aa r X i v : . [ m a t h . OA ] D ec On the Lifting Property for C ∗ -algebras byGilles Pisier ∗ Texas A&M UniversityCollege Station, TX 77843, U. S. A.December 7, 2020
Abstract
We characterize the lifting property (LP) of a separable C ∗ -algebra A by a property of itsmaximal tensor product with other C ∗ -algebras, namely we prove that A has the LP if and onlyif for any family { D i | i ∈ I } of C ∗ -algebras the canonical map ℓ ∞ ( { D i } ) ⊗ max A → ℓ ∞ ( { D i ⊗ max A } )is isometric. Equivalently, this holds if and only if M ⊗ max A = M ⊗ nor A for any von Neumannalgebra M . MSC (2010): 46L06, 46L07, 46L09Key words: C*-algebras, von Neumann algebras, lifting property, tensor products
Contents (0.1) ∗ ORCID 0000-0002-3091-2049 C ∗ -algebra A has the lifting property (LP in short) if any contractive completelypositive (c.c.p. in short) map u : A → C/ I into a quotient C ∗ -algebra admits a c.c.p. liftingˆ u : A → C . In [9] Choi and Effros proved that all (separable) nuclear C ∗ -algebras have theLP. Later on in [19] Kirchberg proved that the full C ∗ -algebra of the free group F n with n > n = ∞ ) generators, which is notoriously non-nuclear, also has the LP. It follows that separableunital C ∗ -algebras with the LP are just the quotients of C ∗ ( F ∞ ) for which the quotient map admitsa unital completely positive (u.c.p. in short) lifting. More generally, as observed by Boca in [7], itfollows from the latter fact that the LP is stable by unital (maximal) free products. Indeed, it isan immediate consequence of Boca’s theorem in [6] (see [11] for a recent simpler proof) that thefree product of a family of unital ∗ -homomorphisms that are liftable by u.c.p. maps is also liftableby a u.c.p. map. However it is well known that the reduced C ∗ -algebra of F n fails the LP.Our main result is a very simple (functorial) characterization of the LP in terms of maximaltensor products (namely (0.4) below) that seems to have been overlooked by previous authors. In[18] Kirchberg gave a tensor product characterizarion of the local version of the LP called the LLP.He showed that a C ∗ -algebra A has the LLP if and only if B ( ℓ ) ⊗ max A = B ( ℓ ) ⊗ min A . He thenwent on to conjecture that for separable C ∗ -algebras the LLP implies the LP, and he showed thata negative solution would imply a negative answer for the Connes embedding problem (see Remark0.17 for more information). We hope that our new criterion for the LP will help to answer thisquestion whether the LLP implies the LP. More specifically, we believe that a modification of theconstruction in [29] might lead to a counterexample.Let ( D i ) i ∈ I be a family of C ∗ -algebras. We will denote by ℓ ∞ ( { D i |∈ I } ) or simply by ℓ ∞ ( { D i } )the C ∗ -algebra formed of the families d = ( d i ) i ∈ I in Q i ∈ I D i such that sup i ∈ I k d i k < ∞ , equippedwith the norm d sup i ∈ I k d i k .Consider the following property of a C ∗ -algebra A :(0.1) For any family ( D i ) i ∈ I of C ∗ -algebras and any t ∈ ℓ ∞ ( { D i } ) ⊗ A we have(0.2) k t k ℓ ∞ ( { D i } ) ⊗ max A ≤ sup i ∈ I k t i k D i ⊗ max A , where t i = ( p i ⊗ Id A )( t ) with p i : ℓ ∞ ( { D i } ) → D i denoting the i -th coordinate projection.Of course, for any i ∈ I we have k p i ⊗ Id A : ℓ ∞ ( { D i } ) ⊗ max A → D i ⊗ max A k ≤
1, and hence wehave a natural contractive ∗ -homomorphism(0.3) ℓ ∞ ( { D i } ) ⊗ max A → ℓ ∞ ( { D i ⊗ max A } ) . Thus (0.2) means that we have a natural isometric embedding(0.4) ℓ ∞ ( { D i } ) ⊗ max A ⊂ ℓ ∞ ( { D i ⊗ max A } ) . More precisely ℓ ∞ ( { D i } ) ⊗ max A can be identified with the closure of ℓ ∞ ( { D i } ) ⊗ A (algebraictensor product) in ℓ ∞ ( { D i ⊗ max A } ).Let us denote C = C ∗ ( F ∞ ) , the full (or “maximal”) C ∗ -algebra of the free group F ∞ with countably infinitely many generators.Using the description of the norm in D ⊗ max ℓ n (for an arbitrary C ∗ -algebra D ), with ℓ n ⊂ C identified as usual with the span of n free generators, it is easy to check that C has this property(0.1) (see Lemma 5.1 below). In fact, as a consequence, any unital separable A with LP has thisproperty. Our main result is that conversely this characterizes the LP.2he fact that (0.1) implies the LP contains many previously known lifting theorems, for instancethe Choi-Effros one. It also gives a new proof of the LP for C ∗ ( F ∞ ).The key step will be a new form of the reflexivity principle. Consider for E ⊂ A finite dimen-sional the normed space M B ( E, C ) defined below in § → max-tensorizingmaps into another C ∗ -algebra C . We will show that if (and only if) A has the property (0.1) thenthe natural map M B ( E, C ∗∗ ) ⊂ M B ( E, C ) ∗∗ is contractive for any finite dimensional E ⊂ A . As a consequence it follows that any u ∈ M B ( E, C )admits an extension in
M B ( A, C ) with the same
M B -norm up to ε >
0. From this extensionproperty we deduce a lifting one: if A assumed unital and separable satisfies (0.1), any unital c.p.map u : A → C/ I ( C/ I any quotient) admits a unital c.p. lifting. In other words this says that(0.1) implies the LP. Remark . Let ( D i ) i ∈ I be a family of C ∗ -algebras. Consider the algebraic tensor product ℓ ∞ ( { D i } ) ⊗ A . We will use the natural embedding ℓ ∞ ( { D i } ) ⊗ A ⊂ Y i ∈ I ( D i ⊗ A ) . Any t ∈ ℓ ∞ ( { D i } ) ⊗ A can be identified with a bounded family ( t i ) with t i ∈ D i ⊗ E for some f.d.subspace E ⊂ A . Thus the correspondence t ( t i ) gives us a canonical linear embedding(0.5) ℓ ∞ ( { D i } ) ⊗ A ⊂ ℓ ∞ ( { D i ⊗ max A } ) . The property in (0.1) is equivalent to the assertion that this map is an isometric embedding when ℓ ∞ ( { D i } ) ⊗ A is equipped with the maximal C ∗ -norm.More precisely, for any f.d. subspace E ⊂ A we have a canonical linear isomorphism ℓ ∞ ( { D i } ) ⊗ E ≃ ℓ ∞ ( { D i ⊗ E } ) . By convention, for any C ∗ -algebra D let us denote by D ⊗ max E the normed space obtained byequipping D ⊗ E with the norm induced on it by D ⊗ max A . Then the property in (0.1) is equivalentto the assertion that for any f.d. E ⊂ A we have an isometric isomorphism(0.6) ℓ ∞ ( { D i } ) ⊗ max E ≃ ℓ ∞ ( { D i ⊗ max E } ) . Notation.
Let
D, A be C ∗ -algebras and let E ⊂ A be a subspace. We will denote( D ⊗ E ) +max = ( D ⊗ max A ) + ∩ ( D ⊗ E ) Theorem 0.2.
Let A be a separable C ∗ -algebra. The following are equivalent: (i) The algebra A has the lifting property (LP). (ii) For any family ( D i ) i ∈ I of C ∗ -algebras and any t ∈ ℓ ∞ ( { D i } ) ⊗ A we have k t k ℓ ∞ ( { D i } ) ⊗ max A ≤ sup i ∈ I k t i k D i ⊗ max A . In other words, the natural ∗ -homomorphism (0.3) is isometric. + For any family ( D i ) i ∈ I of C ∗ -algebras and any t ∈ ℓ ∞ ( { D i } ) ⊗ A , the following implicationholds ∀ i ∈ I t i ∈ ( D i ⊗ A ) +max ⇒ t ∈ ( ℓ ∞ ( { D i } ) ⊗ A ) +max . We prove Theorem 0.2 in § Remark . If one replaces everywhere max by min in (ii) in Theorem 0.2,then the property clearly holds for all C ∗ -algebras. Thus Theorem 0.2 implies as a corollary theChoi-Effros lifting theorem from [9], which asserts that nuclear C ∗ -algebras have the LP. Remark . It is easy to see that if A is the direct sum of finitely many C ∗ -algebras satisfying(0.1) then A also does. Remark . Let A be a C ∗ -algebra. It is classical (see e.g. [28, Prop. 7.19]) that for any (self-adjoint, closed and two sided) ideal I ⊂ A and any C ∗ -algebra D we have an isometric embedding D ⊗ max I ⊂ D ⊗ max A . This shows that if A satisfies (0.1) then any ideal I in A also does.Similarly, for any ideal I ⊂ D we have an isometric embedding I ⊗ max A ⊂ D ⊗ max A . Thus if(0.1) holds for a given family ( D i ) it also holds for any family ( I i ) where each I i is an ideal in D i .Since any D is an ideal in its unitization, one deduces from this that if (0.1) holds for any family( D i ) of unital C ∗ -algebras, then it holds for any family. One also sees that A satisfies (0.1) if andonly if its unitization satisfies it. Remark . We will use the fact due to Kirchberg [18] that for any t ∈ D i ⊗ A there is a separable C ∗ -subalgebra ∆ i ⊂ D i such that t ∈ ∆ i ⊗ A and k t k ∆ i ⊗ max A = k t k D i ⊗ max A . Indeed, by [28, Lemma 7.23] for any ε > D εi ⊂ D i such that t ∈ D εi ⊗ A and k t k D εi ⊗ max A ≤ (1 + ε ) k t k D i ⊗ max A . This implies that the C ∗ -algebra ∆ i generated by { D εi | ε =1 /n, n ≥ } has the announced property. Using this fact in the property (0.1) we may assume thatall the D i ’s are separable (and unital by the preceding remark). Remark . Let C/ I be a quotient C ∗ -algebra and let A be another C ∗ -algebra. It is well known(see e.g. [28, Prop. 7.15]) that(0.7) ( C/ I ) ⊗ max A = ( C ⊗ max A ) / ( I ⊗ max A ) . Equivalently, A ⊗ max ( C/ I ) = ( A ⊗ max C ) / ( A ⊗ max I ).Moreover, for any f.d. subspace E ⊂ A we have (for a detailed proof see e.g. [28, Lem. 4.26])(0.8) ( C/ I ) ⊗ max E = ( C ⊗ max E ) / ( I ⊗ max E ) . Moreover, the closed unit ball of ( C ⊗ max E ) / ( I ⊗ max E ) coincides with the image under the quotientmap of the closed unit ball of C ⊗ max E . This known fact can be checked just like for the min-normin [28, Lem. 7.44].Since any separable unital C ∗ -algebra D can be viewed as a quotient of C , so that say D = C / I we have an isomorphism(0.9) D ⊗ max A = ( C ⊗ max A ) / ( I ⊗ max A ) . Using the Remarks 0.6 and 0.7 one obtains:
Proposition 0.8.
To verify the property (0.1) we may assume without loss of generality that D i = C for any i ∈ I . emma 0.9. Let ( D i ) i ∈ I be a family of C ∗ -algebras. Then for any C ∗ -algebra C we have naturalisometric embeddings (0.10) ℓ ∞ ( { D i ⊗ max C } ) ⊂ ℓ ∞ ( { D ∗∗ i ⊗ max C } ) . (0.11) ℓ ∞ ( { D i } ) ⊗ max C ⊂ ℓ ∞ ( { D ∗∗ i } ) ⊗ max C. Proof.
We will use the classical fact that (0.11) (or (0.10)) holds when ( D i ) i ∈ I is reduced to a singleelement (see Remark 0.15 or [28, Prop. 7.26]), which means that we have for each i ∈ I a naturalisometric embedding D i ⊗ max C ⊂ D ∗∗ i ⊗ max C . From this (0.10) is immediate.To check (0.11), since any unital separable C is a quotient of C , we may assume by Remark 0.7that C = C (see [28, Th. 7.29] for details). Now since C satisfies (0.1) (or equivalently the LP) wehave isometric embeddings ℓ ∞ ( { D i } ) ⊗ max C ⊂ ℓ ∞ ( { D i ⊗ max C } ) and ℓ ∞ ( { D ∗∗ i } ) ⊗ max C ⊂ ℓ ∞ ( { D ∗∗ i ⊗ max C } ) . Thus (0.11) follows from (0.10) for C = C , and hence in general. Proposition 0.10.
To verify the property (0.1) we may assume without loss of generality that D i is the bidual of a C ∗ -algebra for any i ∈ I . A fortiori we may assume that D i is a von Neumannalgebra for any i ∈ I .Proof. This is an immediate consequence of Lemma 0.9.In [19], where Kirchberg shows that the C ∗ -algebra C = C ∗ ( F ∞ ) has the LP, he also statesthat if a C ∗ -algebra C has the LP then for any von Neumann algebra M the nor-norm coincideson M ⊗ C (or on C ⊗ M ) with the max-norm. We will show that the converse also holds. Thenor-norm of an element t ∈ M ⊗ C ( C any C ∗ -algebra) was defined by Effros and Lance [13] as k t k nor = sup {k σ · π ( t ) k} where the sup runs over all H and all commuting pairs of representations σ : M → B ( H ), π : C → B ( H ) with the restriction that σ is normal on M . Here σ · π : M ⊗ C → B ( H ) is the ∗ -homomorphism defined by σ · π ( m ⊗ c ) = σ ( m ) π ( c ) ( m ∈ M , c ∈ C ). The norm k k nor is a C ∗ -normand M ⊗ nor C is defined as the completion of M ⊗ C relative to this norm. One can formulate asimilar definition for C ⊗ nor M . See [28, p. 162] for more on this.We will invoke the following elementary fact (we include its proof for lack of a reference). Lemma 0.11.
