Forcing axioms and coronas of nuclear -algebras
We prove several rigidity results for various large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable -algebras with the metric approximation property and an increasing approximate identity of projections.
Key words and phrases:-algebras; corona algebra; rigidity of quotients; Forcing Axioms; automorphisms
2010 Mathematics Subject Classification:46L40, 46L05, 03E50
Let be a separable Hilbert space, the -algebra of bounded, linear operators on , and its ideal of compact operators. The quotient, , is called the Calkin algebra. The Weyl-von Neumann theorem, one of the most celebrated results of operator theory, classifies the self-adjoint elements of up to unitary equivalence via their spectra. This classification depends heavily on the fact that a self-adjoint element necessarily lifts to a self-adjoint operator on . Consequently, an extension of the Weyl-von Neumann theorem to normal elements of (which do not necessarily lift to normal operators) had to wait for Brown, Douglas and Fillmore in the 1970’s. Their pioneering work [BDF.ext] illuminated deep connections with algebrac topology, -homology, index theory, and extensions of -algebras, and brought to light new questions about , most prominent among them: does have an automorphism which sends the unilateral shift to its adjoint? Or, even weaker, does have an outer automorphism? (Since inner automorphisms of preserve the Fredholm index, they cannot send the unilateral shift to its adjoint.)
These problems remained open for decades, until, in the 2000’s, Phillips and Weaver ([Phillips-Weaver]) showed that the Continuum Hypothesis (CH) implies the existence of outer automorphisms of , and Farah ([Farah.C]) showed that the Open Coloring Axion (OCA) implies that every automorphism of is inner. Together these results show that the existence of outer automorphisms of is independent of ZFC. Whether there can be an automorphism sending the shift to its adjoint in some model of ZFC remains open, though Farah’s result implies that there are models of ZFC where no such automorphism exists.
Farah’s theorem is just one of many results illustrating the effect of the Proper Forcing Axiom (of which OCA is a consequence) on the rigidity of an uncountable structure; similarly, CH has often been found to have the opposite effect on the same structures ([Just, Velickovic.OCAA, Farah.AQ]). In retrospect, the proofs of most of these rigidity results take the following common form: first, one uses PFA to show that every automorphism is, in a certain canonical way, determined by a Borel subset of . These automorphisms are ’absolute’, being present in every model of ZFC by Shoenfield’s absoluteness Theorem ([Kanamori, Theorem 13.15]); we think of them as being the “trivial” automorphisms. Second, one shows, using only ZFC, that such trivial automorphisms must have a certain desired structure. On the other hand, CH can usually be used with back-and-forth arguments to show that there are too many automorphisms for all of them to be determined by a Borel subset of .
In the category of -algebras, the objects relevant to this rigidity / nonrigidity phenomenon are the corona algebras. This class includes both and all algebras of the form , where is locally compact, noncompact, and Hausdorff. The corona algebras form a wide class of -algebras, due to the fact that to every nonunital -algebra one may associated a corona algebra . Moreover, they have been of use to -algebraists since at least the 1960’s, when Busby showed that the extensions of by are determined (up to a certain notion of equivalence) by -homomorphisms from into [Busby].
In [Coskey-Farah], Coskey and Farah formalized the notion of a “trivial” automorphism of a corona algebra , where is separable. Before reproducing their definition we recall some relevant facts. The corona algebra takes the form of a quotient , where is the multiplier algebra of , defined to be the largest -algebra containing as an essential ideal. The quotient map is denoted by . The strict topology on is the topology generated by the seminorms and (). If is separable, then its multiplier algebra is separable and metrizable in its strict topology, and moreover every norm-bounded subset of is Polish in this topology.
Let and be separable -algebras. A function is trivial if its graph
is Borel as a subset of when endowed with the strict topology.
In an effort to capture the rigidity phenomena described above, Coskey and Farah made the following two conjectures:
Suppose is a separable, nonunital -algebra. Then,
CH implies that there is a nontrivial automorphism of , and
PFA implies that every automorphism of is trivial.
