LOCAL THEORY OF FREE NONCOMMUTATIVE FUNCTIONS: GERMS, MEROMORPHIC FUNCTIONS AND HERMITE INTERPOLATION
IGOR KLEP1, VICTOR VINNIKOV2, AND JURIJ VOLCIˇ Cˇ 3
Abstract. Free analysis is a quantization of the usual function theory much like oper- ator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of matrices of all sizes that preserve direct sums and similarities. This paper investigates the local theory of noncommutative functions. The first main ua result shows that for a scalar point Y , the ring OY of uniformly analytic noncommutative germs about Y is an integral domain and admits a universal skew field of fractions, whose elements are called meromorphic germs. A corollary is a local-global rank principle that ua connects ranks of matrix evaluations of a matrix A over OY with the factorization of ua A over OY . Different phenomena occur for a semisimple tuple of non-scalar matrices ua Y . Here it is shown that OY contains copies of the matrix algebra generated by Y . In particular, there exist nonzero nilpotent uniformly analytic functions defined in a ua neighborhood of Y , and OY does not embed into a skew field. Nevertheless, the ring ua OY is described as the completion of a free algebra with respect to the vanishing ideal at Y . This is a consequence of the second main result, a free Hermite interpolation theorem: if f is a noncommutative function, then for any finite set of semisimple points and a natural number L there exists a noncommutative polynomial that agrees with f at the chosen points up to differentials of order L. All the obtained results also have analogs for (non-uniformly) analytic germs and formal germs.
1. Introduction The study of analytic functions in noncommuting variables goes back to the seminal work of Taylor [Tay72, Tay73]. Recently, noncommutative function theory or free analysis saw a rapid development fueled by free probability, dilation theory, operator systems and spaces, control theory and optimization [BGM05, Pop06, Voi10, MS11, HKM11, K-VV14, AM16]. The central objects are noncommutative (nc) functions f defined on tuples of square matrices of finite size that respect basis change and direct sums (see Subsection 2.1 for a precise definition). For example,
−1 f(x1, x2) = x1 exp x2(x1x2 − x2x1)
Date: November 26, 2019. 2010 Mathematics Subject Classification. 32A20, 47A56, 16W60 (Primary); 16R50, 32A05, 16K40 (Secondary). Key words and phrases. Free analysis, noncommutative function, analytic germ, universal skew field of fractions, noncommutative meromorphic function, Hermite interpolation. 1Supported by the Marsden Fund Council of the Royal Society of New Zealand. Partially supported by the Slovenian Research Agency grants J1-8132, N1-0057 and P1-0222. 2Supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1. 3Supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1. 1 2 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ is an nc function defined on all pairs of matrices (X1,X2) such that X1X2 − X2X1 is nonsingular. Thus f is not defined for any pair of scalar matrices, but it is defined on 2 an open set in Mn(C) for n > 1. The key idea of functions respecting basis change and direct sums (albeit in the setting of C∗-algebras) was first addressed by Takesaki [Tak67]. Nc functions admit a differential calculus and posses extraordinary analytic properties. If an nc function f is bounded on a neighborhood of Y , then f is continuous and even an- alytic there, and equals its noncommutative power series expansion about Y determined by its differentials at Y [Voi10, HKM12, K-VV14]. The precise nature of convergence de- pends on the underlying topology on tuples of matrices of all sizes. When using the disjoint union topology, we obtain analytic nc functions. Another natural option is the uniformly open topology generated by noncommutative balls about matrix points (see Subsection 2.3.2), in which case we talk about uniformly analytic nc functions. While the methods for dealing with analytic nc functions derive mainly from complex analysis, uniformly analytic nc functions are closer to operator space theory. In both settings, several classi- cal analytic results have their free analogs, such as the implicit/inverse function theorem [AK-V15, AM16’], the Oka–Weil approximation theorem [AM15], the Nevanlinna–Pick interpolation [Pop08], Choquet theory [DK+], a homogeneous Nullstellensatz [SSS18], the Jacobian conjecture [Pas14] and the Grothendieck theorem [Aug19]. This paper addresses the local behavior of analytic nc functions. Given a matrix point Y , a (uniformly) analytic noncommutative germ about Y is the equivalence class of a (uniformly) analytic nc function on a neighborhood of Y . By the previous paragraph, such a germ is determined by the power series expansion of an nc function. For this reason we also define formal germs as formal noncommutative power series about Y satisfying certain natural linear constraints, called canonical intertwining conditions about Y (Definition 2.2). Roughly speaking, these conditions encode preservation of similarity and direct sums behavior of nc functions, so that a (uniformly) analytic germ is precisely a formal germ given by a (uniformly) convergent power series satisfying canonical intertwining conditions. This paper presents the first systematic study of algebras of noncommutative germs with a view toward functional calculus.
g a Main results and guide to the paper. For a matrix point Y ∈ Ms(C) let OY , OY ua and OY denote the C-algebras of formal, analytic, and uniformly analytic germs in g freely noncommuting variables x = (x1, . . . , xg), respectively. After preliminary results in Section2, our study of these algebras branches in two directions, depending on whether Y is a scalar point or not. If Y is a scalar point (s = 1), then OY is isomorphic to C<
fields of fractions [K-VV12, KVV+]. On the other hand, the universality specifies what it means for a meromorphic expression to be identically zero, which is essential for analysis because it allows us to talk about functions induced by meromorphic expressions. More concretely, we prove the Amitsur–Cohn theorem for meromorphic identities. An algebra A is stably finite if for every n ∈ N and A, B ∈ An×n, AB = I implies BA = I. For ∗ instance, a C -algebra with a faithful trace, such as a type II1 von Neumann algebra, is stably finite, while the full algebra of bounded operators on a separable Hilbert space is not; see [RLL00, Bla06] for details and more examples.
Theorem A. Let m = m(x1, . . . , xg) be a meromorphic expression. The following are equivalent: (1) for every n, m(Ξ(n)) is either 0 or undefined;
(2) for every stably finite algebra A and a1, . . . , ag ∈ A, if there is a homomorphism C<
g a ua Theorem B. Let Y ∈ C . The inner rank of a matrix A over OY (resp. OY ) equals rk A(X) max : n ∈ ,X ∈ M ( )g in a neighborhood of Y . n N n C While TheoremB does not mention meromorphic germs or the universal property, they are crucial for its proof in Theorem 5.7 below. Our last result pertaining to analytic nc functions about the origin concerns the ac- g tion of GLn(C) on Mn(C) via simultaneous conjugation. Then the formal meromor- phic germs are closely related to meromorphic GLn(C)-concomitants (equivariant maps) g q :Mn(C) → Mn(C). More precisely, for every concomitant q there exist f1, f2 ∈ C<
ua Theorem C. If Y is a semisimple point, then OY contains a subalgebra isomorphic to ua ∼ S(Y ). Hence if S(Y ) = Ms(C), then OY = Ms(A) for some algebra A. 4 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
In particular, if a semisimple Y is not similar to a direct sum of scalar points, then there exist uniformly analytic nc functions f about Y such that f 6= 0 and f 2 = 0. The existence of nilpotent analytic nc functions about semisimple points poses questions about the structure of germ algebras that are endemic to the non-scalar case. Our main tool for answering them is the following novel Hermite interpolation result for nc functions. Theorem D. Let f be an nc function, S a finite collection of semisimple points in its domain, and L ∈ N. Then there exists a noncommutative polynomial p such that f and p agree on S up to their noncommutative differentials of order L. A more general version is given in Theorem 6.12, and the degree of the interpolating polynomial can be explicitly estimated. In contrast with other interpolation results for nc functions [Pop02, Pop08, BMV18], TheoremD is the first one to approximate nc functions with polynomials in non-scalar points up to higher order differentials. Also, TheoremD fails without the semisimplicity assumption; see [AM16] for an example where not even a value of an nc function at a non-semisimple point can be attained by a polynomial. The first consequence of our interpolation theorem is Corollary 6.18 which offers a deeper understanding of the formal germs in terms of the free algebra C
2. Preliminaries k Let be a field of characteristic 0 and fix g ∈ N. Let x = {x1, . . . , xg} be freely non- commuting variables. Let
Furthermore, we write ⊕nY for Y ⊕ · · · ⊕ Y with n summands. Also, ⊗ denotes both the tensor product over k and the Kronecker product of matrices.
2.1. Noncommutative functions. Let us follow the terminology and definitions of [K-VV14]. A noncommutative (nc) space over kg is kg G k g G kg n×n nc = Mn( ) = ( ) . n∈N n∈N k F k kg k g In particular, nc = n Mn( ). For Ω ⊆ nc we write Ωn = Ω ∩ Mn( ) . We say that kg Ω ⊆ nc is a noncommutative (nc) set if X ⊕ Y ∈ Ω for every X,Y ∈ Ω. A map f :Ω → knc on an nc set is a noncommutative (nc) function if
(1) f is graded, f(Ωn) ⊆ Mn(k) for all n; (2) f respects direct sums, f(X ⊕ Y ) = f(X) ⊕ f(Y ) for all X,Y ∈ Ω; −1 −1 (3) f respects similarities, f(SXS ) = Sf(X)S for all X ∈ Ωn and S ∈ GLn(k) −1 such that SXS ∈ Ωn. For a more thorough treatment of free analysis and noncommutative function theory see [Voi10, K-VV14, AM16].
