NONCOMMUTATIVE SEMIALGEBRAIC SETS IN NILPOTENT VARIABLES 1 TERRY A. LORING AND TATIANA SHULMAN 1 0 2 Abstract. We solve the lifting problem in C∗-algebras for many sets of relations n thatincludetherelationsxNj j =0oneachvariable. Theremainingrelationsmustbe a of the form kp(x1,...,xn)k≤C for C a positive constantand p a noncommutative J ∗-polynomial that is in some sense homogeneous. For example, we prove liftability 2 for the set of relations 1 x3 =0, y4 =0, z5 =0, xx∗+yy∗+zz∗ ≤1. ] A Thus we find more noncommutative semialgebraic sets that have the topology of noncommutative absolute retracts. O . h t a m 1. Introduction [ Lifting problems involving norms and star-polynomials are fundamental in C∗- 1 algebras. They arise in basic lemmas in the subject, as we shall see in a moment. v 9 They also arise in descriptions of the boundary map in K-theory, in technical lemmas 6 on inductive limits, and have of course been around in operator theory. Much of our 2 understanding of the Calkin algebra comes from having found properties of its cosets 2 . that exist only when some operator in a coset has that property. 1 ∗ 0 Let A denote a C -algebra and let I be an ideal in A. The quotient map will be 1 ∗ denoted π : A → A/I. Of course A/I is a C -algebra, but let us ponder how we know 1 : this. The standard proof uses an approximate unit uλ and an approximate lifting v property. The lemma used is that for any approximate unit u , and any a in A, i λ X limka(1−u )k = kπ(a)k r λ a λ and trivially we obtain as a corollary limk(1−u )b(1−u )k = kπ(b)k. λ λ λ For a large λ, the lift x¯ = a(1−u ) of π(a) approximately achieves two norm condi- λ tions, ∗ ∗ kx¯k ≈ kπ(x¯)k, kx¯ x¯k ≈ kπ(x¯) π(x¯)k. 2 ∗ ∗ The equality kx¯k = kx¯ x¯k upstairs now passes downstairs, so A/I is a C -algebra. We have an eye on potential applications in noncommutative real algebraic geom- etry [7, 8]. What essential differences are there between real algebraic geometry and 2000 Mathematics Subject Classification. 46L85, 47B99 . Key words and phrases. C∗-algebra,relation, projective, lifting. 1 2 TERRYA.LORINGANDTATIANASHULMAN noncommutative real algebraicgeometry? Occam would cut between these fields with the equation xn = 0. Could we just exclude this equation? Probably not. A search of the physics literature finds that polynomials in nilpotent variables are gaining popularity. Two examples to see are [3] in condensed matter physics, and [12] in quantum information. Focusing back on lifting problems, we recall what is known about lifting nilpotents ∗ up from general C -algebra quotients. Akemann and Pedersen [1] showed the relation x2 = 0 lifts, and Olsen and Pedersen [14] did the same for xn = 0. Akemann and Pedersen [1] also showed that if xn−1 6= 0 for some x ∈ A/I then one can find a lift X of x with Xj = xj , (j = 1,...,n−1). If xn = 0 and xn−1 6= 0 then we would like to combine these results, lifting both the (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) nilpotent condition and the n−1 norm conditions. It was not until recently, in [16], that it was shown one could lift just the two relations kxk ≤ C, xn = 0 for C > 0. Here we show how to lift a nilpotent and all these norm conditions, so show the liftablity of the set of relations xj ≤ C , j = 1,...,n, j even