In [Coskey-Farah], Coskey and Farah verified Conjecture B, part (1), for a wide class of -algebras; other -algebras have been dealt with in [V.Nontrivial]. Our focus will be on part (2) of the conjecture, which has been verified for a small class of -algebras in [McKenney.UHF] and [Ghasemi.FDD]. Our assumption will not be PFA but a weakening of it: we will assume Martin’s Axiom for -many dense sets () and a strengthening of the Open Coloring Axiom called . (See §2 for a statement.) Our main result (Theorem LABEL:thm:Borel in the sequel) confirms part (2) of the conjecture for a large class of -algebras. (For the definition of the metric approximation property, see the beginning of §LABEL:sec:cons.RedProd.)
Assume and . Let and be separable -algebras, each with an increasing approximate identity of projections, and suppose has the metric approximation property. Then every isomorphism is trivial.
The metric approximation property is known to hold for a large class of -algebras including, but not limited to, all nuclear -algebras. (For instance, is known to have the metric approximation property [Haagerup.FF2] even though it is not nuclear.) It does not hold for all separable -algebras, however ([Szank:AP]). Combined with the main result of [Coskey-Farah], Theorem C gives the following corollary:
Let be a separable, nuclear, and unital -algebra. Then the existence of a nontrivial automorphism of is independent of .
Given a sequence of unital -algebras (), the quotient is a corona algebra which we call the reduced product of the sequence . In Theorem LABEL:thm:trivialredprod of the sequel, we provide a finer characterization of the possible isomorphisms between reduced products of a certain type:
Assume . Let () be sequences of unital, separable -algebras with no central projections, and suppose that each has the metric approximation property. Suppose
is an isomorphism. Then there are finite sets , a bijection , and maps such that lifts , and is an -isomorphism, where tends to zero.
Here an -isomorphism is a map which preserves all of the -algebraic operations on the unit ball, up to an error in norm of . (See LABEL:defin:apmaps for a precise definition.)
Motivated by the conclusion of Theorem E, in §LABEL:sec:apmaps we study the conditions under which two -algebras and which are -isomorphic must be isomorphic, with a focus on when we can make uniform over all and in some class of -algebras. We obtain positive results when is unital, separable, and AF, and when both and are Kirchberg algebras satisfying the UCT. The combination of these results with Theorem E shows that several rigidity results concerning corona algebras, known already to be false under CH, are true under PFA. (See Corollary LABEL:cor:trivialredprod for the specifics).
The proofs of Theorem C and Theorem E are based on a technical and powerful lifting result, Theorem LABEL:thm:lifting in the sequel, which may be viewed as a noncommutative extension of the “OCA lifting theorem”, Theorem 3.3.5, of [Farah.AQ]. Our version provides well behaved liftings for maps of the form
where is a sequence of finite-dimensional Banach spaces, and preserves a limited set of algebraic operations. The proof of this result makes up the technical core of the paper, and most of our other results are derived from it. Many other lifting results in this area can be viewed as restricted versions of Theorem LABEL:thm:lifting; for instance, the OCA lifting theorem of [Farah.AQ] applies to the case where each is one-dimensional and is commutative with real rank zero; whereas [Ghasemi.FDD] and [Farah.C] provide versions of Theorem LABEL:thm:lifting where each is a matrix algebra and is UHF. Without introducing substantially new techniques, any PFA rigidity result for quotients of metric structures must have at its core some similar lifting result. For this reason, the high generality of our lifting theorem makes it apt for future applications on rigidity of quotient structures.
The paper is structured as follows. §2 is dedicated to preliminaries and notation from both set theory and operator algebras. In §LABEL:sec:apmaps we define the notion of an -isomorphism and prove several results concerning -isomorphisms and their relationship to isomorphisms between reduced products. In §LABEL:sec:liftstatements and §LABEL:sec:lemma_proofs we state and then prove our lifting theorem (Theorem LABEL:thm:lifting). In §LABEL:sec:cons.RedProd, §LABEL:sec:cons.Borel, and §LABEL:sec:cons.NonembLift we provide several consequences of Theorem LABEL:thm:lifting, including Theorems C and E. Finally, in §LABEL:sec:conclusion we offer some open problems.