Let D be another C ∗ -algebra. Then for any c.c.p. map u : C → D the mapping Id M ⊗ u : M ⊗ C → M ⊗ D extends to a contractive (and c.p.) map from M ⊗ nor C to M ⊗ nor D .Proof. Let S max (resp. S nor ) denote the set of states on M ⊗ max C (resp. M ⊗ nor C ). An elementof S max can be identified with a bilinear form f : M × C → C of norm ≤ P f ( x ij , y ij ) ≥ n and all [ x ij ] ∈ M n ( M ) + and [ y ij ] ∈ M n ( C ) + . The set S nor corresponds to the forms f ∈ S max that are normal in the first variable. See [13] or [28, § u : C → D be a c.c.p. map. Clearly, for any f ∈ S nor the form( x, y ) f ( x, u ( y )) is still in S nor . Since k t k nor ≈ sup f ∈ S nor | f ( t ) | for any t ∈ M ⊗ C , the mapping Id M ⊗ u (uniquely) extends to a bounded linear map u M : M ⊗ nor C → M ⊗ nor D . Since u is c.p.we have u M ( t ∗ t ) ∈ ( M ⊗ C ) + := span { s ∗ s | s ∈ M ⊗ C } for any t ∈ M ⊗ C , and hence by density u M ( t ∗ t ) ∈ ( M ⊗ nor C ) + for any t ∈ M ⊗ nor C . This means that u M is positive. Replacing M by M n ( M ) shows that u M is c.p. Since k t k nor = sup f ∈ S nor | f ( t ) | when t ≥
0, we have k u M k ≤ Lemma 0.12.
Let A be a C ∗ -algebra. For any family ( D i ) i ∈ I of C ∗ -algebras we have a naturalisometric embedding ℓ ∞ ( { D ∗∗ i } ) ⊗ nor A ⊂ ℓ ∞ ( { D ∗∗ i ⊗ nor A } ) . Proof.
Let D = c ( { D i } ) so that D ∗∗ = ℓ ∞ ( { D ∗∗ i } ). Consider t ∈ D ∗∗ ⊗ A . Let π : A → B ( H ), σ : D ∗∗ → π ( A ) ′ be commuting non-degenerate ∗ -homomorphisms with σ normal . Thus σ isthe canonical weak* to weak* continuous extension of σ | D : D → π ( A ) ′ . It will be notationallyconvenient to view D i as a subalgebra of D . We observe that σ | D is the direct sum of representationsof the D i ’s. Let σ i : D i → π ( A ) ′ be defined by σ i ( x ) = σ ( x ) (recall D i ⊂ D ). There is anorthogonal decomposition of H of the form Id H = P i ∈ I p i with p i ∈ σ ( D ) ⊂ π ( A ) ′ such that forany c = ( c i ) ∈ D we have σ ( c ) = norm sense P i ∈ I σ i ( c i ) and also σ i ( x ) = p i σ i ( x ) = σ i ( x ) p i for any x ∈ D i . Let ¨ σ i : D ∗∗ i → π ( A ) ′ be the weak* to weak* continuous extension of σ i . Since σ is normal,we have then for any c ′′ = ( c ′′ i ) ∈ ℓ ∞ ( { D ∗∗ i } ) σ ( c ′′ ) = weak* sense X i ∈ I ¨ σ i ( c ′′ i ) . This means σ ≃ ⊕ ¨ σ i . Therefore for any t ∈ D ∗∗ ⊗ A = ℓ ∞ ( { D ∗∗ i } ) ⊗ A we have k σ · π ( t ) k = sup i ∈ I k ¨ σ i · π ( t i ) k ≤ sup i ∈ I k t i k D ∗∗ i ⊗ nor A . Taking the sup over all the above specified pairs ( σ, π ), we obtain k t k nor ≤ sup i ∈ I k t i k D ∗∗ i ⊗ nor A , whence (since the converse is obvious) a natural isometric embedding D ∗∗ ⊗ nor A ⊂ ℓ ∞ ( { D ∗∗ i ⊗ nor A } ) . This completes the proof.The next statement is now an easy consequence of Theorem 0.2.
Theorem 0.13.
Let A be a separable C ∗ -algebra. The following are equivalent: (i) The algebra A has the lifting property (LP). (i)’ The algebra A satisfies (0.1) . (ii) For any von Neumann algebra M we have (0.12) M ⊗ nor A = M ⊗ max A (or equivalently A ⊗ nor M = A ⊗ max M ) . (ii)’ For any C ∗ -algebra D we have D ∗∗ ⊗ nor A = D ∗∗ ⊗ max A (or equivalently A ⊗ nor D ∗∗ = A ⊗ max D ∗∗ ) . Proof.
The equivalence (i) ⇔ (i)’ duplicates for convenience part of Theorem 0.2.(i) ⇒ (ii) boils down to Kirchberg’s result from [19] that C satisfies (ii), for which a simpler proofwas given in [26] (see also [28, Th. 9.10]). Once this is known, if A has the LP then A itself satisfies(ii). Indeed, we may assume A = C / I and by Lemma 0.11 if r : A → C is a c.c.p. lifting then r M : M ⊗ nor A to M ⊗ nor C = M ⊗ max C is contractive, from which (0.12) follows. A priori thisuses the separability of A but we will give a direct proof of (i)’ ⇒ (ii) valid in the non-separablecase in § ⇒ (ii)’ is trivial. Assume (ii)’ (with A possibly non-separable). By Lemma 0.12 we have anatural isometric embedding ℓ ∞ ( { D ∗∗ i } ) ⊗ max A ⊂ ℓ ∞ ( { D ∗∗ i ⊗ max A } ) . By (0.10) and (0.11) we must have (0.1), which means that (i)’ holds. Thus (ii)’ ⇒ (i)’.6e will prove a variant of the preceding theorem in terms of ultraproducts in § Remark . It is known (see [28, Th. 8.22]) that we always have an isometric natural embedding D ∗∗ ⊗ bin A ∗∗ ⊂ ( D ⊗ max A ) ∗∗ . Therefore, for arbitrary D and A , we have an isometric natural embedding(0.13) D ∗∗ ⊗ nor A ⊂ ( D ⊗ max A ) ∗∗ . Thus (ii)’ in Theorem 0.13 can be reformulated as saying that for any D we have an isometricnatural embedding(0.14) D ∗∗ ⊗ max A ⊂ ( D ⊗ max A ) ∗∗ . This is the analogue for the max-tensor product of Archbold and Batty’s property C ′ from [4],which is closely related to the local reflexivity of [12] (see [27, p. 310] or [28, Rem. 8.34] for moreinformation on this topic). However, what matters here is the injectivity of the ∗ -homomorphismin (0.14). Its continuity is guaranteed by (0.13). In sharp contrast for property C ′ continuity iswhat matters while injectivity is automatic since the min-tensor product is injective. Remark . Let A be a C ∗ -algebra. We already used in Lemma 0.9 that fact that for any other C ∗ -algebra D we have an isometric natural morphism D ⊗ max A → D ⊗ max A ∗∗ . Similarly, for anyvon Neumann algebra M , the analogous morphism M ⊗ nor A → M ⊗ nor A ∗∗ is isometric . Thisfollows from the basic observation that if σ : M → B ( H ) and π : A → B ( H ) are representationswith commmuting ranges, then the canonical normal extension ¨ π of π to A ∗∗ takes values in σ ( M ) ′ . Remark . Following Kirchberg [18], a unital C ∗ -algebra A is said to have thelocal lifting property (LLP) if for any unital c.p. map u : A → C/ I ( C/ I being any quotient of aunital C ∗ -algebra C ) the map u is “locally liftable” in the following sense: for any f.d. operatorsystem E ⊂ A the restriction u | E : E → C/ I admits a u.c.p. lifting u E : E → C . The crucialdifference between “local and global” is that a priori u | E does not extend to a u.c.p. map on thewhole of A . If A is not unital we say that it has the LLP if its unitization does. Equivalently ageneral C ∗ -algebra A has the LLP if for any c.c.p. map u : A → C/ I ( C/ I being any quotientof a C ∗ -algebra C ) and any f.d. subspace E ⊂ A the restriction u | E : E → C/ I admits a lifting u E : E → C such that k u E k cb ≤ A has the LLP if and only if B ( ℓ ) ⊗ min A = B ( ℓ ) ⊗ max A or equivalentlyif and only if ℓ ∞ ( { M n | n ≥ } ) ⊗ min A = ℓ ∞ ( { M n | n ≥ } ) ⊗ max A . Using this it is easy to check,taking { D i } = { M n | n ≥ } , that (0.1) implies the LLP. Remark . Clearly the LP implies the LLP. Kirchberg observed in [18] thatif his conjecture that C ⊗ min A = C ⊗ max A for any A with LLP (or equivalently just for A = C )is correct then conversely the LLP implies the LP for separable C ∗ -algebras.Note that if C ⊗ min C = C ⊗ max C , then since C satisfies (0.1) we have ℓ ∞ ( C ) ⊗ min C = ℓ ∞ ( C ) ⊗ max C , and hence for any A with LLP also(0.15) ℓ ∞ ( C ) ⊗ min A = ℓ ∞ ( C ) ⊗ max A. Now (0.15) obviously implies (recalling Proposition 0.8) the property in (0.1), which by Theorem0.2 implies the LP in the separable case, whence another viewpoint on Kirchberg’s observation. according to a recent paper entitled MIP* = RE posted on arxiv in Jan. 2020 by Ji, Natarajan,Vidick, Wright, and Yuen this conjecture is not correct C ∗ -algebra C has the weak expectation property (WEP) (forwhich we refer the reader to [28, p. 188]) if and only if C ⊗ max C = C ⊗ min C . Thus his conjectureis equivalent to the assertion that C has the WEP. He also showed that it is equivalent to theConnes embedding problem (see [28, p. 291]). Remark . For the full C ∗ -algebra C ∗ ( G ) of a discrete group G ,the LP holds both when G is amenable and when G is a free group. It is thus not easy to findcounterexamples, but the existence of C ∗ ( G )’s failing LP has been proved using property (T) groupsby Ozawa in [24]. Later on, Thom [32] gave an explicit example of G for which C ∗ ( G ) fails theLLP. More recently, Ioana, Spaas and Wiersma [16] proved that SL n ( Z ) for n ≥ § Abbreviations and notation
As is customary we abbreviate completely positive by c.p. com-pletely bounded by c.b. unital completely positive by u.c.p. and contractive completely positiveby c.c.p. Analogously we will abbreviate maximally bounded, maximally positive and unital max-imally positive respectively by m.b. m.p. and u.m.p. We also use f.d. for finite dimensional. Wedenote by B E the closed unit ball of a normed space E , and by Id E : E → E the identity operatoron E . We denote by E ⊗ F the algebraic tensor product of two vector spaces. Lastly, by an idealin a C ∗ -algebra we implicitly mean a two-sided, closed and self-adjoint ideal. Let E ⊂ A be an operator space sitting in a C ∗ -algebra A . Let D be another C ∗ -algebra. Recallwe denote (abusively) by D ⊗ max E the closure of D ⊗ E in D ⊗ max A , and we denote by k k max the norm induced on D ⊗ max E by D ⊗ max A . We define similarly E ⊗ max D . We should emphasizethat D ⊗ max E (or E ⊗ max D ) depends on A and on the embedding E ⊂ A , but there will be norisk of confusion. Of course we could also define E ⊗ max F ⊂ A ⊗ max D for a subspace F ⊂ D butwe will not go that far. Let C be another C ∗ -algebra.We will denote by M B ( E, C ) the set of maps u : E → C such that for any C ∗ -algebra D , themap Id D ⊗ u : D ⊗ E → D ⊗ C extends to a bounded mapping u D : D ⊗ max E → D ⊗ max C , andmoreover such that the bound on k u D k is uniform over all D ’s. We call such maps “maximallybounded” or “max-bounded” or simply m.b. for short. (They are called (max → max)-tensorizingin [28].) We denote k u k mb = sup {k u D : D ⊗ max E → D ⊗ max C k} where the sup runs over all possible D ’s.We have clearly k u D k mb ≤ k u k mb .Moreover, if A is unital, assuming that E is an operator system let us denote by M P ( E, C ) thesubset formed of the maps u such that Id D ⊗ u : D ⊗ E → D ⊗ C is positive for any D , by whichwe mean that [ Id D ⊗ u ](( D ⊗ E ) +max ) ⊂ ( D ⊗ C ) +max . Equivalently, u D is positive for all D ’s. We call such maps “maximally positive” (m.p. in short).Replacing D by M n ( D ) shows that the latter maps will be automatically c.p.In passing, recall that for any c.p. u : E → C we have k u k cb = k u k = k u (1) k , and hence k u k mb = k u (1) k for any m.p. u .Obviously we have k u k mb = 1 if E = A and u is a ∗ -homomorphism.Clearly (taking D = M n ) we have M B ( E, C ) ⊂ CB ( E, C ) and(1.1) ∀ u ∈ M B ( E, C ) k u k cb ≤ k u k mb . roposition 1.1. A map u : E → C belongs to M B ( E, C ) if and only if Id C ⊗ u defines a boundedmap from C ⊗ max E to C ⊗ max C and we have k u k mb = k Id C ⊗ u : C ⊗ max E → C ⊗ max C k . Proof.
This follows easily from (0.9).
