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The Carrier Space of a Reflexive Operator Algebra

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The Carrier Space of a Reflexive Operator Algebra PACIFIC JOURNAL OF MATHEMATICS Vol. 81, No. 2, 1979 THE CARRIER SPACE OF A REFLEXIVE OPERATOR ALGEBRA ALAN HOPENWASSER AND DAVID LARSON Many properties of nest algebras are actually valid for reflexive operator algebras with a commutative subspace lattice. In this paper we collect a number of such results related to the carrier space of the algebra. Included among these results are a generalization of Ringrose's criterion, a description of the partial correspondence between lattice homomorphisms of the carrier space and projections in the lattice, the construction of isometric representations of certain quotient algebras, and a direct sum decomposition of the commutant of the core modulo the intersection of the spectral ideals. Let J^ = Alg ^ where if is a commutative subspace lattice and let ^ be the intersection of all the spectral ideals in Jzf. (See §1 for definitions.) In §1 we generalize Ringrose's criterion to the commutative subspace lattice case: A e ^ if, and only if, for each e > 0 there is a finite family {Et} of mutually orthogonal intervals from & such that Σ^ = l and WE.AE.W < ε, i = 1, . , n. We also prove that ^ is the closed linear span of commutators of the form AL — LA, where A e jy and L e £f. In § 2 we describe the partial correspondence between certain projections in Sf and certain lattice homomorphisms in the carrier space X. A necessary (but not sufficient) condition for an operator A to be in the radical of *$/ is given in §3. In §4 we exhibit isometric representations as algebras of operators acting on Hubert space of each quotient algebra J^fjj^φ and of the quotient *S%f\<J\ In the nest algebra case this was done by Lance in [5]. Finally, in §5 we generalize somewhat a theorem from [6] which identifies the ^"-commutant of the core of J^ as the direct sum of the diagonal of j& and 1* Let ^ be a commutative subspace lattice acting on a separable Hubert space g(f, that is to say, £f is a lattice of commuting, orthogonal projections on έ%f which contains 0 and 1 and is closed in the strong operator topology. Let J^ = Alg Sf, the algebra of all operators leaving invariant each projection in £f. Then jy is a reflexive operator algebra whose lattice of invariant subspaces is just Sf [1]. Define the carrier space, X, of Sf to be the set of all lattice homomorphisms of S^ onto the trivial lattice {0,1}. If the carrier space is given the topology in which a net, φu, converges to φ if, and only if, φv{L) —> φ(L) for each L e JZf, then it becomes a 417 418 ALAN HOPENWASSER AND DAVID LARSON compact, Hausdorfϊ topological space. A projection E in <$f is said to be an interval if E = L — M for projections L, Me^f with M < L. If ^e X, we say that E is a ίesί interval for ^ if 0(-2f) — 1 and 0(^0 — 0 (It is easy to check that this is well-defined.) Let ^φ — {E\E is a test interval for φ}. jF'φ is a family of intervals which satisfies the finite in- tersection property and is maximal with respect to this property. Any family of intervals satisfying these conditions is called a basic family; there is a one-to-one correspondence between elements of the carrier space of £f and basic families of intervals from ^ [3]. For each φ in X, define a continuous semi-norm Nφ on J^f by NΦ(T) = inf {\\ETE\\ | JSe JQ, for each Γej^ This, in turn, permits the definition of the spectral ideal, J^$, associated with φ: J^$ = {Tej%f\Nφ(T) — 0}. The spectral ideals are closed two-sided ideals in j^ as is the intersection, Jf, of all the spectral ideals. Proofs of these facts can be found in [3], as well as the fact that ^ is contained in the radical, \^?, of the algebra J^Γ It is known that ^ = & if j^ is a nest algebra [7] and in a number of other cases, and we conjecture that equality always holds. Should it occur that ^ need not equal ^?, it now appears clear that the role played by ^ in the structure of jy is at least as important as the role played by the radical. As evidence in favor of the conjecture, we prove below that the Ringrose criterion for membership in the radical of a nest algebra ([7], Theorem 5.4) is a criterion for membership in ^ in the general case. Each semi-norm Nφ on <$>/ can be identified with the quotient norm on Jzfjjtfφ. Denote this quotient algebra by 3?