zero element
in $X$ the orbit $\mathcal{O}(\mathrm{T};x)=\{T_{a}x : a\in A\}$ is dense in $X$ . A representation is said
totally irreducible (cf [2]), if for each natural and for each -tupleof linearly
elements of $X$ the multiple orbit: $a\in A$ } is dense inin the product topology. Thus
is totally irreducible if
and only if the algebra TA $=\{T_{a} : a\in A\}$ is dense in $L(X)$ in the strong operator
topology (the topology of poinwise convergence on $X$ ). The commutant of is defined
as $\mathrm{T}’=\{T\in L(X) : TT_{a}=T_{a}\tau\forall a\in A\}$ .
say thatis trivial if it consists of scalarmultiples of the odentity operator only. Fell and Doran asked in [2] (ProbIem , p. 329)
the following question.
Problem. Let $X$ be a locally convex space and an irreducible
on$X$ with a trivial$A$
commutant of an algebra over the same field of scalars as $X$ . Does it
follow that is totally irreducible ?
The above problem asks whether the topological version of the celebrated Jacobson
Density Theorem (cf. below) holds true. Let $X$ be a real or complex vector space. Analgebraic representation of an algebra $A$ on $X$ is a homomorphism into the algebra of allhomomorphisms of $X$ and it is said (algebraically) irreducible if all orbits for a non-zeroelement $x\in X$ are equal to $X$ , in particular each non-zero proper subspace of $X$ cannot
be invariant for all operators . Generally speaking, an irreducible representation on a
locally convex space does not need be algebraically irreducible. The following result isa consequence of the Jacobson Density Theorem (cf. [1], p. 123, Theorem 10; [2], pp.
283-286; [3], pp. 271-274).
$A-\mathrm{a}\mathrm{n}$$\mathrm{T}$$\mathrm{T}$$T_{a},$$\mathrm{T}$$x$$\mathrm{T}$$n$$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}.\overline{\mathrm{d}}\mathrm{e}\mathrm{n}\mathrm{t}$$X^{n}$$\mathrm{n}$$x_{1},$$\ldots,$$x_{n}$$\mathcal{O}(\mathrm{T};x_{1}\backslash ’\ldots, x_{n})=\{(T_{a}x_{1}, \ldots, TaXn)\in$$X^{n}$$\mathrm{T}$$\mathrm{T}$$\mathrm{W}$$\mathrm{T}’$$\mathrm{I}\mathrm{I}$$\mathrm{T}$$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}!^{\mathrm{o}\mathrm{n}}$$l$$\mathrm{T}$$T_{a}$

Theorem J. Let $X$ be a real or complex vector space and let be an algebra overthe same field of scalars as $X$ . Let be an algebraically irreducible representation of
on $X$ such that every endomorphism of $X$ commuting with all operators
is a scalarmultiple of the identity operator. Then for each linearly inependent -tuple, of$A$$A$$\mathrm{T}$$T_{a}$$\mathrm{n}$1991 Mathematics Subject Classification:$47\mathrm{A}67,16\mathrm{G}30$.

$x_{1},$ $\ldots$$x_{n}$

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elements of $X$ the orbit$\mathcal{O}(\mathrm{T};x_{1}, \ldots,Xn)$coincides with$X^{n}$.

The above result does not imply a positive answer to the above Problem even when
does not necessarily mean thatis algebraically irreducible, since the triviality ofmust coincide with the scalar
the endomorphisms of $X$ commuting with all operators
multiples of the identity. A positive answer to this problem in the case of an algebraicallyirreducible representation on a space of type $F$ was given in [4]. There is no known
example of an infinite .imensional Banach space $X$ for which the answer to the Problem
is in positive, nor is there an example of a locally convex space for which the answer isthough the general belief is that such an example should exist. In this talk we$\mathrm{T}’$$\mathrm{T}$$T_{a}$$\mathrm{d}.$negative,shall discuss the following topics.
I. A positive answer to the Problem of Fell and Doran in the case when $X=(s)-$the (completely metrizable) locally convex space of all scalar valued sequences, or, moregenerally, when $X$ is an arbitrary product of the field of scalars ([7]). Those are theonly known examples of the positive solution of the Problem in case when the considered
representations are not automatically algebraically irreducible.
II. Necessary and sufficient conditions in order that a representation satisfying theassumptions of the Problem satisfies also the conclusion ([5]). We do not know whether
these conditions are satisfied automatically. If so, they give a way of attacking the Prob, if not, they offer an additional con- dition under which the answer is positive. The
conditions are expressed in terms of intertwinning operators.
. The existence
II. A concept of”order” of a strongly closed proper subalgebra ofof a subalgebra of order 3 implies immediately the existence of a counterexample to the$\mathrm{l}\mathrm{e}\mathrm{m}$$\mathrm{L}(\mathrm{X})$Problem ([6]).

References[1] F.F. Bonsall and J. Duncan, Complete Normed Algebras. Springer Verlag 1973.
[2] J.M.G. Fell and R.S. Doran, Representations of -Algebras, Locally Compact Groups,and Banach *-Algebraic Bundles, vol. I, Academic Press 1988.
$*$

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[3] N. Jacobson, Lectures in Abstract Algebra, vol. , New York 1953.
[4] W. elazko, A density theorem for -spaces, Studia Math. 96 (1990), 159-166.
, On a problem of Fell and Doran, Colloq. Math. 62 (1991), 31-37.
[5], On strongly closed subalgebras of $B(X)$ , ibidem 67 (1994), 289-295.
[6]Math.
, A density theorem for algebra representations on the spaces (s),
[7]
$\Pi$

$\mathrm{F}$$\dot{\mathrm{Z}}$––$\mathrm{S}\mathrm{t}\mathrm{u}\dot{\mathrm{d}}\mathrm{i}\mathrm{a}$–130 (1998), 293-296.

Mathematical Institute of the Polish Academy of Sciences 00-950 Warszawa (POLAND),
Sniadeckich 8, P.O.Box 137-mail adress:
zelazko@impan. impan.

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