2.2. Differential operators. An nc set Ω is (right) admissible if for every X ∈ Ωm, m×n g Y ∈ Ωn and Z ∈ (k ) there exists α ∈ k \{0} such that X αZ ∈ Ω . 0 Y m+n
Let f be an nc function on an admissible set Ω, Y ∈ Ωs and ` ∈ N. Then the `-th order (right) noncommutative (nc) differential operator at Y is the `-linear map ` k g ` k ∆Y f : (Ms( ) ) → Ms( ) determined by
1 1 1 ` 1 ` YZ f(Y ) ∆Y f(Z ) ··· ∆Y f(Z ,...,Z ) .. .. .. .. . . . . . . (2.1) f . = . . .. ` .. 1 ` Z ∆Y f(Z ) Y f(Y ) The particular block structure in (2.1) is due to f being noncommutative; cf. [K-VV14, 0 Theorem 3.11]. For convenience we write ∆Y f = f(Y ).
2.2.1. Ampliations. Let V,W be vector spaces over k and let T : V ` → W be an `-linear map. Then T can be viewed as a linear map T : V ⊗` → W . For every n we can naturally extend it to a linear map (V ⊗`)n×n → W n×n by block-wise application of T . By composing it with the canonical map (V n×n)⊗` → (V ⊗`)n×n n×n ` n×n we obtain an `-linear map Tn :(V ) → W . Whenever n is clear from the context, we simply write T instead of Tn. g ` As a special case, an `-linear map T : (Ms(k) ) → Ms(k) extends to an `-linear map g ` n×n T : (Mns(k) ) → Mns(k) as above using the identification Ms(k) = Mns(k). 6 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
Remark 2.1. In [K-VV14], this amplified map is denoted as 1 ` 1 ` Tn(Z ,...,Z ) = (Z s · · · s Z )T i g g s×s n×n for Z ∈ Mns(k) = ((k ) ) using the faux product s for n × n matrices over the tensor algebra T((kg)s×s). Returning to the differential operators, we have the following relation between ampli- fying points in the nc space and amplifying operators [K-VV14, Proposition 3.3]: ` 1 ` ` 1 ` (2.2) (∆⊕nY f)(Z ,...,Z ) = ∆Y f n (Z ,...,Z ). 2.2.2. Canonical intertwining conditions. Since nc functions respect direct sums and sim- ilarities, their differential operators satisfy certain intertwining conditions, which we de- scribe next. k g ∞ Definition 2.2. Let Y ∈ Ms( ) . A sequence (f`)`=0 of `-linear maps g ` f` : (Ms(k) ) → Ms(k) satisfies the canonical intertwining conditions with respect to Y (shortly IC(Y )) if
f1([S,Y ]) = [S, f0] and for ` ≥ 2, 1 `−1 1 `−1 1 2 `−1 f`([S,Y ],Z ,...,Z ) = Sf`−1(Z ,...,Z ) − f`−1(SZ ,Z ,...,Z ), j j+1 j−1 j j+1 f`(...,Z , [S,Y ],Z ,... ) = f`−1(...,Z ,Z S,Z ,... ) j j+1 j+2 − f`−1(...,Z ,SZ ,Z ,... ), 1 `−1 1 `−2 `−1 1 `−1 f`(Z ,...Z , [S,Y ]) = f`−1(Z ,...,Z ,Z S) − f`−1(Z ,...,Z )S j g for 1 ≤ j ≤ ` − 2 and all S ∈ Ms(k), Z ∈ Ms(k) . Remark 2.3. If Y ∈ kg, then IC(Y ) are void.
Remark 2.4. If f is an nc function on an admissible set Ω and Y ∈ Ωs, then ` ∞ ∆Y f `=0 satisfies IC(Y ) by [K-VV14, Remark 4.3]. 2.3. Topologies on a noncommutative space and analyticity of noncommutative k g functions. In this subsection let = C. We will consider two natural topologies on Cnc. g For X = (X1,...,Xg) ∈ Mn(C) let kXkn denote the maximum of the operator norms of Xj; when n is clear from the context, we simply write k · k. The norms k · kn on g g n×n g Mn(C) = (C ) correspond to C viewed as an operator space via Ruan’s theorem g ` k [Pau02, Pis03]. For an `-linear map T : (Ms(C) ) → Ms( ) we have 1 ` 1 ` kTnkns = max kTn(Z ,...,Z )kns : kZ kns = ··· = kZ kns = 1 . If
kT kcb = sup kTnkns n is finite, then T is completely bounded (in the sense of Christensen and Sinclair [Pau02, Chapter 17]; see also [K-VV14, Proposition 7.49]). When ` = 1, this definition of course agrees with the usual notion of a completely bounded linear map. Actually, the results of LOCAL THEORY OF FREE NC FUNCTIONS 7 this paper apply to any operator space structure on Cg, but we restrict to this standard one for the sake of simplicity.
g 2.3.1. Disjoint union topology. A subset Ω ⊆ Cnc is open in the disjoint union topology g if Ωn is open in the Euclidean topology on Mn(C) for all n ∈ N ([K-VV14, Section 7.1]; cf. fine topology [AM16, Example 3.3]). Let f be an nc function on an open nc set Ω. We say that f is locally bounded on Ω if for every X ∈ Ω, f is bounded on some open neighborhood of X. g Let Y ∈ Ms(C) . If f is a locally bounded nc function on some nc neighborhood of Y , then for every n ∈ N we have
r ` lim sup ` sup k∆⊕nY f(Z,...,Z)kns < ∞ `→∞ kZkns=1
g n and for all X in some Euclidean neighborhood Uns ⊆ Mns(C) of ⊕ Y , ∞ X ` n n (2.3) f(X) = ∆⊕nY f(X − ⊕ Y,...,X − ⊕ Y ) `=0 holds and the series (2.3) converges absolutely and uniformly on Uns [K-VV14, Theorem ∞ 7.8]. On the other hand, if (f`)`=0 is a sequence of multilinear maps satisfying IC(Y ) and r lim sup ` sup k(f`)n(Z,...,Z)kns < ∞ `→∞ kZkns=1 for all n, then the series ∞ X n n (2.4) f`(X − ⊕ Y,...,X − ⊕ Y ) `=0 g converges absolutely and uniformly for X in some open neighborhood Uns ⊆ Mns(C) of F g Y , for all n ∈ N [K-VV14, Theorem 8.8]. Furthermore, if n Uns ⊂ Cnc contains an nc set Ω that is open in the disjoint union topology, then (2.4) is a locally bounded nc function on Ω.
g 2.3.2. Uniformly open topology. For Y ∈ Ms(C) and ε > 0 let G g n g Bε(Y ) = {X ∈ Mns(C) : kX − ⊕ Y kns < ε} ⊂ Cnc n∈N be the noncommutative (nc) ball about Y of radius ε. By [K-VV14, Proposition 7.12], g nc balls form a basis of a topology on Cnc, which is called the uniformly open topology ([K-VV14, Section 7.2]; cf. fat topology [AM16, Example 3.5]). Let Ω be a uniformly open nc set. An nc function f is uniformly locally bounded on Ω if for every X ∈ Ω, f is bounded on some nc ball about X. Similarly to the disjoint union topology case, uniform local boundedness is related to uniform analyticity.
g Theorem 2.5 ([K-VV14, Theorems 7.21 and 8.11]). Let Y ∈ Ms(C) . If f is a uniformly locally bounded nc function on some uniformly open nc neighborhood of Y , then ∞ X ` n n (2.5) f(X) = ∆Y f(X − ⊕ Y,...,X − ⊕ Y ) `=0 8 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ for all X in some nc ball about Y , where the series (2.5) converges absolutely and uni- p` ` formly, and lim sup`→∞ k∆Y fkcb < ∞. ∞ Conversely, let (f`)`=0 be a sequence of multilinear maps satisfying IC(Y ) and
p` lim sup kf`kcb < ∞. `→∞ Then the series ∞ X n n (2.6) f`(X − ⊕ Y,...,X − ⊕ Y ) `=0 converges absolutely and uniformly for all X in some nc ball about Y , and (2.6) is a locally bounded nc function on that nc ball. The infinite sums (2.3) and (2.5) are called Taylor-Taylor series about Y [K-VV14, Chapter 4]; see also [Tay73, Voi10] for earlier accounts. 2.4. Noncommutative germs. To study the local behavior of (uniformly) analytic nc functions we define the following.