Corollary 1.2. If C has the WEP then any c.b. map u : E → C is in M B ( E, C ) and k u k cb = k u k mb .Proof. By the WEP of C , we have C ⊗ max C = C ⊗ min C (see [8, p. 380] or [28, p. 188]), and hence k Id C ⊗ u : C ⊗ max E → C ⊗ max C k = k Id C ⊗ u : C ⊗ max E → C ⊗ min C k≤ k Id C ⊗ u : C ⊗ min E → C ⊗ min C k ≤ k u k cb . Remark . Let u ∈ M B ( E, C ). Let F ⊂ C be such that u ( E ) ⊂ F and let B be another C ∗ -algebra. Then for any v ∈ M B ( F, B ) the composition vu : E → B is in M B ( E, B ) and k vu k mb ≤ k v k mb k u k mb . Remark . When E = A any c.p. map u : A → C is in M B ( A, C ) and k u k mb = k u k (see e.g. [27,p. 229] or [28, Cor. 7.8]). However, this is no longer true in general when E is merely an operatorsystem in A . This distinction is important for the present work.Remark . We will use the following well known elementary fact. Let
A, C be unital C ∗ -algebras.Let E ⊂ A be an operator system. Let u : E → C be a unital c.b. map. Then u is c.p. if and onlyif k u k cb = 1 (see e.g. [25], [27, p. 24] or [28, Th. 1.35]). Remark . In the same situation, any unital map u ∈ M B ( E, C ) such that k u k mb = 1 mustbe c.p. Indeed, (1.1) implies k u k cb = k u (1) k = 1, and u is c.p. by the preceding remark. Moreprecisely, the same reasoning applied to the maps u D (with D unital) shows that u is m.p. Remark . Let u : E → C with E an operator system. Let u ∗ : E → C be defined by u ∗ ( x ) =( u ( x ∗ )) ∗ ( x ∈ E ), so that k u ∗ k = k u k . We claim k u ∗ k mb = k u k mb for any u ∈ M B ( E, C ). Indeed,this is easy to check using ( u D ) ∗ = ( u ∗ ) D .A map u : E → C is called “self-adjoint” if u = u ∗ .The following important result was shown to the author by Kirchberg with permission to includeit in [27]. It also appears in [28, Th. 7.6]. Theorem 1.8 ([20]) . Let
A, C be C ∗ -algebras. Let i C : C → C ∗∗ denote the inclusion map. A map u : A → C is in M B ( A, C ) if and only if i C u : A → C ∗∗ is decomposable. Moreover we have k u k mb = k i C u k dec . Recall that a map u : A → C is called decomposable if it is a linear combination of c.p. maps.See [14] or [28, §
6] for more on decomposable maps and the definition of the dec-norm.Elaborating on Kirchberg’s argument we included in [27] and later again in [28, Th. 7.4] thefollowing variant as an extension theorem.
Theorem 1.9.
Let
A, C be C ∗ -algebras. Let E ⊂ A be a subspace. Consider a map u : E → C in M B ( E, C ) . Then k u k mb = inf k e u k dec , where the infimum runs over all maps e u : A → C ∗∗ such that e u | E = i C u . Moreover, the latterinfimum is attained. emark . A fortiori, since k i C u k dec ≤ k u k dec , we have k u k mb ≤ k u k dec .Moreover, for any e u : A → C ∗∗ we have(1.2) k e u k mb = k e u k dec . Indeed, since there is a c.c.p. projection P : ( C ∗∗ ) ∗∗ → C ∗∗ , we have k e u k mb = k i C ∗∗ e u k dec ≥k P i C ∗∗ e u k dec = k e u k dec . Remark . For any C ∗ -algebras C, D the natural inclusion D ⊗ max C ⊂ D ⊗ max C ∗∗ is isometric(see Remark 0.15). Therefore, for any u : E → C we have k u k mb = k i C u k mb where i C : C → C ∗∗ is the canonical inclusion.Moreover, if there is a projection P : C ∗∗ → C with k P k dec = 1 (for instance if P is c.c.p.) then k u k dec = k i C u k dec . Remark . Let E ⊂ A and let M ⊂ B ( H ) be a von Neumannalgebra. Consider a map u : E → M . Let us denote by ˆ u : M ′ ⊗ E → B ( H ) the linear map suchthat ˆ u ( x ′ ⊗ a ) = x ′ u ( a ) ( x ′ ∈ M ′ , a ∈ E ). Then k u k mb = k ˆ u : M ′ ⊗ max E → B ( H ) k cb = inf {k e u k dec | e u : A → M, e u | E = u } . If either M has infinite multiplicity or M ′ has a cyclic vector, then k ˆ u : M ′ ⊗ max E → B ( H ) k cb = k ˆ u : M ′ ⊗ max E → B ( H ) k . This is Th. 6.20 with Cor. 6.21 and Cor. 6.23 in [28].
Remark . In the situation of the preceding remark, if E is an operator system and if ˆ u : M ′ ⊗ max E → B ( H ) is c.p. (in particular if u ∈ M P ( E, M )) then u admits a c.p. extension e u : A → M with k e u k = k u k (in particular e u ∈ M P ( A, M )). This follows from Arveson’s extensiontheorem for c.p. maps and the argument called “the trick” in [8, p. 87].Let E ⊂ A and C be as in Theorem 1.9. To state our results in full generality, we introduce thespace SB ( E, C ) for which the unit ball is formed of all u : E → C satisfying an operator analogueof (0.1). Definition 1.14.
We call strongly maximally bounded the maps u : E → C such that, for anyfamily ( D i ) i ∈ I , the map Id ⊗ u defines a uniformly bounded map from ℓ ∞ ( { D i } ) ⊗ E equippedwith the norm induced by ℓ ∞ ( { D i ⊗ max E } ) to ℓ ∞ ( { D i } ) ⊗ C equipped with the norm induced by ℓ ∞ ( { D i } ) ⊗ max C . We define(1.3) k u k sb = sup {k ( Id ⊗ u )( t ) k ℓ ∞ ( { D i } ) ⊗ max C | ( t i ) i ∈ I ∈ ℓ ∞ ( { D i } ) ⊗ E, sup i ∈ I k t i k D i ⊗ max C ≤ } , where the sup runs over all possible families ( D i ) i ∈ I of C ∗ -algebras. Definition 1.15.
Let
A, C be C ∗ -algebras. Let E ⊂ A be a subspace. A linear map u : E → C willbe called maximally isometric (or max-isometric in short) if for any C ∗ -algebra D the associatedmap Id D ⊗ u : D ⊗ E → D ⊗ C extends to an isometric map D ⊗ max E → D ⊗ max C . Equivalently,this means that E ⊗ max D → C ⊗ max D is isometric for all C ∗ -algebras D .These maps were called max-injective in [28]. We adopt here “maximally isometric” to empha-size the analogy with completely isometric. 10 emark . For example, for any C ∗ -algebra C the canonical inclusion i C : C → C ∗∗ is max-isometric (see Remark 0.15 or [28, Cor. 7.27]). Thus for any D we have an isometric ∗ -homomorphism D ⊗ max C → D ⊗ max C ∗∗ , which implies(1.4) ( D ⊗ C ) ∩ ( D ⊗ max C ∗∗ ) + ⊂ ( D ⊗ max C ) + . Moreover, the inclusion
I → C of an ideal, in particular the inclusion C → C of C into itsunitization, is max-isometric (see e.g. [28, Prop. 7.19]). Remark . We have k Id A k SB ( A,A ) = 1 if (and only if) A satisfies the property (0.1). In thatcase, by Remark 1.6 for any u ∈ CP ( A, C ) (where C is an arbitrary C ∗ -algebra) we have(1.5) k u k SB ( A,C ) = k u k . Remark . We will use the elementary fact (see [28, Cor. 7.16] for a proof) that D ⊗ max E → D ⊗ max C is isometric for all D (i.e. it is max-isometric) if and only if this holds for D = C . Inparticular checking D separable is enough.As we will soon show (see Theorem 4.3), maximally bounded maps admit maximally boundedliftings when (0.1) holds. To tackle u.c.p. liftings we will need a bit more work. The followingperturbation lemma is the m.p. analogue of Th. 2.5 in [12]. Lemma 1.19.
Let E ⊂ A be an n -dimensional operator system, C a unital C ∗ -algebra. Let < ε < / n . For any self-adjoint unital map u : E → C with k u k mb ≤ ε , there is a unitalmaximally positive (u.m.p. in short) map v : E → C such that k u − v k mb ≤ nε .Here u self-adjoint means that u ( x ) = u ( x ∗ ) ∗ for any x ∈ E .Sketch. The proof is essentially the same as that of the corresponding statement for the c.b. normand c.p. maps appearing in [12, Th. 2.5] and reproduced in [28, Th. 2.28]. However we have touse the “m.p. order” instead of the c.p. one. The only notable difference is that one should useTheorem 1.9 instead of the injectivity of B ( H ). Using the latter we can find a self-adjoint extension e u : A → C ∗∗ with k e u k dec ≤ ε . We can write e u = v − v with v and v in M P ( A, C ∗∗ ) suchthat k v + v k ≤ ε . Since u (1) = 1 we have k v (1) k = k ( v + v )(1) k ≤ ε , and hence k v (1) k ≤ ε/
2. Arguing as in [12, Th. 2.5] or [28, Th. 2.28] there is f ∈ E ∗ with k f k ≤ nε suchthat the mapping w : x f ( x )1 − v ( x ) is in M P ( E, C ∗∗ ). We then set v = v | E + w = u + f ( · )1.Note that v ( E ) ⊂ C . A priori v ∈ M P ( E, C ∗∗ ), but we actually have v ∈ M P ( E, C ) by (1.4).Moreover k v − u k mb ≤ nε . In particular k v (1) − k ≤ nε <
1, which shows that v (1) isinvertible and close to 1 when ε is small. The rest of the proof is as in [12, Th. 2.5] or [28, Th.2.28].The following variant is immediate: Lemma 1.20.
Let E ⊂ A be an n -dimensional operator system, C a unital C ∗ -algebra. For anyself-adjoint map u : E → C such that k u (1) − k < ε and k u k mb ≤ ε , there is a unitalmap v ∈ M P ( E, C ) such that k u − v k mb ≤ f E ( ε ) , where f E is a function of ε ∈ (0 , such that lim ε → f E ( ε ) = 0 .Sketch. Let u ′ ( · ) = u (1) − / u ( · ) u (1) − / . Then k u − u ′ k ≤ f ′ ( ε ) with lim ε → f ′ ( ε ) = 0. We mayapply Lemma 1.19 to u ′ . 11 Arveson’s principle
To tackle global lifting problems a principle due to Arveson has proved very useful. It asserts roughlythat in the separable case pointwise limits of suitably liftable maps are liftables. Its general formcan be stated as follows. Let E be a separable operator space, C unital C ∗ -algebra, I ⊂ C an ideal.A bounded subset of F ⊂ B ( E, C ) will be called admissible if for any pair f, g in F and any σ ∈ C + with k σ k ≤ x σ / f ( x ) σ / + (1 − σ ) / g ( x )(1 − σ ) / belongs to F . This implies that F is convex.Let q : C → C/ I denote the quotient map and let q ( F ) = { qf | f ∈ F } . Then Arveson’s principle (see [5, p. 351]) can be stated like this:
Theorem 2.1 (Arveson’s principle) . Assume E separable and F admissible. For the topology ofpointwise convergence on E we have q ( F ) = q ( F ) . Actually we do not even need to assume F bounded if we restrict to the pointwise convergence ona countable subset of E . One can verify this assertion by examining the presentations [10, p. 266] or [27, p. 46 and p.425] or [28, Th. 9.46].The classical admissible classes are contractions, complete contractions and, when E is anoperator system, positive contractions or completely positive (c.p. in short) contractions. In theunital case, unital positive or unital completely positive (u.c.p. in short) maps form admissibleclasses. Let f, g ∈ B MB ( E,C ) . Then it is easy to see on one hand that the map x ( f ( x ) , g ( x ))is in B MB ( E,C ⊕ C ) . On the other hand the map ψ : C ⊕ C → C defined by v ( a, b ) = σ / aσ / +(1 − σ ) / b (1 − σ ) / is unital c.p. and hence in B MB ( C ⊕ C,C ) (see Remark 1.4). This shows (withRemark 1.3) that the unit ball of M B ( E, C ) is admissible.In the rest of this section, we record a few elementary facts.
Lemma 2.2.
Let A be a unital C ∗ -algebra. Let E ⊂ A be an operator system. Consider a unitalmap u : E → C/ I ( C/ I any quotient C ∗ -algebra). Let v ∈ M B ( E, C ) (resp. v ∈ SB ( E, C ) , rresp. v ∈ CB ( E, C ) ) be a lifting of u . Then for any ε > there is a unital lifting v ′ ∈ M B ( E, C ) (resp. v ′ ∈ SB ( E, C ) , rresp. v ′ ∈ CB ( E, C ) ) with k v ′ k mb ≤ (1 + ε ) k v k mb (resp. k v ′ k sb ≤ (1 + ε ) k v k sb ,rresp. k v ′ k cb ≤ (1 + ε ) k v k cb ).If v is merely assumed bounded, we also have k v ′ k ≤ (1 + ε ) k v k .Proof. Let ( σ α ) be a quasicentral approximate unit in I in the sense of [5] (see also e.g. [10] or [28,p. 454]). Let T α : C ⊕ C → C be the unital c.p. map defined by T α ( x, c ) = (cid:0) (1 − σ α ) / σ / α (cid:1) (cid:18) x c (cid:19) (1 − σ α ) / σ / α ! = (1 − σ α ) / x (1 − σ α ) / + cσ α . Since v (1) − ∈ I we have(2.1) k (1 − σ α ) / [ v (1) − − σ α ) / k → . f : E → C be a state, i.e. a positive linear form such that f (1) = 1. Note that k f k sb = k f k mb = k f k = 1. Also k v k ≥ k u k ≥
1. The map ψ : E → C ⊕ C defined by ψ ( x ) = ( v ( x ) , f ( x ))clearly satisfies k ψ k mb ≤ k v k mb . Let T ′ α ( x ) = T α ψ ( x ). Then k T ′ α k mb ≤ k v k mb (see Remarks 1.3 and1.4), T ′ α : E → C lifts u and by (2.1) we have k T ′ α (1) − k →
0, so that going far enough in the netwe can ensure that T ′ α (1) is invertible in C . We then define v ′ : E → C by v ′ ( x ) = T ′ α (1) − T ′ α ( x )(we could use x T ′ α (1) − / T ′ α ( x ) T ′ α (1) − / ). Now v ′ is unital and k v ′ k mb ≤ k T ′ α (1) − kk T ′ α k mb ≤k T ′ α (1) − kk v k mb . Choosing α “large” enough so that k T ′ α (1) − k < ε , we obtain the desiredbound for the mb-case. The other cases are identical.Let I be a directed set. Assume x i ∈ B ( H ) for all i ∈ I and x ∈ B ( H ). Recall that (bydefinition) x i tends to x for the strong* operator topology (in short x = sot* − lim x i ) if x i h → xh and x ∗ i h → x ∗ h for any h ∈ H .Let M ⊂ B ( H ) be a von Neumann subalgebra of B ( H ). Assume that there is a C ∗ -subalgebra C ⊂ M such that C ′′ = M . It is well known that the unit ball of M is the closure of that of C for the strong* topology (see [31, Th. 4.8. p. 82]). This implies the following well known andelementary fact representing M as a quotient of a natural C ∗ -subalgebra of ℓ ∞ ( { C } ). Lemma 2.3.