The authors are indebted to Ilijas Farah for many useful remarks. Some of these results are contained in second author’s PhD Thesis, see [V.PhDThesis].
2.1. Descriptive set theory and ideals in
A topological space is said to be Polish if it is separable and completely metrizable. As all compact metrizable spaces are Polish, so is when identified with the product topology on . If is Polish, is called meager if it is the union of countably-many nowhere dense sets. is called Baire-measurable is it has meager symmetric difference with an open set and analytic if it is the continuous image of Borel subset of a Polish space. Every analytic subset of a Polish space is Baire-measurable. If and are Polish and we say that is -measurable if for every open , is in the -algebra generated by the analytic sets. -measurable functions are, in particular, Baire-measurable (see [Kechris.CDST, Theorem 21.6]). We will frequently need the following characterization of the comeager subsets of certain compact spaces. (The symbol reads “there exist infinitely-many ”, whereas the symbol reads “for all but finitely-many ”.) The following is a consequence of the work of Jalali-Naini and Talagrand ([Jalali-Naini, Talagrand.Compacts]). Its proof can be found in [Farah.AQ, §3.10].
Let be finite sets. A set is comeager if and only if there is a partition of into intervals, and a sequence , such that whenever .
A set is hereditary if for all and , we have . Proposition 2.1 implies that the intersection of finitely-many hereditary, nonmeager subsets of is also hereditary and nonmeager ([Farah.AQ, §3.10]). This extends to countable intersections when the sets are also assumed to contain the finite subsets of .
We state two descriptive set-theoretic results about uniformization of functions. The first is the Jankov-von Neumann Theorem.
Theorem 2.2 ([Kechris.CDST, Theorem 18.1]).
Let be Polish and be analytic. Let . Then there exists a -measurable function such that for all , . We say that uniformizes .
In general is not possible to uniformize Borel sets with a Borel function. This is however possible when the vertical sections of are well behaved.
Theorem 2.3 ([Kechris.CDST, Theorem 8.6]).
Let be Polish and be a Borel set such that its vertical sections are either empty or nonmeager. Then there is a Borel function uniformizing .
A subset is an ideal on if it is hereditary and closed under finite unions. is proper if , or equivalently . All ideals are, unless otherwise stated, assumed to be proper. An ideal is said to be dense if for every infinite there is an infinite with . Ideals are in duality with filters: if is an ideal, then is a filter. Moreover, is maximal among all proper ideals if and only if is an ultrafilter.
A family of infinite sets is almost disjoint (a.d.) if for every distinct we have that is finite. An a.d. family is treelike if there is a bijection such that for every , is a branch through , i.e., a pairwise comparable subset of . An ideal is ccc/Fin if meets every uncountable, a.d. family . Proposition 2.1 implies the following, which characterizes the ideal of finite sets as minimal, with respect to the Rudin-Blass ordering, among all ideals with the Baire Property. (We will not need the Rudin-Blass ordering in our work.)
Proposition 2.4 ([Jalali-Naini, Talagrand.Compacts]).
Let be an ideal containing the finite sets. Then the following are equivalent.
has the Baire Property.
There is a partition , where each is a finite set, such that for any infinite set , is not in .
An easy argument shows that if satisfies the last of the above equivalent statements, then there is an a.d. family of size continuum which is disjoint from , and the same is true if does not contain the finite sets. Thus every ccc/Fin ideal is nonmeager.
2.2. Forcing axioms and their consequences
Forcing axioms were introduced as extensions of the Baire Category Theorem. Here we just introduce the axioms we will use in this paper. For a comprehensive account on Forcing Axioms, see for example [Todorcevic.FA].