g Definition 2.6. Let Y ∈ Ms(C) . An analytic nc germ about Y is an equivalence class of analytic nc functions about Y , where two nc functions are equivalent if they agree on a disjoint union open nc neighborhood of Y . Analogously we define a uniformly analytic nc germ about Y (using the uniformly open topology). The C-algebras of a ua analytic and uniformly analytic nc germs about Y are denoted OY and OY , respectively. By Theorem 2.5, uniformly analytic nc germs about Y are in one-to-one correspondence with sequences of multilinear maps (f`)` satisfying IC(Y ) and
p` lim sup kf`kcb < ∞. `→∞ Similarly, analytic nc germs about Y embed into the set of sequences of multilinear maps
(f`)` satisfying IC(Y ) and r lim sup ` sup k(f`)n(Z,...,Z)kns < ∞ `→∞ kZkns=1 for all n (this embedding is indeed proper, see [K-VV14, Example 8.6]). More generally, g a sequence of multilinear maps (f`)` satisfying IC(Y ) for some Y ∈ Ms(k) is called a formal nc germ about Y . Formal nc germs are endowed with natural addition and k g convolution multiplication, thus forming a -algebra OY . For Y ∈ Ms(C) we have ua a OY ⊂ OY ⊂ OY , and these inclusions are strict; see [K-VV14, Example 8.14] or [Voi10, Section 17]. g In [K-VV14, Chapter 5] it is described in detail how formal nc germs about Y ∈ Ms(k) g can be viewed as germs of nc functions. We say that Z = (Z1,...,Zg) ∈ Mn(k) is jointly nilpotent if Z1,...,Zg generate a nilpotent k-algebra in Mn(k). Let G k g n kg Nilp(Y ) = {X ∈ Mns( ) : X − ⊕ Y is jointly nilpotent} ⊂ nc. n Then Nilp(Y ) is an admissible nc set, so every nc function on Nilp(Y ) admits nc differential operators, which form a formal nc germ. Conversely, every formal nc germ determines an nc function on Nilp(Y ) via Taylor-Taylor series. LOCAL THEORY OF FREE NC FUNCTIONS 9
kg ∼ k g Remark 2.7. If Y ∈ , then OY = <
3. Universal skew field of fractions of formal power series and the Amitsur-Cohn theorem for meromorphic identities In this section we construct noncommutative formal meromorphic functions as the elements of the universal skew field of fractions of noncommutative formal power series, see Subsection 3.3 and Theorem 3.8. Further, a meromorphic variant of Amitsur’s theorem [Ami66, Theorem 16] and Cohn’s theorem [Coh06, Theorem 7.8.3] for noncommutative rational functions is given in Theorem 3.9: a formal meromorphic expression vanishes under all finite-dimensional representations if and only if it vanishes in every stably finite algebra.
3.1. Meromorphic expressions and identities. The k-algebra k<
Definition 3.1. A formal meromorphic expression over x over k is an expression of the form m = r(s1, . . . , s`), where r is a formal rational expression in the letters y = (y1, . . . , y`) and s1, . . . , s` ∈ k<
φ : k<
φ : k<
Remark 3.3. As opposed to the PI theory, central simple algebras of the same degree do not satisfy the same “series” identities. For example, k and k((t)) are both 1-dimensional (commutative) fields; however, there is only one homomorphism k((t)) → k, while there k k P i k are several homomorphisms ((t)) → ((t)). As a consequence, i≥1 x1 is a MI for but not for k((t)).
3.2. Completion of the ring of generic matrices. Fix n ∈ N and let k((ξ)) be the field of fractions of the polynomial ring in gn2 variables k k k [[ξ]] = [[ξij : 1 ≤ i, j ≤ n, 1 ≤ k ≤ g]]. k k Let Ξk = (ξij)i,j be n × n generic matrices, and let GMn ⊂ Mn( [ξ]) be the algebra of generic matrices [Row80, Definition 1.3.5], i.e., the unital k-algebra generated by Ξ =
(Ξ1,..., Ξg). Let GMd n be the closure of GMn in Mn(k[[ξ]]). Equivalently, GMd n is the completion of GMn with respect to the ideal generated by Ξ, and its elements are formal power series in Ξ; cf. [GMS18] for an analytic tracial version. Since Mn(k((ξ))) is clearly a scalar extension of GMn and hence of GMd n, we conclude that GMd n is a prime ring. Its center is thus a domain; let UDdn be the ring of central quotients of GMd n. Since GMd n is a PI-ring, Posner’s theorem [Row80, Theorem 1.7.9] implies that UDdn is a central simple algebra of degree n. LOCAL THEORY OF FREE NC FUNCTIONS 11
Proposition 3.4. UDdn is a skew field.
Proof. Suppose that UDdn is not a skew field. Since it is a central simple algebra, we P∞ conclude that GMd n contains nilpotents. Let f ∈ GMd n be nilpotent. Write f = i=d fi, n n where fi ∈ GMn is homogeneous of degree i, and fd 6= 0. Then f = 0 implies fd = 0, which is a contradiction since GMn is a domain.
The skew field UDdn has a special role among the division algebras of degree n (Propo- sition 3.6 below), which will be important in subsequent construction of nc germs. First we require the following.
Lemma 3.5. For n ∈ N let k[ξ] ⊂ R ⊂ k[[ξ]] be the ring generated by the entries of ele- ments in GMd n. Let C be a commutative k-algebra. Then every continuous homomorphism GMd n → Mn(C) extends to a homomorphism Mn(R) → Mn(C).
Proof. Let φ : GMd n → Mn(C) be a homomorphism. Clearly there is a homomorphism 0 k 0 k φ :Mn( [ξ]) → Mn(C) such that φ|GMn = φ |GMn . Since Mn( [ξ]) is generated by GMn and Mn(k), this implies m m X X (3.1) a f a ··· f a = 0 ⇒ a φ(f )a ··· φ(f )a = 0 i1 i1 i2 i`i i`i+1 i1 i1 i2 i`i i`i+1 i=1 i=1 k k k for all aij ∈ Mn( ) and fij ∈ GMn. Because Mn(R) is the -subalgebra in Mn( [[ξ]]) 00 generated by GMd n and Mn(k), there is a homomorphism φ :Mn(R) → Mn(C) defined by m m X X a f a ··· f a 7→ a φ(f )a ··· φ(f )a i1 i1 i2 i`i i`i+1 i1 i1 i2 i`i i`i+1 i=1 i=1 k 00 for all aij ∈ Mn( ) and fij ∈ GMd n. Indeed, to show that φ is well-defined it suffices to 00 verify this on homogeneous elements in Mn(R), for which φ is well-defined by (3.1).
Proposition 3.6. Let m be a meromorphic expression and n ∈ N. If m(Ξ) = 0 in UDdn, then m is a MI for every division algebra of degree n. Proof. Let D be a division algebra of degree n. By applying PI theory to homogeneous components it follows that every continuous homomorphism φ : k<
r(ϕ1(s1(Ξ)), . . . , ϕ1(s`(Ξ))) = 0. Consequently,
φ(m) = r(φ(s1), . . . , φ(s`)) = r(ϕ(s1(Ξ)), . . . , ϕ(s`(Ξ))) = 0. 3.3. Construction of the skew field M and its universality. Every nc power series can be evaluated at a tuple of generic matrices, resulting in a matrix of commutative power series. Likewise, one can evaluate a formal rational expression of nc power series on a tuple of generic matrices, which either yields a matrix of fractions of commutative power series or is undefined due to a matrix singularity at some point of the calculation. The following type of construction first originated with noncommutative rational func- tions [HMV06]. Let M0 be the set of formal rational expressions over k<
0 Lemma 3.7. Let m ∈ M be an MI for UDdN for all N ∈ N. Then m represents 0 in the universal skew field of fractions of k<
0 Since U contains a skew field of fractions of GMd n<
d 0 X 0 (3.3) φ(m) tΞ = h! qww(Ξ ) dth t=0 |w|=h P k for every h ∈ N ∪ {0}. Let ph = |w|=h qww ∈ UDdn ⊗
(3) m is an FMI for every stably finite algebra.
Proof. (3) ⇒ (2) is trivial since every skew field is stably finite, and (2) ⇒ (1) follows by the construction of M. d×d (1) ⇒ (3) Let m = r(s1, . . . , s`). By [HMS18, Theorem 4.12] there exist Q ∈ k
4. Meromorphic GLn(C)-invariants As nc functions respect similarities, invariant theory plays an important role in free ˇ g analysis [KS17]. Let n ∈ N and consider the action of GLn(C) on Mn(C) given by a −1 g g X = aXa for X ∈ Mn(C) and a ∈ GLn(C). A map f :Mn(C) → Mn(C) is a GLn(C)-concomitant (or an equivariant map) if it intertwines with the action of g GLn(C) on Mn(C) and Mn(C). In parallel with the classical invariant theory, where UDn is identified with the ring of rational concomitants [Pro76, Sal99], we relate UDdn with meromorphic concomitants; see [Lum91, GMS18] for analytic concomitants. Consider the action of GLn(C) on Mn(C((ξ))) given by (4.1) f a = a−1f(aξa−1)a for f ∈ Mn(C((ξ))) and a ∈ GLn(C). Then f is invariant for this action if and only if it is a GLn(C)-concomitant. Observe that this action preserves Mn(C[[ξ]]) and its homogeneous GLn(C) components. By [Pro76, Theorem 2.1] it follows that elements of Mn(C[[ξ]]) are power series in products of words and traces of words in the tuple of generic matrices Ξ. We say that f ∈ C[[ξ]] is analytic if it converges absolutely and uniformly on some 2 neighborhood of 0 ∈ Cgn . Let O ⊂ C[[ξ]] be the subring of analytic series, and let M be its field of fractions. Let Un(C) ⊂ GLn(C) be the unitary group.
a Lemma 4.1. Let f ∈ C[[ξ]] and f(0) = 0. If f divides f for every a ∈ Un(C), then f = fh˜ for some h ∈ C[[ξ]]∗ and f˜ ∈ C[[ξ]]GLn(C). Moreover, if f ∈ O, then one can choose f,˜ h ∈ O. LOCAL THEORY OF FREE NC FUNCTIONS 15 P∞ Proof. Write f = i=d fi with d ≥ 1 and fi homogeneous of degree i. Consider the map f a λ :U ( ) → ∗, λ(a) = (0). n C C f Since ! f ab f ab f b f a b f b f a f b (0) = (0) = (0) (0) = (0) (0), f f b f f f f f
λ is a continuous group homomorphism. As every 1-dimensional representation of Un(C) factors through the determinant, we have λ(a) = det(a)t for some integer t. By (4.1) we see that ker λ contains all scalar matrices, so t = 0 and λ = 1. For every a ∈ Un(C) there exist homogeneous polynomials ha,` of degree ` for ` ∈ N such that ! a X f = 1 + ha,i f. i≥1 For each ` ≥ 1 we thus have `−1 a X (4.2) ha,`fd = fd+` − fd+` − ha,ifd+`−i. i=1
By induction on ` we see from (4.2) that the map a 7→ ha,` from Un(C) to the space of homogeneous polynomials of degree ` is continuous with respect to the Euclidean topology. Hence we can define homogeneous polynomials of degree ` Z k` = ha,` dµ(a), where µ is the (right) Haar measure on Un(C). Let ∞ Z ˜ X a X f = fi dµ(a) , h = 1 + ki. i=d i≥1 ˜ ˜ By (4.2) we have hf = f and f is Un(C)-invariant by construction. Furthermore, Un(C) ˜ is Zariski dense in GLn(C) by the unitarian trick [Pro07, Corollary 8.6.1], so f is also GLn(C)-invariant. Now suppose f is analytic. Then there is a neighborhood D of the origin such that f a converges absolutely and uniformly on D for all a ∈ Un(C). For every α ∈ D we have Z f˜(α) = f a(α) dµ(a), so f˜ also converges absolutely and uniformly on D. Since fh−f˜ = 0 and f, f˜ are analytic, h is also analytic, e.g. by Artin’s approximation theorem [Art68, Theorem 1.2].