In the preceding situation, for a suitable directed index set I , with which we set C i = C for all i ∈ I , there is a C ∗ -subalgebra L ⊂ ℓ ∞ ( { C i | i ∈ I } ) and a surjective ∗ -homomorphism Q : L → M such that for any ( x i ) ∈ L we have Q (( x i )) = sot* − lim x i .If C is unital, we can get L and Q unital as well.Proof. Choose a directed set I so that for any x ∈ B M there is ( x i ) i ∈ I ∈ Q i ∈ I B C i (recall C i = C for all i ∈ I ) such that x = sot* − lim x i . For instance, the set of neighborhoods of 0 for thesot*-topology can play this role. Let L ⊂ ℓ ∞ ( { C i } ) denote the unital C ∗ -subalgebra formed ofthe elements ( x i ) i ∈ I in L such that sot* − lim x i exists. We then set Q ( x ) = sot* − lim x i for any x = ( x i ) ∈ L . The last assertion is obvious.Let X ⊂ A be a separable subspace of a C ∗ -algebra A . For any C ∗ -algebra C we give ourselvesan admissible class of mappings F ( X, C ) that we always assume closed for pointwise convergence.We will say that X ⊂ A has the F -lifting property if for any u ∈ F ( X, C/ I ) there is a lifting ˆ u in F ( X, C ). Remark . In the sequel, we will assume that F is admissible and in addition that for any u ∈ F ( X, C ) and any ∗ -homomorphism π : C → D between C ∗ -algebras π ◦ u ∈ F ( X, D ). When F is formed of unital maps ( X is then an operator system and C a unital C ∗ -algebra), we assumemoreover that π ◦ u ∈ F ( X, D ) for any u.c.p. map π : C → D . Remark . [Examples] Examples of such classes are those formed of maps that are contractive,positive and contractive, n -contractive, n -positive and n -contractive, completely contractive orc.c.p. When dealing with positive or c.p. maps, we assume that X is an operator system. Thenwe may intersect the latter classes with that of unital ones.To tackle liftings within biduals we will use the following classical fact (for which a proof canbe found e.g. in [28, p.465]): for any C ∗ -algebra C and any ideal I ⊂ C with which we can formthe quotient C ∗ -algebra C/ I , we have a canonical isomorphism:(2.2) C ∗∗ ≃ ( C/ I ) ∗∗ ⊕ I ∗∗ and hence an isomorphism ( C/ I ) ∗∗ ≃ C ∗∗ / I ∗∗ . Theorem 2.6.
Let X ⊂ A and F be as in Remark 2.4. Then the following are equivalent: (i) The space X has the F -LP. (ii) For any von Neumann algebra M , any C ⊂ M such that C ′′ = M and any u ∈ F ( X, M ) there is a net of maps u i ∈ F ( X, C ) tending pointwise sot* to u . (iii) For any C and any u ∈ F ( X, C ∗∗ ) there is a net of maps u i ∈ F ( X, C ) tending pointwise to u for the weak* (i.e. σ ( C ∗∗ , C ∗ ) ) topology.Proof. Assume (i). We apply Lemma 2.3. We have a C ∗ -subalgebra L ⊂ ℓ ∞ ( { C i } ) and a surjective ∗ -homomorphism Q : L → M such that for any ( x i ) ∈ L we have Q (( x i )) = sot* − lim x i (and afortiori Q (( x i )) = weak* − lim x i ). Let u ∈ F ( X, M ) and let ˆ u ∈ F ( X, L ) be a lifting for Q . Then( u i ) i ∈ I such that ˆ u = ( u i ) i ∈ I gives us (ii).(ii) ⇒ (iii) is obvious.Assume (iii). Let u ∈ F ( X, C/ I ). We will use (2.2). The proof can be read on the followingdiagram. C q (cid:15) (cid:15) / / C ∗∗ q ∗∗ (cid:15) (cid:15) X v ' ' v i = = ④④④④④ u / / C/ I / / [ C/ I ] ∗∗ ρ f f By (2.2) we have a lifting ρ : ( C/ I ) ∗∗ ≃ C ∗∗ / I ∗∗ → C ∗∗ which is a ∗ -homomorphism. Let f bea state on ( C/ I ) ∗∗ and let p denote the unit in I ∗∗ ⊂ C ∗∗ . Then ρ : C ∗∗ / I ∗∗ → C ∗∗ defined by ρ = ρ ( · ) + f ( · ) p is a u.c.p. lifting (replacing ρ by ρ is needed only when F is formed of unitalmaps). Therefore, recalling Remark 2.4, we can find a map v ∈ F ( X, C ∗∗ ) such that q ∗∗ v = i C/ I u (where as usual i D : D → D ∗∗ the canonical inclusion). By (iii) there is a net ( v i ) in F ( X, C )tending weak* to v . Then qv i = q ∗∗ v i tends pointwise weak* to q ∗∗ v = u . This means qv i → u pointwise for the weak topology of C/ I . We can then invoke Mazur’s classical theorem to obtain(after passing to suitable convex combinations) a net such that qv i → u pointwise in norm (see e.g.[28, Rem. A.10] for details). By Arveson’s principle, u admits a lifting in F , so we obtain (i).Let X ⊂ A be as above. In addition for any C and any f.d. subspace E ⊂ X we assume givenanother f.d. E ⊃ E and a class F ( E, C ) satisfying the same two assumptions as F ( X, C ). Wealso assume that u ∈ F ( X, C ) implies u | E ∈ F ( E, C ). We will say that u : X → C/ I is locally F -liftable if for any f.d. E ⊂ X there is a f.d. E ⊃ E and a map u E ∈ F ( E, C ) lifting therestriction u | E . We will say that X has the F -LLP if for any C , any u ∈ F ( X, C/ I ) is locally F -liftable. The introduction of E in place of E is a convenient way to include e.g. the class F formed of unital c.p. maps on f.d. operator systems. In the latter case E can be any f.d. operatorsystem containing E . Theorem 2.7.
With the above assumptions and those of Theorem 2.6, the following are equivalent: (i)
The space X has the F -LLP. (ii) For any C , any u ∈ F ( X, C ∗∗ ) and any f.d. E ⊂ X , there is a f.d. E ⊃ E and a net ofmaps u Ei ∈ F ( E, C ) tending pointwise weak* to u | E . roof. The proof of Theorem 2.6 can be easily adapted to prove this.Since the work of T.B. Andersen and Ando [1, 2] it has been known that if a separable Banachspace X has the metric approximation property then any contractive u : X → C/ I admits acontractive lifting. Although we will not use it, we conclude our general discussion by reformulatingthis result in our framework. Let us say that u : X → C/ I is F -approximable if there is a netof finite rank maps u α ∈ F ( X, C/ I ) tending pointwise to u . For the examples of F considered inRemark 2.5 it is obviously equivalent to say that the identity map of C/ I is locally F -liftable orto say that for any X any finite rank map in F ( X, C/ I ) is locally F -liftable.With this terminology, Ando’s well known result [2] can nowadays be reformulated like this: Proposition 2.8.
Assume that any finite rank map in F ( X, C/ I ) is liftable in F ( X, C ) . If u : X → C/ I is F -approximable, then u admits a (global) lifting in F ( X, C ) .Proof. This follows from Arveson’s principle.Let us denote by F ccp (resp. F ucp ) the class of c.c.p. (resp. u.c.p.) maps. Then the LP is justthe F ccp -LP. In the unital case, we show below that the F ccp -LP and the F ucp -LP are equivalent.Of course in the latter case we restrict to lifting quotients of unital C ∗ -algebras.The following statement about the unitization process is a well known consequence of the worksof Choi-Effros and Kirchberg [9, 19]. Proposition 2.9.
Let A be a separable C ∗ -algebra and let A be its unitization. The followingproperties of A are equivalent: (i) The lifting property LP, meaning the F ccp -LP. (ii) The unitization A has the F ucp -LP.Moreover, when A is unital (i) and (ii) are equivalent to (iii) The C ∗ -algebra A has the F ucp -LP.Proof. Assume (i). Let q : C → C/ I be the quotient map. Assuming C unital, let u : A → C/ I be a u.c.p. map. By (i) there is w ∈ CP ( A, C ) with k w k ≤ qw = u | A : A → C/ I . Let w : A → C be the unital extension of w . By [9, Lemma 3.9], w is c.p. Since C is unital there isa unital ∗ -homomorphism π : C → C extending Id C . Let ˆ u = πw : A → C . Then ˆ u is a u.c.p.map such that q ˆ u | A = u | A and q ˆ u (1) = 1. It follows that q ˆ u = u , whence (ii).Assume (ii). Let u : A → C/ I be a c.c.p. map. Let u : A → ( C/ I ) ≃ C / I be theunital map extending u . By [9, Lemma 3.9] again, u is c.p. By (ii) there is a unital c.p. map c u : A → C lifting u . Let q : C → C/ I and Q : C → C / I ≃ ( C/ I ) be the quotient maps. Wehave Q c u = u and u | A = u . A moment of thought shows that Q is the unital extension of q , sothat Q − ( C/ I ) = C . Thus Q c u ( a ) = u ( a ) = u ( a ) ∈ C/ I for any a ∈ A implies that c u ( a ) ∈ C forany a ∈ A . We conclude that c u | A : A → C is a c.p. lifting of u with norm ≤
1, whence (i).Assume (i) with A unital. Let u : A → C/ I be a u.c.p. map. Let v : A → C be a c.c.p. liftingof u . To show (iii) let f be a state on A and let ˆ u = v + (1 − v (1)) f . Then ˆ u is a u.c.p. lifting.This shows (i) ⇒ (iii).Conversely, assume (iii). To show (i) let u : A → C/ I be a c.c.p. map. Let u ′ : A → C / I be the map u composed with the inclusion of C into its unitization C / I . Note that since u ′ takes its values in C/ I , any lifting ˆ u of u ′ must take its values in C , and hence be a lifting of u .Therefore, it suffices to show that u ′ admits a c.c.p. lifting. Assume first that u ′ (1) is invertible in C . We will argue as for Lemma 2.2. Define w ∈ CP ( A, C / I ) by w ( x ) = u ′ (1) − / u ′ ( x ) u ′ (1) − / .Then w is unital. Let ˆ w : A → C be a u.c.p. lifting of w . Let z ∈ C be a lifting of u ′ (1)15ith k z k = k u ′ (1) k ≤
1. Then the mapping ˆ u : A → C defined by ˆ u ( · ) = z / ˆ w ( · ) z / is a c.c.p.lifting of u ′ . Let f be a state on A . To complete the proof note that for any ε > u ε ∈ CP ( A, C / I ) defined by u ε ( · ) = u ′ ( · ) + εf ( · ) is a c.p. perturbation of u ′ with u ε (1) invertible.By what precedes the maps (1 + ε ) − u ε admit c.c.p. liftings. By Arveson’s principle u ′ also does.This shows (iii) ⇒ (i). Remark . If A is unital (i)-(iii) in Proposition 2.9 are also equivalent to:(iv) Any unital ∗ -homomorphism u : A → C / I into a quotient of C admits a u.c.p. lifting.Indeed, (iii) ⇒ (iv) is trivial, and to show (iv) ⇒ (iii) one can realize A as a quotient of C . Then(iv) implies that the identity of A factors via u.c.p. maps through C , so that (iii) follows fromKirchberg’s theorem that A = C satisfies (iii) or (ii).However we deliberately avoid using the equivalence with (iv) to justify our claim that Theorem0.2 yields a new proof of the latter theorem of Kirchberg. Remark . In [18], Kirchberg defines the LP for A by the property (ii) in Proposition 2.9. Weprefer to use the equivalent definition in (i) that avoids the unitization.See [15] for a discussion of lifting properties in the more general context of M -ideals. We start by a new version of the “local reflexivity principle” (see particularly (3.3)). This is theanalogue of [12, Lemma 5.2] for the maximal tensor product.
Theorem 3.1.