if Cn = 0. In the particul(cid:13)ar c(cid:13)ase where the quotient is the Calkin algebra and the lifting is to B(H), we proved (cid:13)this(cid:13)using different methods in [10], as a partial answer to Olsen’s question [13]. More generally, we consider soft homogeneous relations (as defined below) together with relations xNj = 0. In onevariable, another example of such a collection of liftable j relations is ∗ 2 3 kxk ≤ C1, x x−x ≤ C2, x = 0. In two variables, we have such curiosities as (cid:13) (cid:13) (cid:13) (cid:13) 3 3 kxk ≤ 1, kyk ≤ 1, x = 0, y = 0, kx−yk ≤ ǫ which we can now lift. Given a ∗-polynomial in x1,...,xn we have the usual relation p(x1,...,xn) = 0, ∗ where now the x are in a C -algebra. In part due to the shortage of semiprojective j ∗ C -algebras, Blackadar [2] suggested that we would do well to study the relation kp(x1,...,xn)k ≤ C for some C > 0. Following Exel’s lead [6], we call this a soft polynomial relation. Softened relations come up naturally when trying to classifying ∗ C -algebras that are inductive limits, as in [5], when exact relations in the limit lead only to inexact relations in a building block in the inductive system. The homogeneity we need is only that there be a subset (such as) x1,...,xr of the variables and an integer d ≥ 1 so that every monomial in p contains exactly d factors ∗ ∗ from x1,x1...,xr,xr. NONCOMMUTATIVE SEMIALGEBRAIC SETS IN NILPOTENT VARIABLES 3 The relation xN = 0 is“more liftable”than most liftable relations in that it can be added to many liftable sets while maintaining liftability. Other relations that behave ∗ this way are x = x and x ≥ 0. We explored semiagebraic sets (as NC topological spaces) in positive and Hermitian variables in [11]. There are still other relations that are“more liftable”in this sense. We consider in ∗ this note xyx = 0 and xy = 0. This is not the end of the story, as we might have a rare case of too little theory and too many examples. We use many technical results from our previous work [11]. We also have use for the Kasparov Technical Theorem. Indeed we use only a simplified version, but the fully technical version can probably be used to find even more lifting theorems in this realm. For a reference, a choice could be made from [4, 9, 14]. We will use the notation a ≪ b to mean b acts like unit on a, i.e. ab = a = ba. A trick we use repeatedly is to replace a single element c so that 0 ≤ c ≤ 1 and x c = x , cy = 0 j j k for some sequences x and y with two elements a and b with j k (1.1) 0 ≤ a ≪ b ≤ 1 and (1.2) x a = x , by = 0. j j k These are found with basic functional calculus. The simplified version of Kasparov’s technical theorem we need can be stated as follows: for x1,x2,··· and y1,y2,··· in a corona algebra C(A) = M(A)/A (for A σ-unital) with x y = 0 for all j and k, there j k are elements a and b in C(A) satisfying (1.1) and (1.2). 2. Lifting Nilpotents while Preserving Various Norms Lemma 2.1. Suppose A is σ-unital C∗-algebra, n is at least 2, and consider the quotient map π : M(A) → M(A)/A. (1) If x is an element of M(A) so that π(xn) = 0 then there are elements p1,...,pn−1 and q1,...,qn−1 of M(A) with j > k =⇒ p q = 0 j k and n−1 π q xp = π(x). j j ! j=1 X (2) If π(x˜) = π(x) and we set n−1 x¯ = q x˜p , j j j=1 X then π(x¯) = π(x) and x¯n = 0. 