GLn(C) GLn(C) Lemma 4.2. If r ∈ Mn(C((ξ))) , then r = f0/f for some f0 ∈ Mn(C[[ξ]]) and f ∈ C[[ξ]]GLn(C). GLn(C) GLn(C) GLn(C) Moreover, if r ∈ Mn(M) , then one can choose f0 ∈ Mn(O) and f ∈ O .
i j Proof. Because Mn(C((ξ))) has a GLn(C)-invariant basis {Ξ1Ξ2 : 1 ≤ i, j ≤ n}, it suffices to assume r ∈ C((ξ))GLn(C). Since C[[ξ]] is a unique factorization domain, we can write GLn(C) r = f0/f for some coprime f0, f ∈ C[[ξ]]. If r∈ / C[[ξ]] , then f(0) = 0. For every a a a a a ∈ GLn(C) we have r = r and hence f0 f = f f0, so f divides f . By Lemma 4.1 16 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
∗ ˜ GLn(C) ˜ −1 ˜ there exist h ∈ C[[ξ]] and f ∈ C[[ξ]] such that f = hf. Then r = (h f0)/f and −1 GLn(C) h f0 ∈ C[[ξ]] . Exactly the same reasoning applies in the analytic situation since O is also a unique factorization domain [Kra92, Proposition 6.4.9].
Proposition 4.3. There exists a homogeneous polynomial in the center of GMn such GLn(C) −1 that every s ∈ Mn(C[[ξ]]) can be written as s = p q for some q ∈ GMd n. If GLn(C) s ∈ Mn(O) , then q ∈ GMd n ∩ Mn(O).
Proof. Let Tn be the subalgebra of Mn(C[ξ]) generated by GMn and tr(GMn). Then GLn(C) Mn(C[ξ]) = Tn by [Pro76, Theorems 1.3 and 2.1]. By [Row80, Theorem 4.3.1] there 2 is a multilinear polynomial h in n variables that is a central polynomial for GMn and h(GMn,..., GMn) GMn is an ideal in Tn. Since h is central for GMn, there exist homoge- neous r1, . . . , rn2 ∈ GMn such that p := h(r1, . . . , rn2 ) 6= 0. Therefore p is homogeneous, lies in the center of GMn and p Tn ⊂ GMn. GLn(C) If s ∈ Mn(C[[ξ]]) , then its homogeneous components si are also GLn(C)-invariant −1 and thus belong to Tn. Therefore si = p qi for some homogeneous qi ∈ GMn. Hence P −1 q = i≥0 qi ∈ GMd n and s = p q. Furthermore, if entries of s are analytic, then so are the entries of q = ps.
GLn(C) Theorem 4.4. Mn(C((ξ))) = UDdn. Moreover, every meromorphic GLn(C)-conco- −1 mitant equals f1(Ξ)f2(Ξ) for some analytic power series f1(Ξ), f2(Ξ) in Ξ.
GLn(C) Proof. Clearly UDdn ⊆ Mn(C((ξ))) holds since UDdn is the ring of central quotients of GMd n. Conversely, every GLn(C)-invariant r ∈ Mn(C((ξ))) can be written as r = f0/f GLn(C) GLn(C) for f0 ∈ Mn(C[[ξ]]) and f ∈ C[[ξ]] by Lemma 4.2, so r ∈ UDdn by Proposition 4.3. The second statement follows in same manner by the analytic parts of Lemma 4.2 and Proposition 4.3. The results of this section complement the earlier research of Luminet [Lum91] concern- ing the rings of analytic concomitants on matrices of fixed size n. Therein it is shown that g meromorphic concomitants arising from germs about an irreducible point X ∈ Mn(C) (see Section6 for the definition of irreducibility) are isomorphic to M n(K), where K is the field of fractions of analytic commutative power series in (g − 1)n2 + 1 indeterminates [Lum91, Propositions 5.3 and 6.2]. On the other hand, the results of this section connect meromorphic concomitants arising from germs about the origin (which is far from being irreducible) with the restrictions of nc meromorphic germs to n × n matrices. The rest of this papers deals with analytic nc germs in the size-independent setting.
5. Universal skew field of fractions of analytic germs In this section we show that the ring of (uniformly) analytic germs about a scalar point Y ∈ Cg admits a universal skew field of fractions, which we call the skew field of nc (uniformly) meromorphic germs; see Subsection 5.2. This theory is used in the local- a global rank principle, Theorem 5.7, to relate the intrinsic rank of matrices over OY or ua OY with the ranks of their matrix evaluations. We note that there is no commutative analog of this statement. LOCAL THEORY OF FREE NC FUNCTIONS 17
g In Section6 we will see that for Y ∈ Ms(C) with s ≥ 2, the algebras of germs about Y depend strongly on Y . However, for every Y ∈ Cg we have ua ∼ ua a ∼ a ua a OY = O0 , OY = O0, O0 ⊂ O0 ⊂ C<
a ua Lemma 5.1. O0 and O0 are local rings; in both cases, the maximal ideal consists of functions vanishing at the origin.
a Proof. The statement clearly holds for O0 because commutative analytic germs form a ua local ring. Now let f ∈ O0 be such that f(0) 6= 0. Since f is continuous with respect to 1 the uniformly open topology, there exists ε > 0 such that kf(0) − f(X)k < 2|f(0)| for all X ∈ Bε(0). Then ∞ ` X f(0) − f(X) f −1(X) = f(0)−1 f(0) `=0 −1 ua is an nc function that converges absolutely and uniformly on Bε(0), so f ∈ O0 . 5.1. Semifir property and inertness. For j = 1, . . . , g define linear operators ! X X Lj : C<
A composite of Lj’s is called a right transduction [Coh06, Section 2.5] or a left back- ward shift [K-VV12, Section 4.2].
a ua Lemma 5.2. Right transductions preserve O0 and O0 .
a g Proof. If f ∈ O0 and α ∈ C , then X 0 f(X) 0 (5.1) f = P α 0 j αjLj(f)(X) f(0) whenever defined. Since f is analytic on some open nc neighborhood of 0, (5.1) implies a ua Lj(f) ∈ O0 for every j. The same reasoning applies to O0 . An embedding of rings R ⊂ S is totally inert [Coh06, Section 2.9] if for every d ∈ N, 1×d d×1 −1 U ⊂ S , V ⊂ S satisfying UV ⊂ R there exists P ∈ GLd(S) such that for u ∈ UP and 1 ≤ i ≤ d, either ui ∈ R or vi = 0 for all v ∈ PV ; and analogously for v ∈ PV .