Assume that A satisfies (0.1) . Let E ⊂ A be any f.d. subspace. Then for any C ∗ -algebra C we have a contractive inclusion (3.1) M B ( E, C ∗∗ ) → M B ( E, C ) ∗∗ . In other words any u in the unit ball of M B ( E, C ∗∗ ) is the weak* limit of a net ( u i ) in the unitball of M B ( E, C ) .Proof. This will follow from the bipolar theorem. We first need to identify the dual of
M B ( E, C ).As a vector space
M B ( E, C ) ≃ C ⊗ E ∗ and hence M B ( E, C ) ∗ ≃ C ∗ ⊗ E (or say ( C ∗ ) dim( E ) ). Weequip C ∗ ⊗ E with the norm α defined as follows. Let K ⊂ M B ( E, C ) ∗ denote the set of those f ∈ M B ( E, C ) ∗ for which there is a C ∗ -algebra D , a functional w in the unit ball of ( D ⊗ max C ) ∗ and t ∈ B D ⊗ max E , so that ∀ u ∈ M B ( E, C ) f ( u ) = h w, [ Id D ⊗ u ]( t ) i . We could rephrase this in tensor product language: note that we have a natural bilinear map( D ⊗ max C ) ∗ × ( D ⊗ max E ) → C ∗ ⊗ E associated to the duality D ∗ × D → C , then K can be identified with the union (over all D ’s) ofthe images of the product of the two unit balls under this bilinear map.We will show that K is the unit ball of M B ( E, C ) ∗ . Let D , D be C ∗ -algebras. Let D = D ⊕ D with the usual C ∗ -norm. Using the easily checked identities (here the direct sum is in the ℓ ∞ -sense) D ⊗ max E = ( D ⊗ max E ) ⊕ ( D ⊗ max E ) , and D ⊗ max C = ( D ⊗ max C ) ⊕ ( D ⊗ max C ) , D ⊗ max C ) ∗ = ( D ⊗ max C ) ∗ ⊕ ( D ⊗ max C ) ∗ (direct sum in the ℓ -sense), it is easy tocheck that K is convex and hence that K is the unit ball of some norm α on M B ( E, C ) ∗ .Our main point is the claim that K is weak* closed. To prove this, let ( f i ) be a net in K convergingweak* to some f ∈ M B ( E, C ) ∗ . Let D i be C ∗ -algebras, w i ∈ B ( D i ⊗ max C ) ∗ and t i ∈ B D i ⊗ max E suchthat we have ∀ u ∈ M B ( E, C ) f i ( u ) = h w i , [ Id D i ⊗ u ]( t i ) i . Let D = ℓ ∞ ( { D i } ) and let t ∈ D ⊗ E be associated to ( t i ). By (0.2) we know that k t k max ≤
1. Let p i : D → D i denote the canonical coordinate projection, and let v i ∈ ( D ⊗ max C ) ∗ be the functionaldefined by v i ( x ) = w i ([ p i ⊗ Id C ]( x )). Clearly v i ∈ B ( D ⊗ max C ) ∗ and f i ( u ) = h v i , [ Id D ⊗ u ]( t ) i . Let w be the weak* limit of ( v i ). By weak* compactness, w ∈ B ( D ⊗ max C ) ∗ . Then f ( u ) = lim f i ( u ) = h w, [ Id D ⊗ u ]( t ) i . Thus we conclude f ∈ K , which proves our claim.By the very definition of k u k MB ( E,C ) we have k u k mb = sup {| f ( u ) | | f ∈ K } . This implies that the unit ball of the dual of
M B ( E, C ) is the bipolar of K , which is equal toits weak* closure. By what precedes, the latter coincides with K . Thus the gauge of K is theannounced dual norm α = k k MB ∗ .Let u ′′ ∈ M B ( E, C ∗∗ ) with k u ′′ k mb ≤
1. By the bipolar theorem, to complete the proof itsuffices to show that u ′′ belongs to the bipolar of K , or equivalently that | f ( u ′′ ) | ≤ f ∈ K . To show this consider f ∈ K taking u ∈ M B ( E, C ) to f ( u ) = h w, [ Id D ⊗ u ]( t ) i with w ∈ B ( D ⊗ max C ) ∗ and t ∈ B D ⊗ max E . Observe that [ Id D ⊗ u ′′ ]( t ) ∈ D ⊗ C ∗∗ ⊂ ( D ⊗ max C ) ∗∗ .Recall that M B ( E, C ∗∗ ) ≃ M B ( E, C ) ∗∗ ≃ ( C ∗∗ ) dim( E ) as vector spaces. Thus we may view f ∈ M B ( E, C ) ∗ as a weak* continuous functional on M B ( E, C ∗∗ ) to define f ( u ′′ ). We claim that(3.2) f ( u ′′ ) = h w, [ Id D ⊗ u ′′ ]( t ) i , where the duality is relative to the pair h ( D ⊗ max C ) ∗ , ( D ⊗ max C ) ∗∗ i . From this claim the conclusionis immediate. Indeed, we have k [ Id D ⊗ u ′′ ]( t ) k D ⊗ max C ∗∗ ≤ k u ′′ k mb ≤
1, and by the maximality ofthe max-norm on D ⊗ C ∗∗ we have a fortiori k [ Id D ⊗ u ′′ ]( t ) k ( D ⊗ max C ) ∗∗ ≤ k [ Id D ⊗ u ′′ ]( t ) k D ⊗ max C ∗∗ ≤ . Therefore | f ( u ′′ ) | = |h w, [ Id D ⊗ u ′′ ]( t ) i| ≤ k w k ( D ⊗ max C ) ∗ ≤
1, which completes the proof moduloour claim (3.2).To prove the claim, note that the identity (3.2) holds for any u ∈ M B ( E, C ). Thus it suffices toprove that the right-hand side of (3.2) is a weak* continuous function of u ′′ (which is obvious forthe left-hand side). To check this one way is to note that t ∈ D ⊗ E can be written as a finite sum t = P d k ⊗ e k ( d k ∈ D, e k ∈ E ) and if we denote by ˙ w : D → C ∗ the linear map associated to w wehave h w, [ Id D ⊗ u ′′ ]( t ) i = X k h w, [ d k ⊗ u ′′ ( e k )] i = X k h ˙ w ( d k ) , u ′′ ( e k ) i , and since ˙ w ( d k ) ∈ C ∗ the weak* continuity as a function of u ′′ is obvious, completing the proof. Remark . The preceding proof actually shows that, without any assumption on A or C , we havea contractive inclusion(3.3) SB ( E, C ∗∗ ) → SB ( E, C ) ∗∗ Of course if A satisfies (0.1) then SB ( E, C ) =
M B ( E, C ) isometrically and we recover Theorem3.1. 17 emark . The converse inclusion to (3.1) holds in general : we claim that we have a contractiveinclusion(3.4)
M B ( E, C ) ∗∗ → M B ( E, C ∗∗ ) . Indeed, let u i : E → C be a net with k u i k mb ≤ u : E → C ∗∗ . Let t ∈ C ⊗ E with k t k max ≤
1, say with t ∈ F ⊗ E with F ⊂ C f.d. Then [ Id C ⊗ u ]( t ) ∈ B [ F ⊗ max C ] ∗∗ . Note thatsince F is f.d. [ F ⊗ max C ] ∗∗ ⊂ [ C ⊗ max C ] ∗∗ ∩ [ F ⊗ C ∗∗ ]. We have clearly an isometric inclusion[ F ⊗ max C ] ∗∗ ⊂ [ C ⊗ max C ] ∗∗ . Moreover (see [28, Th. 8.22]) we have an isometric embedding C ∗∗ ⊗ bin C ∗∗ ⊂ [ C ⊗ max C ] ∗∗ . Since F ⊂ C , the norm induced by C ∗∗ ⊗ bin C ∗∗ on F ⊗ C ∗∗ coincideswith that induced by C ⊗ nor C ∗∗ . By Kirchberg’s theorem to the effect that (0.12) holds when A = C (see [28, Th. 9.10]) C ⊗ nor C ∗∗ = C ⊗ max C ∗∗ . Thus we find k u k MB ( E,C ∗∗ ) ≤
1. This provesthe claim.
Remark . We will use an elementary perturbation argument as follows. Let
A, C be C ∗ -algebrasand let E ⊂ A be a f.d. subspace. Let v : E → C . For any δ > ε > E ) satisfying the following: for any map v ′ : A → C be such that k v ′| E − v k ≤ ε , there is a map v ′′ : A → C such that v ′′| E = v and k v ′′ − v ′ k mb ≤ δ, and hence k v ′′ k mb ≤ k v ′ k mb + δ.A v ′ ! ! ❈❈❈❈ E ?(cid:31) O O v ′| E ≈ v / / C A v ′′ ≈ v ′ ! ! ❈❈❈❈ E ?(cid:31) O O v / / C Let ∆ = v − v ′| E . Let k ∆ k N denote the nuclear norm (in the Banach space sense) of ∆ : E → C .By definition (here E and C are Banach spaces with E f.d.), this is the infimum of P d k f j k E ∗ k c j k C over all the possible representations of ∆ as ∆( x ) = P d f j ( x ) c j ( x ∈ E ). Let k E = k Id E k N . Itis immediate that for any ∆ : E → C we have k ∆ k N ≤ k E k ∆ k . By Hahn-Banach, ∆ admits anextension e ∆ : A → E with k e ∆ k N ≤ k ∆ k N ≤ k E k ∆ k ≤ εk E . Let v ′′ = v ′ + e ∆. Then v ′′| E = v and k v ′′ − v ′ k mb ≤ k v ′′ − v ′ k N = k e ∆ k N ≤ εk E . Whence the announced result with δ = εk E . Theorem 3.5.
Assume that A satisfies (0.1) (or merely the conclusion of Theorem 3.1). Let C bea C ∗ -algebra. Let E ⊂ X be a f.d. subspace of a separable subspace X ⊂ A . Then for any ε > ,any map v : E → C admits an extension e v : X → C such that k e v k mb ≤ (1 + ε ) k v k mb . X e v " " ❊❊❊❊ E ?(cid:31) O O v / / C Proof.
Let E ⊃ E be any finite dimensional superspace with E ⊂ X . Our first goal will be to showthat for any ε > w : E → C such that w | E = v with k w k mb ≤ k v k mb + ε .By Theorem 1.9 and by (1.2), there is a decomposable map e v : A → C ∗∗ such that k e v k D ( A,C ∗∗ ) = k e v k MB ( A,C ∗∗ ) ≤ k v k mb v in the sense that e v | E = i C v where i C : C → C ∗∗ is the canonical inclusion.Let v : E → C ∗∗ be the restriction of e v so that v = e v | E . A fortiori k v : E → C ∗∗ k mb ≤ k v k mb .By Theorem 3.1 there is a net of maps v i : E → C with k v i k mb ≤ k v k mb that tend weak* to v .It follows that the restrictions v i | E tend weak* to v | E = e v | E = i C v . This means that v i | E ( x ) tendsweakly in C (i.e. for σ ( C, C ∗ )) to v ( x ) for any x ∈ E . By a well known application of Mazur’stheorem (see e.g. [28, Rem. A.10] for details), passing to suitable convex combinations of the v i ’s,we can get a similar net such that in addition v i | E tends pointwise in norm to v . Since E is f.d. thisimplies k v i | E − v k →
0. By perturbation (see Remark 3.4), for any ε > w : E → C such that w | E = v with k w k mb ≤ k v k mb + ε , so we reach our first goal.Lastly we use the separability of X to form an increasing sequence ( E n ) of f.d. subspaces of X with dense union such that E = E . Let w = v : E → C . By induction, we can find a sequenceof maps w n : E n → C such that w n +1 | E n = w n and k w n +1 k mb ≤ k w n k mb + 2 − n − δ . By densitythis extends to a linear operator w : X → C such that k w k mb ≤ k w k mb + δ = k v k mb + δ . Thiscompletes the proof. Remark . Actually, it suffices by Remark 3.2 to assume that X ⊂ A satisfies (0.1) (in place of A ) for the preceding proof to be valid. We rephrase this in the next statement. Theorem 3.7.
Let E ⊂ X be a f.d. subspace of a separable subspace X ⊂ A of a C ∗ -algebra. Forany ε > , any map v : E → C into a C ∗ -algebra C admits an extension e v : X → C such that k e v k sb ≤ (1 + ε ) k v k sb .Remark . Let
A, C be unital C ∗ -algebras. Let E ⊂ A be an operator system and let u : E → C be a unital map. Recall (see Remark 1.6) that u is m.p. if and only if k u k mb = 1.We will denote by F ump ( E, C ) the class of u.m.p. maps u : E → C . This is clearly an admissibleclass and it is pointwise closed.We will need the following consequence of Theorem 3.1 for u.m.p. maps: Corollary 3.9.
Let
A, C be unital C ∗ -algebras. Assume that A satisfies the conclusion of Theorem3.1. Let E ⊂ A be a f.d. operator system. Let u : E → C ∗∗ be a u.m.p. map. There is a net ofu.m.p. maps u i : E → C tending pointwise weak* to u .Proof. By Theorem 3.1 there is a net of maps v i : E → C with k v i k mb ≤ u . Replacing v i by ( v i + ( v i ) ∗ ) /
2, we may assume each v i self-adjoint (see Remark 1.7).Moreover, since u (1) = 1, we may observe that v i (1) − → σ ( C, C ∗ ))of C . Passing to convex combinations, we may assume (by Mazur’s theorem) that k v i (1) − k → u i ∈ M P ( E, C ) such that k u i − v i k →
0. Since v i → u pointwise weak*, we also have u i → u pointwise weak*. Theorem 3.10.
Let
A, C be unital C ∗ -algebras. Assume that A satisfies the conclusion of Theorem3.1. Let E ⊂ X be a f.d. operator subsystem of a separable operator system X ⊂ A . Let u : E → C be a u.m.p. map. Then for any ε > there is a u.m.p. map v : X → C such that k v | E − u k < ε .Proof. We may assume that we have an increasing sequence of f.d. operator systems ( E n ) with E = E such that ∪ E n = X . We give ourselves ε n > P n ≥ ε n < ε . The plan is toconstruct a sequence of u.m.p. maps v n : E n → C such that v = u and k v n +1 | E n − v n k < ε n +1 forall n ≥
0. Then the map v defined by v ( x ) = lim v n ( x ) for x ∈ ∪ E n extends by density to a u.m.p.map v : X → C such that k v | E − u k ≤ P n ≥ ε n < ε .The sequence ( v n ) will be constructed by induction starting from v = u . Assume that wehave constructed v , · · · , v n ( n ≥
0) satisfying the announced properties and let us produce v n +1 .19ince k v n k mb = 1 by Theorem 1.9 and (1.2) there is a (still unital) map e v n : A → C ∗∗ such that f v n | E n = i C v n with k f v n k mb = 1. By Remark 1.6, the map f v n is u.m.p. (see also Remark 1.13). Afortiori f v n | E n +1 : E n +1 → C ∗∗ is u.m.p. By Corollary 3.9 there is a net ( u i ) of u.m.p. maps from E n +1 to C such that u i − f v n | E n +1 tends weak* to 0. Since E n ⊂ E n +1 , the restrictions ( u i − f v n ) | E n also tend weak* to 0. Thus u i | E n − v n tends weak* to 0 when i → ∞ . Since u i | E n − v n takes valuesin C , this tends to 0 weakly in C . Passing to convex combinations of the maps u i we may assumethat lim i k u i | E n − v n k = 0. Choosing i large enough, we have k u i | E n − v n k < ε n +1 , so we may set v n +1 = u i . This completes the induction step. We start by what is now a special case of Theorem 2.6.Let C/ I be a quotient C ∗ -algebra with quotient map q : C → C/ I Lemma 4.1.