One of the most studied forcing axioms is the Proper Forcing Axiom (), introduced by Shelah in [Shelah.PF]. We will focus on two consequences of : , an infinitary version of introduced by Farah in [Farah.LG], and , a local version of Martin’s Axiom.
If is a set, denotes the set of unordered pairs of elements of . is the following statement. For every separable metric space and every sequence of partitions , if every is open in the product topology on , and for every , then either
there are () such that and for every , or
there is an uncountable and a continuous bijection such that for all we have
is the restriction of to the case where for every . It is not known whether the two are equivalent, but is sufficient to contradict . In particular, implies that , where is the minimal cardinality of a family of functions in that is unbounded with respect to the relation defined by .
Let be a partially ordered set (poset). Two elements of are called incompatible if there is no element of below both of them. A set of pairwise incompatible elements is an antichain. If all antichains of are countable, is said to have the countable chain condition (ccc). A set is called dense if with . A filter is a subset of such that for all and with , , and for any , there is some such that .
Martin’s Axiom at the cardinal (written ) states: for every poset that has the ccc, and every family of dense subsets (), there is a filter such that for every . is a theorem of , as is the negation of . In particular, contradicts .
For many of the results of this paper we will assume . We note here that every model of has a forcing extension which has the same and satisfies .
For the basics of -algebras we refer the reader to [Dixmier]. Our notation, for the most part, will be standard; in particular, if is a -algebra, then we will write , , and for the closed unit ball of , the set of positive elements, and of unitaries in respectively. A subalgebra will always refer to a -subalgebra of , and an ideal in will mean a -subalgebra of satisfying for all . An ideal in will be called essential if the only satisfying for all is . An isomorphism between -algebras is assumed to preserve the adjoint operation in addition to all other -algebraic operations.
Multipliers, coronas and lifts
The main -algebraic object we study is the multiplier algebra and the associated corona algebra.
Let be a -algebra. The multiplier algebra is the unique, up to isomorphism, unital -algebra which contains as an essential ideal and which has the property that whenever is an essential ideal of a -algebra , there is a unique embedding which is the identity on . The quotient is called the corona algebra of , and we write for the quotient map .
The construction of from is nontrivial. We refer the reader to [Blackadar.OA, II.7.3] for a discussion. For our purposes we will find very useful the following alternative characterization of , which the reader may take as a concrete description of . Recall that any -algebra may be realized as a -subalgebra of , for some Hilbert space , by the Gelfand-Naimark-Segal construction. In this setting, we define the strict topology on to be the the topology generated by the seminorms and , where ranges over . is the closure of with respect to the strict topology.
If is nonunital, then will nonseparable in its norm topology. However, if is separable, then is separable in the strict topology, and moreover every norm-bounded subset of is Polish in the strict topology.
We take the opportunity here to point out a few examples of particular multiplier algebras and corona algebras.
If is unital, ;
If , then and ;
If each is unital, then . In this case, the corona is called a reduced product.
If each is unital and is an ideal, the algebra is defined as follows:
The multiplier algebra of is . The corresponding corona algebra is known as the -reduced product of the ’s or, if is maximal, the ultraproduct.
The following lemma provides a stratification of into subspaces analogous to reduced products. Various forms of this stratification have been used in the literature already: see, for instance, [Elliott.Der2, Theorem 3.1], [Farah.C, Lemma 1.2] and [Arveson.Notes].
Let be a -algebra with a countable approximate identity such that , , and . Given an interval we write . Let be given. Then there are finite intervals , for each and , and , such that
for each , the intervals () are pairwise disjoint and consecutive,
commutes with for each , and
For each , and are both in , and hence we may find for each some such that and are both less than . Applying this recursively we may construct a sequence such that
Define , and let
We note that these sums converge in the strict topology since for any interval with . Moreover, since
it follows that . Finally, we have
Then, setting , we have the required intervals. ∎
The main concern of this paper is the study of isomorphisms , where and are nonunital separable -algebras. Given such , a map is a lift of if the following diagram commutes;