a ua a Proposition 5.3. O0 and O0 are semifirs. Moreover, the embeddings O0 ⊂ C<
a ua Proof. Since O0 and O0 are local rings whose invertible elements are precisely functions a ua non-vanishing at the origin, and right transductions preserve O0 and O0 by Lemma 5.2, they are semifirs by [Coh06, Proposition 2.9.19]. In particular, they are semihereditary rings [Coh06, Section 2.1], their maximal ideals m are finitely generated as right ideals, and T n a ua n m = 0. Therefore O0 ⊂ C<
5.2. Meromorphic noncommutative germs. Next we construct universal skew fields a ua a ua of fractions M0 and M0 of O0 and O0 , respectively. Note that we already know they a a exist since O0 and O0 are semifirs. Since these constructions are nearly identical, we consider in detail only the case of analytic germs. One can consider evaluations of formal rational expressions of analytic germs on tuples a0 of matrices near the origin. Let M0 be the set of those expressions that are defined at a0 g some tuple of matrices. Observe that if m ∈ M0 is well defined at some X ∈ Mn(C) , then g the restriction of m to Mn(C) is an n × n matrix of commutative meromorphic functions whose numerators and denominators are analytic about the origin. Then we impose a a0 relation ∼ on M0 such that m1 ∼ m2 if m1(X) = m2(X) for all X in some neighborhood of the origin where m1(X) and m2(X) are defined. By analyticity we see that ∼ is a well- defined equivalence relation; the equivalence classes are called meromorphic nc germs. Since meromorphic commutative germs embed into the field of fractions of commutative power series, meromorphic nc germs form a skew field by Proposition 3.4; we denote it a M0. a By Proposition 5.3, O0 ⊂ C<
a ua a ua Corollary 5.4. M0 (resp. M0 ) is a universal skew field of fractions of O0 (resp. O0 ). a ua a ua Since O0 (resp. O0 ) is a semifir, every element of M0 (resp. M0 ) can be represented as (5.2) utQ−1v d a d×d ua d×d for some u, v ∈ C and a full matrix Q ∈ (O0) (resp. Q ∈ (O0 ) ) as a consequence of [Coh06, Corollary 7.5.14]. As we will see in Theorem 5.7 below, such a Q is of full rank if g and only if Q(X) is invertible for some X ∈ Cnc in a (uniformly) open neighborhood of the origin. If (uniformly) analytic nc germs are given in the form (5.2), then their arithmetic operations can be defined in the same way as for realizations (or linear representations) of nc rational functions [CR94, BGM05, Vol18, HMS18]. In the case of nc rational functions, Q can be chosen to be affine, and realizations (5.2) with an affine Q of minimal size d exhibit good properties: they are efficiently computable, essentially unique, and the domain of an nc rational function is given as the invertibility set of Q. On the other hand, in the (uniformly) analytic case it is unclear whether any of these properties carry over. Next we show that evaluations of uniformly meromorphic nc germs make sense in arbitrary stably finite Banach algebras, e.g. C∗-algebras with a faithful trace. g ⊗` Let h :(C ) → C be a linear map. Then khkcb is the norm of this functional with respect to the Haagerup norm on (Cg)⊗`; see [Pau02, Chapter 17] or [Pis03, Chapter 5]. ⊗` Now let A be a Banach algebra. If µ : A → A is given by µ(a1 ⊗ · · · ⊗ a`) = a1 ··· a`, then kµk = 1, where A⊗` is endowed with the projective cross norm [Rya02, Section 2.1]. Let A g ` g ⊗` ⊗` h⊗µ h :(A ) → (C ) ⊗ A −−→ C ⊗ A = A. LOCAL THEORY OF FREE NC FUNCTIONS 19
g g ⊗` If A is endowed with the `∞ norm (with respect to the norm on A) and (A ) is endowed with the projective cross norm, then (Ag)⊗` → (Cg)⊗` ⊗ A⊗` is a contraction. Hence the `-linear map hA satisfies
A (5.3) kh (a1, . . . , a`)k ≤ khkcbka1k · · · ka`k for ai ∈ A. ua For f ∈ O0 denote 1 δ(f) := > 0. p` ` lim sup`→∞ k∆0fkcb ` If A is a Banach algebra, then by applying (5.3) to h = ∆0f we see that f converges absolutely and uniformly on {a ∈ Ag : kak < δ(f)} .
Corollary 5.5. Let m be a meromorphic expression built of uniformly analytic germs ua about the origin. If m represents 0 in M0 , then there exists ε > 0 such that for every stably finite Banach algebra A and X ∈ Ag satisfying kXk < ε, m(X) is either undefined or m(X) = 0.
ua Proof. Let m = r(s1, . . . , s`) for sk ∈ O0 . As in the proof of Theorem 3.9, there exists A ∈ C
ε := min{δ(sk), δ(Bij), δ(Cij): i, j, k} > 0.
g Let X ∈ A be such that kXk < ε and m(X) is defined. Then sk and the entries of B,C converge at X. Since (A◦s)(X) = B(X)C(X) is not full, m(X) = 0 by the previous paragraph. Remark 5.6. Let Ω 3 0 be an nc set, open and connected in the disjoint union (uniformly a ua open) topology. Then the (uniformly) analytic nc functions on Ω embed into O0 (O0 ), a ua so they generate a skew field of fractions inside M0 (M0 ), whose elements deserve to be called (uniformly) meromorphic nc functions; cf. [AM15, Section 10].
5.3. Local-global rank principle. As a consequence of our construction of the universal a ua skew fields of O0 and OY we obtain the following theorem relating the inner rank of a matrix over a germ algebra with the maximal ranks of its evaluations on a neighborhood of the origin.
a ua Theorem 5.7. The inner rank of a matrix A over O0 (resp. O0 ) equals rk A(X) (5.5) max : n ∈ ,X ∈ M ( )g in a neighborhood of 0 . n N n C 20 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
Proof. Let A be a d×e matrix and let r denote (5.5). Clearly we have ρ(A) ≥ r. Without loss of generality assume d ≤ e. a First we deal with the case ρ(A) = d, i.e., A is a full matrix. Since O0 is a semifir a a and M0 is its universal skew field of fractions by Corollary 5.4, A has full rank over M0. 0 a 0 Therefore there exists a d × (d − e) matrix A over M0 such that (AA ) is invertible, 0 a so there is a d × d matrix B such that (AA )B = I. By the construction of M0, there exists n ∈ N such that each entry of A0 and B is well-defined at some tuple of n × n g 0 matrices. Furthermore, when restricted to Mn(C) , A and B are matrices of commutative 0 meromorphic functions, so there exists X ∈ Mn(C) such that A (X),B(X) are well- defined. Therefore (AA0)(X)B(X) = I implies rk A(X) = dn and hence r = ρ(A). a Now suppose ρ(A) < d. Then there exist full matrices B and C over M0 of dimensions d × ρ(A) and ρ(A) × e, respectively, such that A = BC. By the previous paragraph g g there exist X ∈ Mm(C) and Y ∈ Mn(C) such that rk B(X) = mρ(A) and rk C(Y ) = nρ(A). Then rk B(⊕nX) = (m + n)ρ(A) = rk C(⊕mY ). Since the restrictions of B g and C to Mm+n(C) are matrices of commutative meromorphic functions, there exists g Z ∈ Mm+n(C) such that rk B(Z) = (m + n)ρ(A) = rk C(Z). Since d > ρ(A), we have ker B(Z) = {0} and therefore rk A(Z) = (m + n) rk C(Z) = (m + n)ρ(A). Hence r = ρ(A). Remark 5.8. There is no commutative analog of Theorem 5.7. Consider the matrix 0 t3 −t2 k 3×3 A = −t3 0 t1 ∈ [t1, t2, t3] t2 −t1 0 from [Coh06, Section 5.5]. Its inner rank over k[[t1, t2, t3]] equals 3. Indeed, suppose that 3×2 2×3 A = BC for B ∈ k[[t1, t2, t3]] and C ∈ k[[t1, t2, t3]] . Then B(0)C(0) = 0 because A is linear, so at least one of the scalar matrices B(0),C(0) is of rank at most 1. Without loss of generality let rk B(0) ≥ 1. Then there exist U ∈ k2×3 of full rank and v ∈ k3 \{0} such that UB(0) = 0 and C(0)v = 0. Then A = BC and linearity of A imply UAv = 0. However, a short calculation shows that this is impossible. On the other hand, we claim that
rk A(T ) 54 ≤ < 3 n 19 k 3 for all n ∈ N and triples T ∈ Mn( ) of commuting matrices (this bound is presumably 9 not optimal). Write κ = 19 and let T be an arbitrary triple of commuting n × n matrices. κn If κn ≤ rk T1 + rk T2 + rk T3, then maxi rk Ti ≥ 3 . Since T1 A(T ) T2 = 0, T3
κn 54 it follows that rk A(T ) ≤ 3n − 3 = 19 n. Now suppose κn ≥ rk T1 + rk T2 + rk T3. Then dim ker Ti ≥ (1 − κ)n for all i, so
dim (ker Ti ∩ ker Tj) ≥ 2(1 − κ)n − n = (1 − 2κ)n. LOCAL THEORY OF FREE NC FUNCTIONS 21
Since ker T2 ∩ ker T3 A(T ) ⊇ ker T1 ∩ ker T3 , ker T1 ∩ ker T2 54 it follows that rk A(T ) ≤ 3n − 3(1 − 2κ)n = 19 n. Corollary 5.9. If A is a matrix over k<
Proof. Apply the same arguments as in the proof of Theorem 5.7. 5.4. Level-wise meromorphic functions. One might be tempted to assert that every a uniformly analytic level-wise meromorphic function is an element of M0. However, in this subsection we provide an example of an nc function f with the following properties: 3 (1) f is defined on Cnc ∩ {det X3 6= 0} and uniformly bounded on some nc ball about every point therein; 3 (2) f is level-wise rational; that is, when restricted to Mn(C) , f equals pn/qn for a matrix polynomial pn and a scalar polynomial qn; ua (3) f∈ / M0 . For s ∈ N let X π(1)−1 π(2)−1 π(s+1)−1 hs = sign(π)x1 x2x1 x2 ··· x1 x2. π∈Ss+1 s(s+1) Then hs is a homogeneous polynomial of degree 2 + s with (s + 1)! terms and hs 2 vanishes on Ms(C) by [Row80, Proposition 1.1.33]. Define ∞ X 1 f = h (x , x )x−s. (s + 1)!(s2)! s 1 2 3 s=1 3 3 Then f is an nc function on Cnc ∩ {det X3 6= 0} and for X ∈ Mn(C) , n−1 X 1 (5.6) f(X) = h (X ,X )X−s. (s + 1)!(s2)! s 1 2 3 s=1 Note that the denominator of (5.6) is a homogeneous scalar polynomial of degree n(n−1). The factor (s + 1)!(s2)! ensures uniform convergence. m ua m Now let ∈ M0 be arbitrary. By the induction on the inversion height of it is m 3 easy to see that there exists d ∈ N such that the denominator of restricted to Mn(C) ua has order at most dn, for all n ∈ N. Therefore f∈ / M0 .