Let E ⊂ A a f.d. subspace. Then any u : E → C/ I admits a lifting ˆ u : E → C suchthat k ˆ u k sb = k u k sb . Proof.
By (iii) ⇒ (i) in Theorem 2.6 it suffices to show that the classes F ( E, C ) = { u : E → C | k u k sb ≤ } satisfy the assertion (iii) in Theorem 2.6 with X = E . This is precisely what (3.3) says. Remark . In the situation of Lemma 4.1, consider u : E → C/ I . We claim that if the map M B ( E, C ∗∗ ) → M B ( E, C ) ∗∗ is contractive then u admits a lifting ˆ u : E → C such that k ˆ u k mb = k u k mb . Indeed, just as in the preceding proof, we may apply (iii) ⇒ (i) in Theorem 2.6 to the class F ( E, C ) = { u : E → C | k u k mb ≤ } . Theorem 4.3.
Let X ⊂ A be a separable subspace of a C ∗ -algebra A . Then any u ∈ SB ( X, C/ I ) admits a lifting ˆ u : X → C with k ˆ u k SB ( X,C ) = k u k SB ( X,C/ I ) . If A satisfies the property (0.1) , then any u ∈ M B ( X, C/ I ) admits a lifting ˆ u : X → C with k ˆ u k MB ( X,C ) = k u k MB ( X,C/ I ) . Proof.
Let u ∈ SB ( X, C/ I ). Let E ⊂ X be a f.d. subspace. By Lemma 4.1, there is u E : E → C lifting u | E with k u E k SB ( E,C ) ≤ k u | E k SB ( E,C/ I ) ≤ k u k SB ( X,C/ I ) . Let ε >
0. By Theorem 3.5,the map u E admits an extension f u E : X → C such that k f u E k SB ( X,C ) ≤ (1 + ε ) k u E k SB ( E,C ) ≤ (1+ ε ) k u k SB ( X,C/ I ) . Since for any x ∈ X we have x ∈ F for some f.d. F ⊂ X , we have q f u E ( x ) = u ( x )whenever E ⊃ F . Therefore, the map u : X → C/ I is in the pointwise closure of the liftable maps { (1 + ε ) − q f u E | E ⊂ X } . By normalization, the liftings { (1 + ε ) − f u E | E ⊂ X } are in the unit ballof SB ( X, C ) and the latter is an admissible class. Thus the conclusion of the first part follows fromArveson’s principle. The second part is obvious since (0.1) implies that
M B ( E, C ) = SB ( E, C )isometrically for any C .Let A, C be unital C ∗ -algebras. Let E ⊂ A be an operator system. Recall that F ump ( E, C )denotes the class of unital maps in
M P ( E, C ) (see Remarks 1.6 or 3.8).20 heorem 4.4.
Assume that A satisfies the conclusion of Theorem 3.1. Let X ⊂ A be a separableoperator system. Any u ∈ F ump ( X, C ∗∗ ) is the pointwise weak* limit of a net in F ump ( X, C ) .Proof. Let E ⊂ X be a f.d. operator system. By Corollary 3.9 there is a net u i ∈ F ump ( E, C )such that u i → u | E pointwise weak*. By Theorem 3.10 there are maps v i ∈ F ump ( X, C ) such that k v i | E − u i k →
0, and hence ( v i − u ) | E → u is in the pointwiseweak* closure of F ump ( X, C ).By Theorem 2.6 we may state:
Corollary 4.5. If A satisfies (0.1) any separable operator system X ⊂ A has the F ump -LP. Corollary 4.6.
Assume
A, C unital and A separable. If A satisfies (0.1) or merely the conclusionof Theorem 3.1, any unital u ∈ CP ( A, C/ I ) admits a unital lifting ˆ u ∈ CP ( A, C ) .Proof. By Remarks 1.4 and 1.6, a unital map u : A → C is c.p. if and only if u ∈ F ump ( A, C ). We first note that C = C ∗ ( F ∞ ) satisfies the property (0.1). Lemma 5.1.
For any free group F (in particular for F = F ∞ ) the C ∗ -algebra C ∗ ( F ) satisfies (0.1) .Proof. We outline the argument already included in [28, Th. 9.11]. By the main idea in [26] itsuffices to check (0.2) for any t ∈ ℓ ∞ ( { D i } ) ⊗ E when E is the span of the unit and a finite numberof the free unitary generators, say U , · · · , U n of C ∗ ( F ) (and actually n = 2 suffices). Let U = 1for convenience. It is known (see [27, p. 257-258] or [28, p. 130-131]) that for any C ∗ -algebra D and any ( x j ) ≤ j ≤ n ∈ D n +1 , with respect to the inclusion E ⊂ C , we have k X x j ⊗ U j k D ⊗ max E = inf {k X a ∗ j a j k / k X b ∗ j b j k / } where the infimum runs over all possible decompositions x j = a ∗ j b j in D . Using this characterization,it is immediate that k t k ℓ ∞ ( { D i } ) ⊗ max E ≤ sup i ∈ I k t i k D i ⊗ max E . Proof of Theorem 0.2.
Assume (i). We may assume that A is unital so that A = C / I for some I and there is a unital c.p. lifting r : A → C . Then using k ( Id D i ⊗ r )( t i ) k max ≤ k t i k max for any t i ∈ D i ⊗ E (see Remark 1.4), we easily pass from “ C satisfies (ii)” (which is Lemma 5.1)to “ A satisfies (ii)”. This proves (i) ⇒ (ii).Conversely, assume (ii). By Corollary 4.6, A has the F ucp -LP, and hence (i) holds by Proposition2.9.To prove (ii) ⇒ (ii) + , we use another fact: for any x = x ∗ in C where C is any unital C ∗ -algebra,we have x ∈ C + ⇔ k − x k ≤ . Using this one easily checks (ii) ⇒ (ii) + . Now assume (ii) + . Using k x k = k x ∗ x k / one can reducechecking (ii) when all t i ’s are self-adjoint. Then using the same fact we obtain (ii) + ⇒ (ii).Note: an alternate route is to use k x k ≤ ⇔ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) xx ∗ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) M ( C ) ≤ . Using this with M ( D i ) in place of D i , one easily checks (ii) + ⇒ (ii).21ctually, the preceding proof can be adapted to use only the conclusion of Theorem 3.1 toobtain the LP, therefore: Theorem 5.2.
A separable C ∗ -algebra A has the LP (or equivalently the property in (0.1) ) if andonly if the natural map M B ( E, C ∗∗ ) → M B ( E, C ) ∗∗ is contractive for all f.d. subspaces E ⊂ A and all C ∗ -algebras C . In fact, it suffices to have this in the case C = C .Proof. Assume A unital for simplicity. By Theorem 3.1 it suffices to show the “if” part. The latteris contained in Corollary 4.6. To justify the last assertion, we reduce the LP to the case when C isseparable, and view C as a quotient of C . Let U be an ultrafilter on a set I . Recall that the ultraproduct ( D i ) U of a family ( D i ) i ∈ I of C ∗ -algebras is defined as the quotient ℓ ∞ ( { D i } ) /c U ( { D i } )where c U ( { D i } ) = { x ∈ ℓ ∞ ( { D i } ) | lim U k x i k D i = 0 } .It is usually denoted as Q i ∈ I D i / U . For short we will also denote it just by ( D i ) U .Recall that, for each i ∈ I , we have trivially a contractive morphism ℓ ∞ ( { D i } ) → D i and hencealso ℓ ∞ ( { D i } ) ⊗ max A → D i ⊗ max A , whence a contractive morphism(6.1) Ψ : ℓ ∞ ( { D i } ) ⊗ max A → ℓ ∞ ( { D i ⊗ max A } ) , which, after passing to the quotient gives us a contractive morphism ℓ ∞ ( { D i } ) ⊗ max A → ( D i ⊗ max A ) U . The latter morphism obviously vanishes on c U ( { D i } ) ⊗ A , and hence also on its closure in ℓ ∞ ( { D i } ) ⊗ max A , i.e. on c U ( { D i } ) ⊗ max A . Thus we obtain a natural contractive morphism( ℓ ∞ ( { D i } ) ⊗ max A ) / ( c U ( { D i } ) ⊗ max A ) → ( D i ⊗ max A ) U . By (0.7) we have for any C ∗ -algebra A ( D i ) U ⊗ max A = ( ℓ ∞ ( { D i } ) ⊗ max A ) / ( c U ( { D i } ) ⊗ max A ) , whence a natural contractive morphism(6.2) Φ : ( D i ) U ⊗ max A → ( D i ⊗ max A ) U . In this section we will show that (6.2) is isometric for any ( D i ) U if and only if A has the LP. Theorem 6.1.
Let A be a separable C ∗ -algebra. The following are equivalent: (i) The algebra A satisfies (0.1) (i.e. A has the LP). (ii) For any family ( D i ) i ∈ I of C ∗ -algebras and any ultrafilter on I we have ∀ t ∈ hY i ∈ I D i / U i ⊗ A k t k max = lim U k t i k D i ⊗ max A . In other words we have a natural isometric embedding [ Y i ∈ I D i / U ] ⊗ max A ⊂ Y i ∈ I [ D i ⊗ max A ] / U . roof. Assume (i). If A satisfies (0.1), Ψ in (6.1) is isometric. We claim that (6.2) must alsobe isometric, or equivalently injective. Indeed, let z ∈ ( D i ) U ⊗ max A such that Φ( z ) = 0. Let z ′ ∈ ℓ ∞ ( { D i } ) ⊗ max A lifting z . Then Ψ( z ′ ) ∈ c U ( { D i ⊗ max A } ). This means that for any ε > α ∈ U such that sup i ∈ α k Ψ( z ′ ) i k D i ⊗ max A < ε . Let P α : ℓ ∞ ( { D i } ) → ℓ ∞ ( { D i } ) be theprojection onto α defined for any x ∈ ℓ ∞ ( { D i } by P α ( x ) i = x i if i ∈ α and P α ( x ) i = 0 otherwise.Since (6.1) is isometric we have k [ P α ⊗ Id A ]( z ′ ) k = k Ψ([ P α ⊗ Id A ]( z ′ )) k = sup i ∈ α k Ψ( z ′ ) i k D i ⊗ max A ≤ ε. Denoting by q : ℓ ∞ ( { D i } ) ⊗ max A → ( D i ) U ⊗ max A the quotient map, since q ( z ′ ) = z , this implies k z k = k q ( z ′ ) k = k q ([ P α ⊗ Id A ]( z ′ )) k ≤ k [ P α ⊗ Id A ]( z ′ ) k ≤ ε. Since ε > z = 0, proving our claim. This proves (i) ⇒ (ii).Assume (ii). Consider the set ˆ I formed of all the finite subsets of I , viewed as directed withrespect to the inclusion order. Let U be a non trivial ultrafilter on ˆ I refining this net. Let d = ( d i ) i ∈ I ∈ ℓ ∞ ( { D i | i ∈ I } ). For any J ∈ ˆ I we set D J = ℓ ∞ ( { D i | i ∈ J } ). Using naturalcoordinate projections we have a natural morphism d = ( d i ) i ∈ I ( d J ) J ∈ ˆ I ∈ ℓ ∞ ( { D J | J ∈ ˆ I } ).Clearly k ( d J ) U k = lim U k d J k = k d k ℓ ∞ ( { D i | i ∈ I } ) . Let ψ : ℓ ∞ ( { D i | i ∈ I } ) → ( D J ) U be the corresponding isometric ∗ -homomorphism. We claimthat we have a c.p. contraction ψ : ( D J ) U → ℓ ∞ ( { D ∗∗ i | i ∈ I } ) such that the composition ψ ψ coincides with the inclusion ℓ ∞ ( { D i | i ∈ I } ) ⊂ ℓ ∞ ( { D ∗∗ i | i ∈ I } ). Indeed, consider d = ( d J ) U ∈ ( D J ) U . For any J ∈ ˆ I we may write d J = ( d J ( i )) i ∈ J ∈ ℓ ∞ ( { D i | i ∈ J } ). Fix i ∈ I .We will define ψ ( d ) by its i -th coordinate ψ i ( d ) taking values in D ∗∗ i . Note that i ∈ J for all J far enough in the net. We then set ψ i ( d ) = weak* lim U d J ( i ) . It is then easy to check our claim.By (0.11) the composition ψ ψ is max-isometric, and k ψ k mb ≤ ψ ismax-isometric. In particular (taking C = A ) ψ ⊗ Id A defines an isometric embedding ℓ ∞ ( { D i | i ∈ I } ) ⊗ max A ⊂ ( D J ) U ⊗ max A. By (ii) we have an isometric embedding ( D J ) U ⊗ max A ⊂ Q J ∈ ˆ I [ D J ⊗ max A ] / U . It follows that forany t = ( t i ) ∈ ℓ ∞ ( { D i | i ∈ I } ) ⊗ max A we may write k t k ℓ ∞ ( { D i | i ∈ I } ) ⊗ max A = lim U k t J k D J ⊗ max A , and since k t J k D J ⊗ max A = sup i ∈ J k t i k D i ⊗ max A for any finite J , the inequality in (0.1) follows. Thisproves (ii) ⇒ (i). Let us say that a (not necessarily separable) C ∗ -algebra has the LP if for any separable subspace E there is a separable C ∗ -subalgebra A s with the LP that contains E .For example all nuclear C ∗ -algebras and also C ∗ ( F ) for any free group F have the LP.To tackle the non-separable case the following statement will be useful.23 roposition 7.1. Let A be a C ∗ -algebra. Let E ⊂ A be a separable subspace. There is a separable C ∗ -subalgebra A s satisfying E ⊂ A s ⊂ A such that, for any C ∗ -algebra D , the ∗ -homomorphism D ⊗ max A s → D ⊗ max A is isometric. In other words, the inclusion A s → A is max -isometric.Proof. By Remarks 1.18 it suffices to prove this for D = C . The latter case is proved in detail in[28, Prop. 7.24] (except that the factors are flipped). Remark . Let A s ⊂ A be a C ∗ -subalgebra. Assume that the inclusion i : A s → A is max-isometric. We claim that for any von Neumann algebra M the map Id M ⊗ i : M ⊗ A s → M ⊗ A extends to an isometric morphism M ⊗ nor A s → M ⊗ nor A .Indeed, our assumption implies that there is a c.c.p. map (a so called “weak expectation”) T : A → A ∗∗ s such that T | A s : A s → A ∗∗ s coincides with the canonical inclusion. See [8, p. 88] or [28, Th.7.29] for a detailed proof. By Lemma 0.11 and Remark 0.15 the claim follows. Remark . If A s → A is max-isometric, for any E ⊂ A s the spaces M B E ⊂ A s ( E, C ) and
M B E ⊂ A ( E, C ) associated to the respective embeddings are (isometrically) identical for any C .By Theorem 5.2, this shows that A has the LP if and only if k M B ( E, C ∗∗ ) → M B ( E, C ) ∗∗ k = 1for any f.d. subspace E ⊂ A .Moreover, if A satisfies (0.1) then A s also does.More precisely: Corollary 7.4. A C ∗ -algebra A satisfies the property in (0.1) if and only if for any separablesubspace E ⊂ A there is a separable C ∗ -subalgebra A s satisfying (0.1) such that E ⊂ A s ⊂ A .Proof. Let t ∈ ℓ ∞ ( { D i } ) ⊗ E with E ⊂ A s ⊂ A as in the proposition. Then t satisfies (0.2) withrespect to A if and only if it does with respect to A s .It is now easy to extend Theorems 0.2 and 0.13 to the non separable case: Theorem 7.5.