φ and the can- onical quotient map by qφ. Later, in §4, we shall exhibit an iso- metric representation of &φ as an algebra of operators acting on Hubert space. PROPOSITION 1. For each Tej^,Nφ(T) = ||g,(T)||. Proof. Let Γej^ If E is a test interval for φ, then ETE- Γej^. Hence \\qΦ{T) \\ £ \\ T + (ETE - Γ)|| - \\ETE\\. Thus, \\qΦ(T)\\ ^ inf {\\ETE\\ \Eej?~φ} - NΦ(T). For the opposite inequality, let Se^fφ. Let ε be an arbitrary positive number and choose a test interval E for φ such that \\ESE\\ < ε. Then || T + S\\ ^ \\E(T + S)E\\ = \\ETE + ESE\\ ^ H^ΪΈΊI - \\ESE\\ ^ NΦ(T) - e . Since ε is arbitrary, || T+ S\\ ^NΦ(T) for all Se j*fφ. Hence THE CARRIER SPACE OF A REFLEXIVE OPERATOR ALGEBRA 419 NΦ(T). A similar result holds for the quotient algebra J^/^ where ^ == Γ\φex*Stfφ If & = J&Ί^ and if q\\Szf -> £& is the canonical quotient map, then supiV^ ) = ||?( )ll Further, for each Γej^ the supremum is attained. In the case of a nest algebra this is just Lemma 1.6 of [6] expressed in terms of the carrier space of £f. The ingredients needed for this result also yield the validity of the Ringrose criterion as a description of the elements of ^" a conse- quence of this criterion is the identification of <J^ as the closed linear span of all commutators AL — LA, with A e jzf and L e Sf. The first preliminary result which we need is the assertion that each such commutator lies in LEMMA 2. J/4GJ/ and Le^f then AL — LAe Proof. Let φeX. If φ(L) = 1 then L e J^ and NΦ(AL - LA) ^ \\L(AL - LA)L\\ = 0. If φ(L) = 0 then 1 - L e &\ and NΦ(AL - LA) ^ || (1 - L)(AL - LA)(X - L) \\ = 0. Thus, AL - LA e J^$, for all φex; i.e., AL — LEMMA 3. If φeX and iej/ then NΦ{A) ^ Proof. Since ^ Q Jϊή,, we have || qφ(A) || ^ || g(A) ||, and the lemma follows from Proposition 1. LEMMA 4. Let A e *s/. Then there exists an element φ e X such that Nψ(A) ^ NΦ(A), for all ψeX. Proof. In the remark on p. 379 of [3] it is shown that the mapping ψ —> Nψ(A) of X into R is upper semi-continuous. Since X is a compact Hausdorff space, this mapping achieves its supremum at some point φ in X. DEFINITION. A projection in Jzf is said to be simple if it is 0 or it is a finite sum of intervals from £f. PROPOSITION 5. The set of simple projections is a complemented lattice. Proof. We must show that if E and F are simple, then E Λ F = EF, E1 = I - E, and E V F = E + F - EF are simple. First suppose that E and ί7 are intervals from JZf, say E—L — M and F = N- P, where L, M, N, Pejϊf and M ^ L and P ^ iSΓ. Then 420 ALAN HOPENWASSER AND DAVID LARSON E A F = L A N - ((L Λ P) V (N A M)) . Since £? is a lattice, L A N and (L Λ P) V (N A M) both lie in £f\ further (L A P) V (N A M) <* L A N, so # Λ -P is an interval. Now 7 suppose that # = V?=i ^ and i* = VJU ^ > where {JS'J and {Fd} are each families of mutually orthogonal intervals from £f. Then E A F = Vi,ί (Et A Fj) and {Ei A Fά) is a family of mutually orthogonal intervals from £f\ thus E A F is simple. Next observe that if E is an interval then E1 is the sum of two orthogonal intervals and hence is simple. If E is merely simple, then E = \f^1Ei1 where {Et} is a mutually orthogonal family of intervals from £f. Since E1 — Λ?=i^i" and eacl]L ^ is simple, the paragraph above implies that EL is simple. It remains only to show that if E and F are simple, then EvF is simple. When E and F are orthogonal, this is obvious; for the general case use the facts that F A E1 is simple and E V F = E V (F A E1). For the following lemma, and for §2, recall that a subset is said to be an ideal if it satisfies the two properties: (a) K19K2e^Tlf =>K,V K2e^: (b) K e JΓ19 L G j^ L ^ K => L 6 J2ίς. Similarly, a subset J^ is said to be a co-ideal if it satisfies the dual properties: (a') (b') An ideal is prime if its complement is a co-ideal. LEMMA 6. Every interval from £? is a test interval for some lattice homomorphism in X. Proof. Let E be an interval from £?. Then there exist projec- tions L and M in Sf with M < L such that E = L - M.
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