6. Germs about semisimple points and Hermite interpolation In this section we turn our attention to germs about semisimple (non-scalar) points. We establish a noncommutative Hermite interpolation result, Theorem 6.12, which states that values and finitely many differentials of an arbitrary nc function at a finite set of semisimple points can be interpolated by an nc polynomial. Furthermore, we identify O as an inverse limit O = lim k
1 YZ . . .. .. k 1 ` I`(Y ) = f ∈
Remark 6.1. The following hold if Y is semisimple:
(i) C(Y ) and S(Y ) are semisimple algebras, and the centralizer of C(Y ) in Ms(k) equals S(Y ) by the double centralizer theorem [Pro07, Theorem 6.2.5];
(ii) every C(Y )-bimodule homomorphism Ms(k) → Ms(k) is given by
m X X 7→ batXat t=1 q for some bat, at ∈ S(Y ). This follows from (i) by a standard argument. Finally, semisimple points Y 1,...,Y h are separated if none of the irreducible blocks q in Y i is similar to an irreducible block in Y j, for 1 ≤ i, j ≤ h. In this case we have
S(Y 1 ⊕ · · · ⊕ Y h) = S(Y 1) ⊕ · · · ⊕ S(Y h), C(Y 1 ⊕ · · · ⊕ Y h) = C(Y 1) ⊕ · · · ⊕ C(Y h).
At this point we can state the first structural result on germs about matrix points (cf. Corollaries 6.18 and 7.5 below).
a ua Proposition 6.2. If Y is irreducible then OY is a prime ring (and likewise for OY , OY if k = C).
g Proof. Let Y ∈ Mn(k) and a = (a`)`, b = (b`)` ∈ OY \{0}. Then there exist minimal 0 00 m , m ≥ 0 such that am0 6= 0 and bm00 6= 0. Since am0 and bm00 are nonzero multilinear 0 00 maps into Mn(k), there exists f0 ∈ Mn(k) such that am0 f0bm00 is a nonzero (m + m )- linear map. Since Y is irreducible, there exists f ∈ k
(afb)m0+m00 = am0 f0bm00 6= 0 and therefore afb 6= 0. Hence OY is a prime ring. The same proof applies in the analytic ua a case since C
6.1. Truncated canonical intertwining conditions. Next we define canonical inter- twining conditions for finite sequences of multilinear maps. k g L Definition 6.3. Let Y ∈ Ms( ) and L ∈ N. A sequence (f`)`=0 of `-linear maps g ` f` : (Ms(k) ) → Ms(k) satisfies the truncated canonical intertwining conditions of order L with respect to j g Y (shortly ICL(Y )) if for all Z ∈ Ms(k) ,
f1([S,Y ]) = [S, f0] and 1 `−1 1 `−1 1 2 `−1 f`([S,Y ],Z ,...,Z ) = Sf`−1(Z ,...,Z ) − f`−1(SZ ,Z ,...,Z ), j j+1 j−1 j j+1 f`(...,Z , [S,Y ],Z ,... ) = f`−1(...,Z ,Z S,Z ,... ) j j+1 j+2 − f`−1(...,Z ,SZ ,Z ,... ), 1 `−1 1 `−2 `−1 1 `−1 f`(Z ,...Z , [S,Y ]) = f`−1(Z ,...,Z ,Z S) − f`−1(Z ,...,Z )S for all 2 ≤ ` ≤ L, 1 ≤ j ≤ ` − 2, S ∈ Ms(k), and 1 1 (6.1) SfL(Z ,... ) = fL(SZ ,... ), j j+1 j j+1 (6.2) fL(...,Z S,Z ,... ) = fL(...,Z ,SZ ,... ), `−1 `−1 (6.3) fL(...,Z S) = fL(...,Z )S for all 1 ≤ j ≤ L − 1, S ∈ C(Y ).
We say that (f0) satisfies IC0(Y ) if [f0, C(Y )] = 0.
∞ L Remark 6.4. A sequence (f`)`=0 satisfies IC(Y ) if and only if (f`)`=0 satisfies ICL(Y ) for all L ∈ N ∪ {0}. g For Y ∈ Ms(k) we consider Ms(k) as a C(Y )-bimodule in a natural way. Since C1[S,Y ]C2 = [C1SC2,Y ] for S ∈ Ms(k) and Ci ∈ C(Y ), [Ms(k),Y ] is a sub-bimodule g in Ms(k) . k g k g ` k Definition 6.5. Let Y ∈ Ms( ) and ` ∈ N. An `-linear map f : (Ms( ) ) → Ms( ) is Y -admissible if it induces a C(Y )-bimodule homomorphism
g ⊗C(Y )` Ms(k) /[Ms(k),Y ] → Ms(k). Remark 6.6. By comparing Definitions 6.3 and 6.6 we see that an `-linear map f is Y - admissible if and only if (0,..., 0, f) satisfies IC`(Y ). 6.2. Noncommutative algebra intermezzo. Throughout this subsection let C be a semisimple k-algebra. When addressing properties of C-bimodules, we can identify them as (left) C ⊗ Cop-modules, where Cop is the opposite algebra of C. Here Cop agrees with C as a vector space over k, and multiplication satisfies aop · bop = (b · a)op. Since tensor product is distributive over direct sum, the k-algebra C ⊗ Cop is also semisimple, so every C ⊗ Cop-module is semisimple by [Pro07, Proposition 6.2.2], i.e., a direct sum of simple (or irreducible) modules. Furthermore, there are only finitely many simple C ⊗ Cop- modules up to isomorphism, say W1,...,Wd. By Schur’s lemma [Pro07, Theorem 6.1.7], EndC⊗Cop (Wi,Wi) is a finite dimensional division algebra over k, and HomC⊗Cop (Wi,Wj) = {0} for i 6= j. 24 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
Let U, V be finitely generated C ⊗ Cop-modules. Then
∼ M mi ∼ M ni U = Wi ,V = Wi i i
mi ni for some mi, ni, and Ui = Wi and Vi = Wi are isotypic components of type i of U and V , respectively [Pro07, Subsection 6.2.3]. By [Pro07, Proposition 6.2.3.1] we have
M ∼ M ni×mi (6.4) HomC⊗Cop (U, V ) = HomC⊗Cop (Ui,Vi) = EndC⊗Cop (Wi,Wi) . i i
Lemma 6.7. Let U, V be finitely generated C-bimodules. Let T ⊆ HomC−C(U, V ) be a subspace such that: (1) for every 0 6= u ∈ U there exists T ∈ T such that T u 6= 0;
(2) Φ ◦ T ⊆ T for every Φ ∈ EndC−C(V,V ).
Then T = HomC−C(U, V ).
Proof. First assume that k is algebraically closed. Then EndC⊗Cop (Wi,Wi) = k for all ni×mi i. By (6.4) it suffices to show that L ∈ T for every L ∈ HomC−C(Ui,Vi) = k and i = 1, . . . , d. Denote n = rk L. Then there exist u1, . . . , un ∈ Ui such that L(u1),...,L(un) are linearly independent in Vi. Clearly u1, . . . , un are linearly independent. By (2) it suffices to find T ∈ T such that T (u1),...,T (un) are linearly independent. To simplify the notation we without loss of generality assume U = Ui and V = Vi for a fixed i. Suppose T (u1),...,T (un) are linearly dependent for all T ∈ T . For i = 1, . . . , n let
φi : T → V,T 7→ T (ui).
Then φ1(T ), . . . , φn(T ) are linearly dependent for all T , so by [BS99, Theorem 2.2] there exist α1, . . . , αn ∈ k, not all 0, such that ! X (6.5) rk αiφi ≤ n − 1. i P Let u = i αiui ∈ V . If u 6= 0, then for every v ∈ V there exists T ∈ T such that T u = v by (1) and (2). However, this contradicts (6.5) since n − 1 < dim V . Therefore u = 0 and u1, . . . , un are linearly dependent, a contradiction. Hence there exists T ∈ T such that T (u1),...,T (un) are linearly independent. Finally, let k be an arbitrary field of characteristic 0. Suppose the conclusion of the lemma fails, i.e.,
(6.6) dimk T < dimk HomC−C(U, V ). Let k be the algebraic closure of k. Then the k ⊗ C-bimodules k ⊗ U, k ⊗ V and the subspace k ⊗ T satisfy the assumptions of the lemma, so k k k (6.7) dimk( ⊗ T ) = dimk Homk⊗C−k⊗C( ⊗ U, ⊗ V ). However, (6.6) and (6.7) contradict k k k dimk( ⊗ T ) = dimk T , dimk Homk⊗C−k⊗C( ⊗ U, ⊗ V ) = dimk HomC−C(U, V ).
LOCAL THEORY OF FREE NC FUNCTIONS 25
Lemma 6.8. Let U, V be finitely generated C-bimodules, and let A be a simple algebra containing C as a subalgebra. For every φ ∈ HomC−C(U ⊗C V, A) there exist m ∈ N and φbt ∈ HomC−C(U, A), φt ∈ HomC−C(V, A) for 1 ≤ t ≤ m such that m q X (6.8) φ(u ⊗C v) = φbt(u)φt(v) t=1 for all u ∈ U and v ∈ V . q
Proof. By distributivity of ⊗C and HomC−C over direct sum it suffices to assume that U and V are simple C −C-bimodules. Moreover, by [Pro07, Corollary 6.1.9.1] we can further op op assume that U = L1 ⊗ L2 and V = L3 ⊗ L4 for some minimal left ideals Li ⊂ C. By [Pro07, Theorem 6.3.1(2)] we have Li = Cci for some idempotents ci ∈ C. We distinguish op two cases. If c2c3 = 0, then L2 ⊗C L3 = {0} and U ⊗C V = {0}, so the lemma trivially holds. Hence assume c2c3 6= 0, and let a = φ (c1 ⊗ c2) ⊗C (c2 ⊗ c4) ∈ A.