A (not necessarily separable) C ∗ -algebra has the LP if and only if it satisfies theproperty (0.1) . Moreover Theorem 0.13 remains valid in the non-separable case.Proof. The first part follows from the separable case by Corollary 7.4. For the second part, assumethat A satisfies (0.1). Let t ∈ M ⊗ E with E f.d. Let A s be as in Proposition 7.1. Then k t k M ⊗ max A = k t k M ⊗ max A s . By Remark 7.2 we also have k t k M ⊗ nor A = k t k M ⊗ nor A s . By the separablecase of Theorem 0.13 we know that k t k M ⊗ nor A s = k t k M ⊗ max A s . Therefore k t k M ⊗ nor A = k t k M ⊗ max A .This shows that (i)’ ⇒ (ii) in Theorem 0.13 remains valid in the non-separable case. The otherimplications have already been proved there. Remark . We will not enumerate all the other non-separable variants of the equivalent forms ofthe LP, except for the following one, that is in some sense the weakest lifting requirement sufficientfor the LP: a C ∗ -algebra A has the LP if and only if for any separable operator system E ⊂ A andany ∗ -homomorphism u : A → C/ I , the restriction u | E admits a c.c.p. lifting. Indeed, if the latterholds and if A is assumed unital for simplicity, writing A = C ∗ ( F ) / I for some large enough freegroup F , it is easy to deduce the desired LP for A from that of C ∗ ( F ). This proves the if part. Theconverse is clear, say, by Proposition 2.9 applied to some separable A s ⊂ A .24 On the OLP
Let X be an operator space. The associated universal unital C ∗ -algebra C ∗ u < X > is characterizedby the following property: it contains X completely isometrically, is generated by X and the unitand any complete contraction u : X → C into a unital C ∗ -algebra (uniquely) extends to a unital ∗ -homomorphism from C ∗ u < X > to C (see e.g. [28, Th. 2.25]).Following [22] we say that X has the OLP if C ∗ u < X > has the LP. By the universal propertyof C ∗ u < X > , it is easy to check that X has the OLP if and only if any u ∈ CB ( X, C/ I ) into anarbitrary quotient C ∗ -algebra admits a lifting ˆ u ∈ CB ( X, C ) such that k ˆ u k cb = k u k cb . See [22, Th.2.12] for various examples of X with the OLP.Note that, with respect to the inclusion X ⊂ A = C ∗ u < X > , the norm induced on D ⊗ X by the max-norm on D ⊗ A can be identified with the so-called δ -norm (see [27, p. 240]). Wemerely recall that the associated completion D ⊗ δ X can be identified as an operator space withthe quotient [ D ⊗ h X ⊗ h D ] / ker( Q ) where Q : D ⊗ h X ⊗ h D → D ⊗ δ X is associated to the productmap D ⊗ D → D . We have then isometrically when we view X as sitting in A = C ∗ u < X > :(8.1) D ⊗ δ X = D ⊗ max X. This implies that we have also isometrically for any C ∗ -algebra C .(8.2) M B ( X, C ) = CB ( X, C ) . Consider E ⊂ X f.d. Let C be a C ∗ -algebra. Recall (see Theorem 1.9) that the unit ball ofthe space M B ( E, C ) is formed of the maps u : E → C that extend to some ˙ u : A → C ∗∗ with k ˙ u k dec ≤
1. Let e u = ˙ u | X : X → C ∗∗ . Then k e u k cb ≤
1, so u extends to a complete contraction e u : X → C ∗∗ . Conversely, if u extends to a complete contraction e u : X → C ∗∗ , then e u extends to a ∗ -homomorphism π : A → C ∗∗ , and this implies k u k MB ( E,C ) ≤ k u k MB ( E,C ) = k u k ext ( E,C ) where (recall i C : C → C ∗∗ denotes the canonical inclusion) k u k ext ( E,C ) = inf {k e u k cb | e u : X → C ∗∗ , e u | E = i C u } . Whence the following characterization of the OLP. The equivalence (i) ⇔ (iii) already appearsin [22, Prop. 2.9]. Theorem 8.1.
Let X be a separable operator space. The following are equivalent: (i) X has the OLP. (ii) For any C ∗ -algebra C and any f.d. subspace E ⊂ X we have an isometric identity ext ( E, C ∗∗ ) = ext ( E, C ) ∗∗ . (iii) For any C ∗ -algebra C , every complete contraction u : X → C ∗∗ is the pointwise-weak* limit ofa net of complete contractions u i : X → C . (iv) For any family ( D i ) i ∈ I of C ∗ -algebras, the natural mapping ℓ ∞ ( { D i } ) ⊗ δ X → ℓ ∞ ( { D i ⊗ δ X } ) is isometric. roof. (i) ⇒ (ii) follows from Theorem 3.1 and (8.3). (ii) ⇔ (iii) is easy. (i) ⇔ (iii) follows fromTheorem 2.6 with F the class of complete contractions. (i) ⇒ (iv) is clear by (8.1) since (byTheorem 0.2) the LP for C ∗ u < X > implies that it satisfies the property in (0.1).Assume (iv). Then by (8.1) and (8.2) we have SB ( X, C ) =
M B ( X, C ) = CB ( X, C ) isometricallyfor any C . By Theorem 4.3, (i) holds.In particular we may apply this when X is a maximal operator space. Then k e u k cb = k e u k forany e u : X → C (or any e u : X → C ∗∗ ). This case is closely related to several results of Ozawa in hisPhD thesis and in [22]. See also Oikhberg’s [21]. The property (ii) in Theorem 8.1 is reminiscentof Johnson and Oikhberg’s extendable local reflexivity from [17].It gives some information on the existence of bounded liftings in the Banach space setting. To illustrate the focus our paper gives to the property in (0.1) and its variants, we turn to Kazhdan’sproperty (T), following [22, 32, 16]. The proof of Theorem 9.1 below is merely a reformulation ofan argument from [16, Th. G], but appealing to (0.1) is perhaps a bit quicker.
Theorem 9.1 ([16]) . Let G be a discrete group with property (T). Let ( N n ) be an increasingsequence of normal subgroups of G and let N ∞ = ∪ N n . If C ∗ ( G/N ∞ ) has the LP then N ∞ = N n for all large enough n .Proof. Let S ⊂ G be a finite unital generating subset. Let G n = G/N n and G ∞ = G/N ∞ . Let q n : G → G n and q ∞ : G → G ∞ denote the quotient maps. Let M n = λ G n ( G n ) ′′ denote the vonNeumann algebra of G n , and M U its ultraproduct with respect to a non-trivial ultrafilter U . Wewill define unitary representations π n and π U on G . We set π n ( t ) = λ G n ( q n ( t )) for any t ∈ G . Let Q U : ℓ ∞ ( {M n } ) → M U be the quotient map. We set π U ( t ) = Q U (( π n ( t ))). Since the kernel of G ∋ t π U ( t ) contains N ∞ , it defines a unitary representation on G/N ∞ and hence we have | S | = k X S π U ( s ) ⊗ π U ( s ) k M U ⊗ max M U ≤ k X S π U ( s ) ⊗ U G/N ∞ ( q ∞ ( s )) k M U ⊗ max C ∗ ( G/N ∞ ) . Let A = C ∗ ( G/N ∞ ). Using (0.1) for A we find(9.1) | S | ≤ lim n, U k X S π n ( s ) ⊗ U G/N ∞ ( q ∞ ( s )) k M n ⊗ max A . Also note that the LP of A implies M n ⊗ max A = M n ⊗ nor A. For some Hilbert space H n there is a faithful representation σ n : M n ⊗ nor A → B ( H n ) that isnormal in the first variable. By a well known argument (9.1) implies that the sequence { π n ⊗ U G/N ∞ ◦ q ∞ } , viewed as unitary representations of G in B ( H n ), admits (asymptotically) almostinvariant vectors. By property (T) for all n large enough, π n ⊗ U G/N ∞ ◦ q ∞ must admit an invariantunit vector in H n . This means there is a unit vector ξ ∈ H n such that ∀ t ∈ G σ n ( π n ( t ) ⊗ U G/N ∞ ◦ q ∞ ( t ))( ξ ) = ξ. Recall π n = λ G n ◦ q n . Let ∀ g ∈ G n f ( g ) = h ξ, σ n ( λ G n ( g ) ⊗ ξ ) i . f is a normal state on M n such that f ( g ) = 1 whenever g ∈ N ∞ /N n ⊂ G n . As is well knownwe may rewrite f as f ( g ) = h η, λ G n ( g ) η i for some unit vector η ∈ ℓ ( G n ). Now f ( g ) = 1 whenever g ∈ N ∞ /N n ⊂ G n means that λ G n ( g ) η = η whenever g ∈ N ∞ /N n ⊂ G n . From this follows that N ∞ /N n must be finite and hence N ∞ is the union of finitely many translates of N n by points say t , · · · , t k in N ∞ . Choosing m ≥ n so that t , · · · , t k ∈ N m we conclude that N m = N ∞ . Corollary 9.2 ([16]) . Let Γ be a discrete group with (T). If C ∗ (Γ) has the LP then Γ is finitelypresented.Proof. We quickly repeat the proof in [16]. By a result of Shalom [30] any property (T) discretegroup Γ is a quotient of a finitely presented group G with property (T). Enumerating the (a prioricountably many) relations that determine Γ, we can find a sequence N n and N ∞ as in Theorem9.1 with G/N n finitely presented such that Γ = G/N ∞ . By Theorem 9.1, if C ∗ (Γ) has the LP thenΓ = G/N n for some n and hence Γ is finitely presented. Remark . Let U be a non-trivial ultrafilter on N . If we do not assume the sequence ( N n )nested, let N ∞ be defined by t ∈ N ∞ ⇔ lim U t ∈ N n = 1. Then the same argument shows that | q n (ker( q ∞ )) | < ∞ for all n “large enough” along U .As pointed out in [16] there is a continuum of property (T) groups that are not finitely presented,namely those considered earlier by Ozawa in [24]. By the preceding corollary C ∗ ( G ) fails the LPfor all those G s.
10 Some uniform estimates derived from (0.1)
In this section our goal is to show that the property (0.1) has surprisingly strong “local” conse-quences. The results are motivated by the construction in [29] and our hope to find a separable A with LLP but failing LP (see Remark 0.17). Definition 10.1.
Let
B, C be C ∗ -algebras. Let E ⊂ B be a self-adjoint subspace and let ε >
0. Alinear map ψ : E → C will be called an ε -morphism if(i) k ψ k ≤ ε ,(ii) for any x, y ∈ E with xy ∈ E we have k ψ ( xy ) − ψ ( x ) ψ ( y ) k ≤ ε k x kk y k , (iii) for any x ∈ E we have k ψ ( x ∗ ) − ψ ( x ) ∗ k ≤ ε k x k . Lemma 10.2.
Let A be any unital C ∗ -algebra and let D be another C ∗ -algebra. Let t ∈ D ⊗ A . (i) Then for any δ > there is ε > and a f.d. operator system D ⊂ A such that t ∈ D ⊗ D and for any C ∗ -algebra C and any v ∈ CP ε ( D , C ) where CP ε ( D , C ) = { v : D → C, ∃ v ′ ∈ CP ( D , C ) with k v ′ k ≤ and k v − v ′ k ≤ ε } we have (10.1) k ( Id D ⊗ v )( t ) k max ≤ (1 + δ ) k t k max . (ii) Assume t ∈ ( D ⊗ A ) +max . Then for any δ > there is ε > and a f.d. operator system D ⊂ A such that t ∈ D ⊗ D and for any C ∗ -algebra C and any v ∈ CP ε ( D , C ) we have (10.2) d (( Id D ⊗ v )( t ) , ( D ⊗ max C ) + ) ≤ δ, where the distance d is meant in D ⊗ max C . For any δ > there is ε > and a f.d. subspace E ⊂ A such that t ∈ D ⊗ E and for any ε -morphism ψ : E → C ( C any other C ∗ -algebra) we have k ( Id D ⊗ ψ )( t ) k D ⊗ max C ≤ (1 + δ ) k t k D ⊗ max A . Proof.