4 TERRYA.LORINGANDTATIANASHULMAN Proof. This is the essential framework that assists the lifting of nilpotents, going back to [14]. Other than a change of notation, this is an amalgam of Lemmas 1.1, 8.1.3, 12.1.3 and 12.1.4 of [9]. (cid:3) Theorem 2.2. If x is an element of a C∗-algebra A, and I is an ideal and π : A → A/I is the quotient map, then for any natural number N, there is an element x¯ in A so that π(x¯) = π(x) and kx¯nk = kπ(xn)k, (n = 1,...,N). Proof. If π xN 6= 0, then this is the first statement in Theorem 3.8 of [1]. Assume then that π xN = 0. Standard reductions (Theorem 10.1.9 of [9]) allow (cid:0) (cid:1) ∗ us to assume A = M(E) and I = E for some separable C -algebra E. The first part (cid:0) (cid:1) of Lemma 2.1 provides elements p1,...,pN−1 and q1,...,qN−1 in M(E) with j > k =⇒ p q = 0 j k and N−1 π q xp = π(x). j j ! j=1 X Let C = kπ(xn)k. Each norm condition n N−1 n q x˜p ≤ C (n = 1,...,N −1) j j n (cid:13) ! (cid:13) (cid:13) Xj=1 (cid:13) (cid:13) (cid:13) is a norm-restriction(cid:13)of a NC polyn(cid:13)omial that is homogeneous in x˜. We can apply (cid:13) (cid:13) Theorem 3.2 of [11] to find xˆ in M(E) with π(xˆ) = π(x˜) and N−1 n q xˆp ≤ C (n = 1,...,N −1). j j n (cid:13)(cid:13) Xj=1 ! (cid:13)(cid:13) (cid:13) (cid:13) Since π(xˆ) = π(x) w(cid:13)e may apply th(cid:13)e second part of Lemma 2.1 to conclude that (cid:13) (cid:13) N−1 x¯ = q xˆp j j j=1 X is a lift of π(x), is nilpotent of order N, and kx¯nk ≤ C = kπ(xn)k n for n = 1,...,N −1. (cid:3) There was nothing special about the homogeneous ∗-polynomials xn, and we can deal with more than one nilpotent variable x at a time. We say a ∗-polynomial is homogeneous of degree r for some subset S of the variables when the total number of ∗ times either x or x for x ∈ S appears in each monomial is r. Staying consistent with the notation in [11], we use p(x,y) = p(x1,...xr,y1,y2,...) as so keep to the left the variables in subset where there is homogeneity. NONCOMMUTATIVE SEMIALGEBRAIC SETS IN NILPOTENT VARIABLES 5 Theorem 2.3. Suppose p1,...,pJ are NC ∗-polynomials in infinitely many variables that are homogeneous in the set of the first r variables, each with degree of homo- geneity d at least one. Suppose C > 0 are real constants and N ≥ 2 are integer j j k constants, k = 1,...,r. For every C∗-algebra A and I ⊳ A an ideal, given x1,...,xr and y1,y2,... in A with N (π(x )) k = 0 k and kp (π(x,y))k ≤ C , j j there are z1,...,zr in A with π(z) = π(x) and zNk = 0 k and kp (z,y)k ≤ C . j j Proof. Again we use standard reductions to assume A = M(E) and I = E for some ∗ separableC -algebraE.NowweapplyLemma 2.1toeachxk andfindpk,1,...,pk,N −1 k and qk,1,...,qk,N −1 in M(E) with k b > c =⇒ p q = 0 k,b k,c and N −1 k π q x p = π(x ). k,b k k,b k ! b=1 X We know that any x˜ we take with π(x˜) = π(x) will give us N −1 k π q x˜ p = π(x ) k,b k k,b k ! b=1 X and N −1 Nk k q x˜ p = 0, k,b k k,b ! b=1 X so we need only fix the relations N1−1 Nr−1 pj q1,bx˜1p1,b,... qr,bx˜rpr,b,y ≤ Cj. (cid:13) !