Since A is simple, there exist bat, at ∈ A such that X (6.9) a = a c c a q bt 2 3 t t q Define φbt ∈ HomC−C(U, A) and φt ∈ HomC−C(V, A) by
φbt(c1 ⊗ c2q) = c1batc2, φ(c2 ⊗ c4) = c3atc4. Since φ is a C-bimodule homomorphism and c are idempotents, we have c ac = a and i q q 1 4 thus (6.8) holds by (6.9). In Section7 we will also require the following fact.
Lemma 6.9. Let A be a central simple k-algebra containing C as a subalgebra. Then
HomC−C(U, A) 6= {0} for every nonzero C-bimodule U.
op Proof. Since A is a central simple algebra over k, we have A⊗A ∼= Endk(A). Therefore A is a faithful left A ⊗ Aop-module, i.e., for every a ∈ A ⊗ Aop \{0} there exists m ∈ A such that a · m 6= 0. Then A is also a faithful left C ⊗ Cop-module. Every simple C ⊗ Cop- module is isomorphic to a minimal left ideal in C ⊗ Cop by [Pro07, Corollary 6.1.9.1]. On the other hand, every minimal left ideal in C ⊗ Cop is isomorphic to a C ⊗ Cop-submodule of A since A is faithful. Since every C ⊗ Cop-module U is a direct sum of simple modules op by semisimplicity, there exists a nonzero C ⊗ C -homomorphism U → A.
6.3. Hermite interpolation. We prove our main interpolation result, Theorem 6.12, using the algebraic tools derived in the previous subsection.
g g Lemma 6.10. Let Y ∈ Ms(k) be a semisimple point and Z ∈ Ms(k) \[Ms(k),Y ]. Then there exists f ∈ k
Proof. Suppose YZ f(Y ) = 0 =⇒ f = 0 0 Y for all f ∈ k
Therefore [Pi1,Y ] = 0 for i ∈ {1, 2}. Since P is invertible, there exists A ∈ Ms(k) such that P21 + AP11 ∈ GLs(k). Moreover, since P11,P21 ∈ C(Y ) and C(Y ) is semisimple, one can choose A ∈ C(Y ). Then we can replace P with I 0 P P P = 11 12 AI P21 + AP11 P22 + AP12 and the relation (6.10) still holds. So we can without loss of generality assume that P21 −1 is invertible. Furthermore, (6.10) implies P21Z = [Y,P22]. Therefore Z = [Y,P21 P22], a contradiction. i k g Proposition 6.11. Let ` ∈ N. For i = 1, . . . , h let Y ∈ Msi ( ) be separated semisimple g ` i points, and let fi : (Ms(k) ) → Ms(k) be Y -admissible `-linear maps. Then there exists f ∈ k
i 1 Therefore for each i there exists fbi ∈ I0(Y ) such that fi = ∆Y i fbi. Furthermore, since Y 1,...,Y h are separated, the algebra S(Y 1 ⊕ · · · ⊕ Y h) contains
(⊕i−10) ⊕ I ⊕ (⊕h−i0)
i0 for i = 1, . . . , h. Therefore there exist fi ∈ k
1 ` X 1 2 ` q fi(Z ,...,Z ) = fbit(Z )fit(Z ,...,Z ) t for all 1 ≤ i ≤ h. By the basis of induction andq the induction hypothesis there exist T i T i fbt ∈ i I0(Y ) and ft ∈ i I`−2(Y ) such that
1 `−1 ∆ i f = f , ∆ f = f q Y bt bit Y i t it for all i. Then q q Y i Z1 0 ··· 0 . i 2 . 0 Y Z . . . . (fbtft) . .. .. 0 . . . .. Z` q 0 ······ 0 Y i equals
1 0 fbit(Z ) ∗ · · · ∗ ∗········· ∗ . . 2 ` . 0 ∗ · · · . 0 0 ··· 0 fit(Z ,...,Z ) . ...... . .. . . . . . . 0 . . q . . .. . .. . . . ∗ 0 ··· 0 ··· 0 0 ········· 0
1 2 ` 0 ······ 0 fbit(Z )fit(Z ,...,Z ) . . . .. 0 . q . = . . . .. . . . . 0 ········· 0
P k by Remark 2.4. Therefore f = t fbtft ∈
` (i) (6.12) ∆Y i f = f` for all 1 ≤ i ≤ h and 0 ≤ ` ≤ L. 28 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
Proof. We prove the statement by induction on L. The basis of induction L = 0 holds (i) i (i) i because [f0 , C(Y )] = 0 if and only if f0 ∈ S(Y ) by Remark 6.1(i). Now assume that the statement holds for L − 1. Then there exists fb ∈ k
f = 3(x1x2 − x2x1) + 2(x2x1x2x1 − x1x2x1x2)
` ` satisfies ∆Y f = ∆Y r for ` ∈ {0, 1}. By brute force one can also check that a minimal- ` ` degree polynomial f satisfying ∆Y f = ∆Y r for ` ∈ {0, 1, 2} has degree 8. Remark 6.14. One can also derive polynomial bounds on the degree of f as in Theorem ` 6.12. By Remark 2.4, the maps f` = ∆Y f for ` ≤ L are determined by the action of f on 2 the Lgsi tuples Y i Z1 . . .. .. k g ∈ M(L+1)si ( ) .. L . Z Y i 1 L k g ⊗L for all Z ⊗ · · · ⊗ Z in some basis for (Msi ( ) ) and 1 ≤ i ≤ h. Write
h X 3 N = L(L + 1)g si i=1 g and let T ∈ MN (k) be the direct sum of these tuples. Let A ⊂ MN (k) be the algebra generated by T1,...,Tg. By [Shi19, Theorem 3], A is generated by polynomials in T k of degree 2N log2 N + 4N − 4. Therefore there exists fe ∈
Remark 6.16. For arbitrary points Y 0 and Y 00 there is a canonical k-algebra homomor- phism
(6.13) OY 0⊕Y 00 → OY 0 × OY 00 . Indeed, in Subsection 2.4 we saw that every formal nc germ about Y 0 ⊕ Y 00 determines an nc function on Nilp(Y 0 ⊕ Y 00). Since nc functions respect direct sums and similarities, it is easy to see that for all
0 00 g X ∈ Nilp(Y ⊕ Y ) ∩ Mn(k) ⊗ (Ms0 (k) ⊕ Ms00 (k)) LOCAL THEORY OF FREE NC FUNCTIONS 29 we have
f(X) ∈ Mn(k) ⊗ (Ms0 (k) ⊕ Ms00 (k)). Consequently, if K is the permutation matrix corresponding to the canonical shuffle of blocks
Mns0 (k) ⊕ Mns00 (k) → Mn(k) ⊗ (Ms0 (k) ⊕ Ms00 (k)), then for all X0 ⊕ X00 ∈ Nilp(Y 0 ⊕ Y 00), (6.14) f(K(X0 ⊕ X00)K−1) = K(f 0(X0) ⊕ f 00(X00))K−1 for some nc functions f 0 and f 00 on Nilp(Y 0) and Nilp(Y 00), respectively. Thus (6.13) is given by f 7→ (f 0, f 00). If k = C and f is (uniformly) analytic, then f 0 and f 00 are also (uniformly) analytic by (6.14). Thus the homomorphism (6.13) restricts to homomor- phisms a a a ua ua ua (6.15) OY 0⊕Y 00 → OY 0 × OY 00 , OY 0⊕Y 00 → OY 0 × OY 00 . We refer to [K-VV14, Chapter 9] for further discussion. Corollary 7.7 below demonstrates that homomorphisms (6.13) and (6.15) are not necessarily injective. 6.4. Completions of the free algebra. In this subsection we apply Hermite interpo- lation for nc functions to investigate the ring structure of nc germs about semisimple points.
`+1 Proposition 6.17. If Y is a semisimple point, then I`(Y ) = I0(Y ) for all ` ∈ N.
Proof. By Proposition 6.11, there is a natural isomorphism between I`(Y )/I`+1(Y ) and all Y -admissible (` + 1)-linear maps. Therefore
I`(Y ) ≡ I0(Y ) ·I`−1(Y ) mod I`+1(Y ) follows by Lemma 6.8 as in the last part of the proof of Proposition 6.11. Furthermore,
I0(Y )I`−1(Y ) ⊆ I`(Y ) ⊆ I`−1(Y ), so I`−1(Y )/I`(Y ) = (I`−1(Y )/I`+1(Y )) (I`(Y )/I`+1(Y )) = (I`−1(Y )/I`+1(Y )) ((I0(Y )I`−1(Y ))/I`+1(Y ))
= I`−1(Y )/(I0(Y )I`−1(Y )), implies I`(Y ) = I0(Y ) ·I`−1(Y ). Corollary 6.18. If Y is a semisimple point, then O = lim k
OY
` m k
g ∼ Lemma 6.19. If Y ∈ Ms(k) and P ∈ GLs(k), then OPYP −1 = OY . Furthermore, for 1 h kg arbitrary Y ,...,Y ∈ nc and m1, . . . , mh ∈ N we have ∼ (6.17) OL mi i OL i . i(⊕ Y ) = i Y
Proof. The first claim is obvious. Now let m = max{m1, . . . , mh}. As in Remark 6.16, there are canonical homomorphisms
φ : O m L i → OL mi i , ψ : OL mi i → OL i . ⊕ ( i Y ) i(⊕ Y ) i(⊕ Y ) i Y Their composition ψ ◦ φ is again a canonical homomorphism of the same kind, and is an isomorphism by (2.2). By the construction of φ, ψ as in Remark 6.16 it is also straight- −1 forward to see that φ(ψ ◦ φ) ψ is the identity map, so ψ is an isomorphism. ∼ g The following theorem greatly generalizes the observation OY = O0 for Y ∈ k used in Section5, and classifies OY in terms of Y . See also [SSS18] for results about correspon- dences between noncommutative varieties and algebras of nc functions on them.