By Remark 0.5 we may assume D unital. Let t ∈ D ⊗ A . Let E ⊂ A f.d. such that t ∈ D ⊗ E . First note that (see Remark 0.6 or Proposition 7.1) there is a separable C ∗ -subalgebra B with E ⊂ B ⊂ A such that k t k D ⊗ max B = k t k D ⊗ max A . Moreover, assuming k t k D ⊗ max A = 1 (forsimplicity), if t ∈ ( D ⊗ A ) +max = ( D ⊗ A ) ∩ ( D ⊗ max A ) + (which means that t is of finite rank with t = t ∗ and k − t k D ⊗ max A ≤
1) there is a B such that in addition t ∈ ( D ⊗ B ) +max . It follows thatwe may assume A separable without loss of generality. Then let ( E n ) be an increasing sequence off.d. subspaces (or operator systems if we wish) such that E ⊂ E and A = ∪ E n .(i) We may assume by homogeneity that k t k max = 1. We will work with the sequence ( E n , /n ).The idea is that this sequence “tends” to ( A, n ∈ N we define δ n by1 + δ n = sup k ( Id D ⊗ v )( t ) k max where the sup runs over all C ’s and v ∈ CP /n ( E n , C ). Note that this is finite because the rank of t is so. For any n there is C n and v n ∈ CP /n ( E n , C n ) such that(10.3) k ( Id D ⊗ v n )( t ) k max ≥ δ n − /n. Let L = ℓ ∞ ( { C n } ) and I = c ( { C n } ). Consider the quotient space L = L/ I so that for any family x = ( x n ) ∈ ℓ ∞ ( { C n } ) the image of x under the quotient map Q : ℓ ∞ ( { C n } ) → L satisfies k Q ( x ) k = lim sup k x n k . Note that by (0.7) we have(10.4) D ⊗ max L = [ D ⊗ max L ] / [ D ⊗ max I ] . The net ( v n ) defines a c.p. map v from A to L with k v k ≤
1. It follows (see Remark 1.4) that k ( Id D ⊗ v )( t ) k D ⊗ max L ≤
1. By (0.7) we have k ( Id D ⊗ v )( t ) k D ⊗ max L = k ( Id D ⊗ v )( t ) k ( D ⊗ max L ) / ( D ⊗ max I ) . We have a natural morphism D ⊗ max L → D ⊗ max C n for each n and hence a natural morphism D ⊗ max L → ℓ ∞ ( { D ⊗ max C n } ), taking D ⊗ max I to c ( { D ⊗ max C n } ). This implieslim sup n k ( Id D ⊗ v n )( t ) k D ⊗ max C n ≤ k ( Id D ⊗ v )( t ) k D ⊗ max L ≤ . Therefore by (10.3) lim sup n δ n ≤ . Thus for all n large enough we have δ n < δ . This proves (i).(ii) For any n , let δ n denote the supremum of the left-hand side of (10.2) over all C ’s and all v ∈ CP /n ( E n , C ). We introduce v n and v ∈ CP ( A, L ) by proceeding as in the preceding argument.Now ( Id D ⊗ v )( t ) ∈ ( D ⊗ L ) +max , which, in view of (10.4), implieslim sup d (( Id D ⊗ v n )( t ) , ( D ⊗ max C n ) + ) = 0 . We conclude as for part (i).Lastly, (iii) is proved similarly as (i). 28 emma 10.3.
Let A be any unital C ∗ -algebra satisfying (0.1) . Let E ⊂ A be a f.d. subspace. Thenfor any δ > there is ε > and a f.d. operator system D ⊂ A containing E such that the followingholds.(i) For any C ∗ -algebras C, D and any v ∈ CP ε ( D , C ) (in particular for any v ∈ CP ( D , C ) with k v k ≤ ) we have (10.5) ∀ t ∈ D ⊗ E k ( Id D ⊗ v )( t ) k max ≤ (1 + δ ) k t k max . In other words (10.6) k v | E k MB ( E,C ) ≤ δ. (ii) For any C ∗ -algebras C , D and any v ∈ CP ε ( D , C ) we have (10.7) ∀ t ∈ ( D ⊗ C ) +max d (( Id D ⊗ v )( t ) , ( D ⊗ max C ) + ) ≤ δ k t k max , where the distance d is meant in D ⊗ max C .(iii) For any C ∗ -algebras C , D and ε -morphism ψ : D → C we have k ( Id D ⊗ ψ )( t ) k D ⊗ max C ≤ (1 + δ ) k t k D ⊗ max A . Proof.
It is easy to see (by Remark 0.6 or Proposition 7.1) that we may restrict to separable D ’s. Wethen assemble the set I that is the disjoint union of the unit balls of D ⊗ max E , when D is an arbitraryseparable C ∗ -algebra (say viewed as quotient of C ). Let t i ∈ D i ⊗ E be the element correspondingto i ∈ I . By (0.2) we know that k t k ℓ ∞ ( { D i } ) ⊗ max A ≤ , and hence k t k ℓ ∞ ( { D i } ) ⊗ max E ≤
1. Let D = ℓ ∞ ( { D i } ) so that t ∈ D ⊗ E . Let ε > D ⊃ E be associated to t as in Lemma 10.2. Forany v ∈ CP ε ( D , C ) we have k ( Id D ⊗ v )( t ) k max ≤ (1 + δ ) k t k max ≤ δ. A fortiori, using the canonical morphisms D → D i sup i ∈ I k ( Id D i ⊗ v )( t i ) k max ≤ δ, equivalently we conclude k v k MB ( E,C ) ≤ δ .(ii) and (iii) are proved similarly. A priori they lead to distinct D ’s but since replacing D by alarger f.d. one preserves the 3 properties, we may obtain all 3 for a common D . Remark . In the converse direction, assume that for any E and δ > D ⊃ E such that,for any C , any v ∈ CP ( D , C ) satisfies (10.6). If A (separable unital for simplicity) has the LLPthen A satisfies (0.1). Indeed, assuming the LLP any quotient morphism q : C → A will admitunital c.p. local liftings on arbitrarily large f.d. operator systems D in A . By (10.6), we have locallifings u E : E → C with k u E k mb ≤ δ on arbitrarily large f.d. subspaces E in A . From the latterit is now easy to transplant (0.1) from C to A (as in the proof of (i) ⇒ (ii) in Theorem 0.2).Thus the property described in part (i) of Lemma 10.3 can be interpreted as describing what ismissing in the LLP to get the LP.Our last result is a reformulation of the equivalence of the LP and the property in (0.1). Underthis light, the LP appears as a very strong property. Theorem 10.5.
Let A be a C ∗ -algebra. The following are equivalent: (i) The algebra A satisfies (0.1) (i.e. it has the lifting property LP). For any f.d. E ⊂ A there is P = P E ∈ C ⊗ E with k P k C ⊗ max A ≤ such that for any unitalseparable C ∗ -algebra D and any t ∈ B D ⊗ max E there is a unital ∗ -homomorphism q D : C → D such that [ q D ⊗ Id A ]( P ) = t. (ii)’ For any f.d. E ⊂ A there is a unital separable C ∗ -algebra C and P = P E ∈ C ⊗ E with k P k C ⊗ max A ≤ such that such that the same as (ii) holds with C in place of C . (iii) For any f.d. E ⊂ A and ε > there is P = P E ∈ C ⊗ E with k P k C ⊗ max A ≤ such that forany unital separable C ∗ -algebra D and any t ∈ B D ⊗ max E there is a unital ∗ -homomorphism q D : C → D such that k [ q D ⊗ Id A ]( P ) − t k max < ε. (iii)’ For any f.d. E ⊂ A and ε > there is a unital separable C ∗ -algebra C and P ∈ C ⊗ E suchthat the same as (iii) holds with C in place of C . (iv) Same as (iii)’ restricted to D = C .Proof. Assume (i). We view the set of all possible D ’s as the set of quotients of C . Let I be thedisjoint union of the sets B D ⊗ max E . Let t = ( t i ) i ∈ I be the family of all possible t ∈ B D ⊗ max E . By(0.2) we have k t k ℓ ∞ ( { D i } ) ⊗ max E ≤ . Let q : C ∗ ( F ) → ℓ ∞ ( { D i } ) be a surjective ∗ -homomorphism. Let t ′ ∈ C ∗ ( F ) ⊗ E be a lifting of t such that k t ′ k C ∗ ( F ) ⊗ max E ≤
1. There is a copy of F ∞ , say F ′ ⊂ F such that t ′ ∈ C ∗ ( F ′ ) ⊗ E . Then C ∗ ( F ′ ) ⊂ C ∗ ( F ). Let C = C ∗ ( F ′ ). We can take for q D the restriction of q i to C ∗ ( F ′ ) ≃ C , and weset P = t ′ viewed as sitting in C ⊗ E . This gives us (i) ⇒ (ii) and (ii) ⇒ (ii)’ is trivial.(ii) ⇒ (iii) and (iii) ⇒ (iii)’ ⇒ (iv) are trivial.Assume (iii)’. Let Q : C → C be a quotient unital morphism. Let P ∈ C ⊗ E be as in (iii)’. Let P ′ ∈ C ⊗ E with k P ′ k max ≤ Q ⊗ Id E ]( P ′ ) = P (this exists by the assertion following(0.8)). Then P ′ has the property required in (iii). Thus (iii)’ ⇒ (iii). Similarly (ii)’ ⇒ (ii).The implication (iv) ⇒ (iii)’ is easy to check similarly using the fact that any D is a quotient of C . We skip the details. It remains to show (iii) ⇒ (i).Assume (iii). We will show (0.2). We need a preliminary elementary observation. Let ( e j ) bea normalized algebraic basis of E . Then it is easy to see that there is a constant c (depending on E and ( e j )) such that for any D and any d j , d ′ j ∈ D we have(10.8) sup j k d j − d ′ j k ≤ c k X ( d j − d ′ j ) ⊗ e j k D ⊗ max E . Now let t i ∈ B D i ⊗ max E . Let P ∈ B C ⊗ max E be as in (iii). For some quotient morphism q i : C → D i we have k t i − [ q i ⊗ Id A ]( P ) k C ⊗ max E ≤ ε. Let P i = P and C i = C for all i ∈ I . Clearly (since P ( P i ) is a ∗ -homomorphism on C ⊗ A ) k ( P i ) k ℓ ∞ ( { C i } ) ⊗ max E ≤ k P k C ⊗ max E ≤ . Note ( q i ) : ℓ ∞ ( { C i } ) → ℓ ∞ ( { D i } ) is a ∗ -homomorphism. Let ( t ′ i ) = [ q i ⊗ Id A ]( P i ). We have k ( t ′ i ) k ℓ ∞ ( { D i } ) ⊗ max E ≤ i k t i − t ′ i k C ⊗ max E ≤ ε, here, actually, we could use instead of a ∗ -homomorphism a u.c.p. map or a q with k q k mb ≤
30y the triangle inequality and (10.8) we have k ( t i ) − ( t ′ i ) k ℓ ∞ ( { D i } ) ⊗ max E ≤ c dim( E ) ε . Therefore k ( t i ) k ℓ ∞ ( { D i } ) ⊗ max E ≤ c dim( E ) ε . Since ε > E , this means that (0.2)holds whence (i) (and the LP by Theorem 0.2).In passing we note : Corollary 10.6.
For the property in (0.1) it suffices to check the case when I = N (and D i = C for all i ∈ I ).Proof. We already observed that D i = C for all i ∈ I suffices (see Proposition 0.8). Assume that(0.2) holds for I = N and D i = C . In the proof of (i) ⇒ (ii) in Theorem 10.5 we may use for the set I the disjoint union of dense sequences in B D k ⊗ max E with k ∈ N and D k = C . Then I is countableand with the latter I the proof of (i) ⇒ (ii) in Theorem 10.5 gives us (iii) in Theorem 10.5. By (iii) ⇒ (i) in Theorem 10.5, the corollary follows. Remark . Let P denote the (non-closed) ∗ -algebra generated by the free unitary generators of C . Let P ∈ C ⊗ E be as in (ii) in Theorem 10.5. Note for any ε > Q ∈ P ⊗ E with k Q − P k ∧ < ε (where k k ∧ denotes the projective norm), and hence also k q D ( Q ) − t k ∧ < ε . Aftera suitable normalization we may also assume k Q k max <
1. Then Q has a factorization in productsin ∪ M n ( C ) and ∪ M n ( A ) and a further Blecher-Paulsen type factorization in the style described in[27, § common pattern in the approximate factorization of all the elements of B D ⊗ max E .The simplest illustration of this phenomenon is the case when A = C and E is the span of theunitary generators ( U i ). In that case we have t = P d i ⊗ U i ∈ B D ⊗ max E if and only if for any ε > d i = a i b ∗ i with a i , b i ∈ D such that k P a i a ∗ i k / = k P b i b ∗ i k / < ε (see [28, p. 130-131]). Equivalently, this holds if and only if for any ε > a i , b i ∈ D such that k P a i a ∗ i k / = k P b i b ∗ i k / = 1 and P i k d i − a i b ∗ i k < ε . Let us assume that the latterholds. Consider first the column operator space C n , then its universal C ∗ -algebra C ∗ u < C n > ,and lastly the free product C = C ∗ u < C n > ∗ C ∗ u < C n > . Let e (1) i ∈ C and e (2) i ∈ C denotethe natural basis of the column space C n of each free factor. Let P E = P e (1) i e (2) ∗ i ⊗ U i ∈ C ⊗ E .By our assumption on ( d i ), there is a ∗ -homomorphism q D : C ⊗ E such that q D ( e (1) i ) = a i and q D ( e (2) i ) = b i and hence k q D ( P E ) − P d i ⊗ U i k ≤ ε . This illustrates the property (iii)’ fromTheorem 10.5. Note in passing that since the column space C n has the OLP, the free product C = C ∗ u < C n > ∗ C ∗ u < C n > has the LP. The latter algebra can be substituted with C in manyquestions involving tensor products.Using the preceding remark, one can give an independent proof of Lemma 10.3. Acknowledgement.
I am grateful to Jean Roydor and Mikael de la Salle for stimulating anduseful comments.
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