(cid:13) (cid:13) Xb=1 Xb=1 (cid:13) (cid:13) (cid:13) These are homogen(cid:13)eous in {x˜1,...,x˜r} so we are done, by(cid:13)Theorem 3.2 of [11]. (cid:3) (cid:13) (cid:13) We could add various relations on the variables y1,y2,..., and include in the pj ∗ ∗-polynomials, in various ways that ensure that there is an associated universal C - algebra which is then projective. For example, we could zero them out the extra variables (so just omit them) and impose a soft relation know for imply all the x are j contractions. Let us give one specific class of examples. 6 TERRYA.LORINGANDTATIANASHULMAN Example 2.4. WehaveprojectivityfortheuniversalC∗-algebraonx1,...,xn subject to the relations xNk = 0, xkx∗k ≤ 1, kpj(x1,...,xn)k ≤ Cj for Cj > 0 and the pj all NC(cid:13)(cid:13)X∗-polyno(cid:13)(cid:13)mials that are homogeneous in x1,...,xn. (cid:13) (cid:13) 3. The Relation xyx∗ = 0. ∗ Wenow exploresetting xyx to zero. This wordis unshrinkable, inthesense of [17]. ∗ We show that many sets of relations involving xyx = 0 are liftable. One example, chosen essentially at random, is the set consisting of the relations ∗ kxk ≤ 1, kyk ≤ 1, kxy +yxk ≤ 1, xyx = 0. Lemma 3.1. Suppose A is σ-unital and C(A) = M(A)/A. If x and y are elements of M(A) so that xyx∗ = 0, then there are elements 0 ≤ e ≪ f ≪ g ≤ 1 so that x(1−g) = x and ey +(1−e)yf = y. Proof. We apply Kasparov’s technical theorem to the product x(yx∗) = 0 to find 0 ≤ d ≤ 1 in C(A) with (3.1) xd = x, ∗ (3.2) dyx = 0. We rewrite (3.1) as ∗ (3.3) (1−d)x = 0 and apply Kasparov’s technical theorem to (3.2) and (3.3) to find 0 ≤ f ≪ g ≤ 1 in C(E) with (1−d)f = (1−d) (3.4) dyf = dy ∗ gx = 0. Thus we have xg = 0 and 0 ≤ 1−d ≪ f ≪ g ≤ 1. We are done, with e = 1−d, since (3.4) gives us ey +(1−e)yf = (1−d)y +dyf = y. (cid:3) NONCOMMUTATIVE SEMIALGEBRAIC SETS IN NILPOTENT VARIABLES 7 Lemma 3.2. Suppose A is σ-unital and consider the quotient map π : M(A) → M(A)/A. (1) If x and y are elements of M(A) so that π(xyx∗) = 0, then there are elements e, f and g in M(A) with (3.5) 0 ≤ e ≪ f ≪ g ≤ 1, π(x(1−g)) = π(x) and π(ey +(1−e)yf) = π(y). (2) If π(x˜) = π(x) and π(y˜) = π(y) then, if we set x¯ = x˜(1−g) and y¯= ey˜+(1−e)y˜f, we have π(x¯) = π(x), π(y¯) = π(y) and x¯y¯x¯∗ = 0. Proof. In C(A), the product π(x)π(y)π(x)∗ is zero, so Lemma 3.1 produces e0, f0 and g0 in C(A) with 0 ≤ e0 ≪ f0 ≪ g ≤ 1, π(x)(1−g0) = π(x) and e0π(y)+(1−e0)π(y)f0 = π(y). Lemma 1.1.1 of [9] tells us there are lifts e, f and g in M(A) of e0, f0 and g0 satisfying (3.5). Then π(x(1−g)) = π(x)(1−g0) = π(x) and π(ey +(1−e)yf) = e0π(y)+(1−e0)π(y)f0 = π(y). As for the second statement, π(x¯) = π(x˜(1−g)) = π(x)(1−g0) = π(x), π(y¯) = π(ey˜+(1−e)y˜f) = e0π(y)+(1−e0)π(y)f0 = π(y) and ∗ ∗ ∗ x¯y¯x¯ = x˜(1−g)ey˜(1−g)x˜ +x˜(1−g)(1−e)y˜f(1−g)x˜ = 0 since (1−g)e = 0 and (1−g)f = 0. (cid:3) Theorem 3.3. Suppose p1,...,pJ are NC ∗-polynomials in infinitely many variables that are homogeneous in the set of the first 2r variables, each with degree of homo- geneity d at least one. Suppose C > 0 are real constants and N ≥ 2 are integer j j j constants. For every C∗-algebra A and I ⊳ A an ideal, given x1,...,xr and y1,...,yr and z1,z2,... in A with ∗ π(x )π(y )π(x ) = 0, (k = 1,...,r) k k k and kp (π(x,y,z))k ≤ C , (j = 1,...,J) j j 8 TERRYA.LORINGANDTATIANASHULMAN there are x¯1,...,x¯r and y¯1,...,y¯r in A with π(x¯) = π(x) and π(y¯) = π(y) and ∗ x¯ y¯ x¯ = 0, (k = 1,...,r) k k k and kp (x¯,y¯,¯z)k ≤ C , (j = 1,...,J). j j Proof. Without loss of generality, assume A = M(E) and I = E for some separable ∗ C -algebra E. Now we apply Lemma 3.2 to each pair x and y and find e , f and j j j j g in M(E) so that, given any lifts x˜ and y˜ of π(x ) and π(y ), setting j j j j j x¯ = x˜ (1−g ) j j j and y¯ = e y˜ +(1−e )y˜ f j j j j j j produces again lifts of the π(x ) and π(y ) with j j ∗ x¯ y¯ x¯ = 0. j j j The needed norm conditions kpj(x˜1(1−g1),...,x˜r(1−gr),e1y˜1 +(1−e1)y˜1f1,...,ery˜r +(1−er)y˜rfr,¯z)k ≤ Cj involve NC ∗-polynomials that are homogeneous in {x1,...,xr,y1,...,yr}, so Theo- rem 3.2 of [11] again finishes the job. (cid:3) Example 3.4. For any r, the C∗-algebra ∗ x y x = 0, C∗ x1,...xr,y1,...,yr x jx∗j+jy y∗ ≤ 1 (cid:28) (cid:12) j j j j (cid:29) (cid:12) is projective. In particular, since projective implies residually finite dimensional, if (cid:12) (cid:13)P (cid:13) (cid:12) (cid:13) (cid:13) one could show that the ∗-algebra ∗ x y x = 0, C x1,...xr,y1,...,yr x jx∗j+jy y∗ ≤ 1 (cid:28) (cid:12) j j j j (cid:29) ∗ (cid:12) is C -representable (as in [15]), then it w(cid:12)o(cid:13)uPld have a sepa(cid:13)rating family of finite di- (cid:12) (cid:13) (cid:13) mensional representations. 4. The Relations x x = 0. j k We can work with variables that are“half-orthogonal”in that any product x x is j k zero. The ∗-monoid here contains only monomials of the forms ∗ ∗ ∗ x x ···x x , x x ···x x j1 j2 j2N−1 j2N j1 j2 j2N j2N+1 and their adjoints. Lemma 4.1. Suppose A is σ-unital and C(A) = M(A)/A. If x1...,xr are elements of M(A) so that x x = 0 for all j and k then there are elements 0 ≤ f,g ≤ 1 so that j k fg = 0 and fx g = x j j for all j. NONCOMMUTATIVE SEMIALGEBRAIC SETS IN NILPOTENT VARIABLES 9 Proof. We apply Kasparov’s technical theorem to find a and b with 0 ≤ a ≪ b ≤ 1 and x a = a, bx = 0. j j Let f = 1−b and g = a. (cid:3) Lemma 4.2. Suppose A is σ-unital and consider the quotient map π : M(A) → M(A)/A. (1) If x1,...,xr are elements of M(A) so that π(xjxk) = 0 for all j and k, then there are elements f and g in M(A) with (4.1) 0 ≤ f,g ≤ 1, (4.2) fg = 0 and π(fx g) = π(x ). j j (2) If π(x˜ ) = π(x ) then, if we set j j x¯ = fx˜ g, j j we have π(x¯ ) = π(x ) and j j x¯ x¯ = 0 j k for all f and g. Proof. The productsπ(xj)π(xk) arezero, so Lemma 4.1 gives us elements 0 ≤ f0,g0 ≤ 1 in C(A) with f0g0 = 0 and f0π(xj)g0 = π(xj). Orthogonal positive contractions lift to orthogonal positive contractions, so there are f and g in M(A) satisfying (4.1) and (4.2) that are lifts of f0 and g0, which means π(fxjg) = f0π(xj)g0 = π(xj). With x¯ as indicated, j π(x¯j) = π(fx˜jg) = f0π(xj)g0 = π(xj) and x¯ x¯ = fx˜ gfx˜ g = 0. j k j k (cid:3) Theorem 4.3. Suppose p1,...,pJ are NC ∗-polynomials in infinitely many variables that are homogeneousin the set of the firstr variables, eachwith degree of homogeneity d at least one. Suppose C > 0 are real constants. For every C∗-algebra A and I ⊳ A j j an ideal, given x1,...,xr and y1,y2,... in A with π(x )π(x ) = 0, (k,l = 1,...,r) k l 10 TERRYA.LORINGANDTATIANASHULMAN and kp (π(x,y))k ≤ C , (j = 1,...,J) j j there are x¯1,...,x¯r in A with π(x¯) = π(x) and x¯ x¯ = 0, (k,l = 1,...,r) k l and kp (x¯,y¯)k ≤ C , (j = 1,...,J). j j Proof. The proof is essentially the same as that of Theorem 3.3. (cid:3) References [1] Charles A. Akemann and Gert K. Pedersen. 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