0 Theorem 6.20. Let Y and Y be semisimple points. Then the rings OY and OY 0 are isomorphic if and only if S(Y ) ∼= S(Y 0). The same conclusion holds for (uniformly) analytic nc germs about Y and Y 0 if k = C.
Proof. (⇒) The description of OY given by Corollary 6.18 implies that OY admits h maximal ideals, where h is the number of simple factors in S(Y ), and their intersection 0 equals I0(Y ). Thus an isomorphism OY → OY 0 maps I0(Y ) to I0(Y ), and so it induces an isomorphism ∼ 0 ∼ 0 S(Y ) = k
0 g (⇐) By Lemma 6.19 it suffices to assume that Y,Y ∈ Ms(k) are direct sums of pairwise non-similar irreducible points. Moreover, since S(Y ) ∼= S(Y 0), each irreducible block of Y is similar to an irreducible block of Y 0, we can further replace Y 0 by a similar matrix point to obtain S(Y ) = S(Y 0). Then also C(Y ) = C(Y 0), so there is a C(Y )-isomorphism
0 0 (6.18) [Ms(k),Y ] → [Ms(k),Y ], [S,Y ] 7→ [S,Y ].
g Since Ms(k) is a semisimple C(Y )-bimodule, the isomorphism (6.18) extends to a C(Y )- g g g bimodule isomorphism L :Ms(k) → Ms(k) . Write L = (L1,...,Lg) for Lj :Ms(k) → k 0 k Ms( ). Then (Yj ,Lj) satisfy IC1(Y ) for all j, so there exist F1,...,Fg ∈
Since L is an isomorphism, the nc polynomial map F = (F1,...,Fg) admits an inverse 0 nc map G = (G1,...,Gg) about Y by the inverse function theorem for nc functions LOCAL THEORY OF FREE NC FUNCTIONS 31
[AK-V15, Theorem 1.7], which is uniformly analytic if k = C by [AK-V15, Theorem 1.4]. Also note that Gj ∈ OY 0 . By Corollary 6.18, the homomorphisms
φ : k
ψ : k
Φ: OY → OY 0 , Ψ: OY 0 → OY .
Since F and G are inverse maps, Φ and Ψ are inverse homomorphisms. Remark 6.21. In the proof of Theorem 6.20 we saw that for any two irreducible points 0 g Y,Y ∈ Ms(k) , there exist an nc polynomial map F and a uniformly analytic nc map G on a neighborhood of Y 0 such that F (Y ) = Y 0, G(Y 0) = Y and F ◦ G = G ◦ F = id. It is natural to ask whether we can choose F,G in such a way that G is also polynomial, that is, whether we can find an nc polynomial automorphism F of the noncommutative space kg 0 nc such that F (Y ) = Y . The answer is positive if g ≥ s + 1 or if Y and Y 0 are saturated (meaning that Y and Y 0 without last components are already irreducible points) by [Rei93, Theorems 4.3 and 4.4]. However, in general there might not be any nc polynomial automorphism F 0 mapping Y to Y . For example, let g = 2. To a point Y = (Y1,Y2) we assign the span of its commutator LY = k · [Y1,Y2] ⊂ Ms(k). By [Dic82, Theorem], every nc polynomial automorphism F preserves LY . On the other hand, there clearly exist irreducible points 0 2 Y,Y ∈ Ms(k) such that LY 6= LY 0 if s ≥ 2. Polynomial automorphisms of the noncommutative space are well-understood through the solution of the free Jacobian conjecture and the free Grothendieck theorem [Pas14, Aug19].
7. Minimal propagation and nilpotent noncommutative functions In this section we describe a particular propagation of a sequence satisfying truncated canonical intertwining conditions about a semisimple point, into a uniformly analytic function, which is quite distinct from the Hermite interpolation with nc polynomials ua described earlier. We construct an embedding of S(Y ) into OY , and thus the first example of a nilpotent uniformly analytic nc function. Lastly, we also provide an nc function that vanishes on uniformly open neighborhoods of Y 0 and Y 00 but not of Y 0 ⊕ Y 00. Before an auxiliary lemma we observe the following. Let D : R[t] → R[t] be the linear map D(p) = (p − p(0))/t. For `, m > 0 we have
(7.1) Dm−1 (t + 1)`−1 + Dm (t + 1)`−1 = Dm (t + 1)` and ` (7.2) Dm (t + 1)` − tDm+1 (t + 1)` = m by the binomial coefficient formulas. 32 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
Lemma 7.1. Let α, β > 0. For ` ∈ N ∪ {0} and −1 ≤ m ≤ ` let c`,m ∈ R≥0 satisfy
c0,0 = 1,
c`,` = 0 and c`,−1 = c`,0 for ` > 0,
c`,m ≤ β max {c`,m+1, α(c`−1,m−1 + c`−1,m)} for − 1 < m < `. Then √ lim sup ` c`,0 < ∞. `→∞ Proof. It suffices to assume
c`,m = β max {c`,m+1, α(c`−1,m−1 + c`−1,m)} for − 1 < m < ` and β ≥ 2. First we compute c1,0 = 2αβ. Then we claim that ` ` m−1 `−2 (7.3) c`,m = 2α β D (t + 1) |t=β for 0 < m, ` `+1 `−2 (7.4) c`,0 = 2α β (β + 1) for ` ≥ 2. First observe that (7.4) follows from (7.3) since
` ` `−2 `−1 ` `−3 c`,0 = β max {c`,1, 2αc`−1,0} = β max 2α β (β + 1) , 2α · 2α β (β + 1) and β ≥ 1 if ` > 2, and
2 2 c2,0 = β max {c2,1, 2αc1,0} = β max 2α β , 2α · 2αβ since β ≥ 2. Moreover, (7.3) clearly holds for m = `. Next we prove (7.3) by increasing induction on ` and decreasing induction on m. By definition we have
2 2 c2,1 = βα(c1,0 + c1,1) = 2α β , so (7.3) holds for ` = 2. Now let 2 < ` and 1 < m < `. By the induction hypothesis we have c`,m = β max {c`,m+1, α(c`−1,m−1 + c`−1,m)} ` ` m `−2 m−2 `−3 m−1 `−3 = 2α β max βD (t + 1) |t=β,D (t + 1) |t=β + D (t + 1) |t=β ` ` m `−2 m−1 `−2 = 2α β max βD (t + 1) |t=β,D (t + 1) |t=β ` ` m−1 `−2 = 2α β D (t + 1) |t=β by (7.1) and (7.2). Furthermore,
c`,1 = β max {c`,2, α(c`−1,0 + c`−1,1)} ` ` `−2 `−3 `−3 = 2α β max D (t + 1) |t=β, β(β + 1) + (β + 1) = 2α`β` max (β + 1)`−2 − 1 /β, (β + 1)`−2 = 2α`β`(β + 1)`−2, so (7.3) holds. g Let Y ∈ Ms(k) be a semisimple point. Recall that C(Y )-bimodules are semisimple, g and that Ms(k) and [Ms(k),Y ] are C(Y )-bimodules in a natural way. Hence there exists g a C(Y )-bimodule projection π :Ms(k) → [Ms(k),Y ]. LOCAL THEORY OF FREE NC FUNCTIONS 33
k g L Theorem 7.2. Let Y ∈ Ms( ) be a semisimple point and let (f`)`=0 satisfy ICL(Y ) for ∞ some L ∈ N ∪ {0}. Then there exists a unique propagation (f`)`=0 satisfying IC(Y ) and
(7.5) f`|(ker π)` = 0 for ` > L. p` k Moreover, lim sup`→∞ kf`kcb < ∞ if = C.
g Proof. Since Ms(k) is a direct sum of [Ms(k),Y ] and ker π, uniqueness follows from the definition of IC(Y ).
Since Ms(k) is a semisimple C(Y )-bimodule and the map Ms(k) → [Ms(k),Y ] given by S 7→ [S,Y ] is a surjective C(Y )-bimodule homomorphism, it admits a C(Y )-bimodule right inverse φ : [Ms(k),Y ] → Ms(k). Moreover, (7.6) φ([S,Y ]) − S ∈ C(Y )
g holds for every S ∈ Ms(k). Let σ :Ms(k) → ker π be the projection onto ker π along g [Ms(k),Y ], so Z = π(Z) + σ(Z) for all Z ∈ Ms(k) . For ` > L and 0 ≤ m ≤ ` we recursively define `-linear maps
m g `−m f`,m : (ker π) × (Ms(k) ) → Ms(k) by f`,` := 0 and `−1 ` `−1 ` f`,`−1(...,W ,Z ) := f`−1,`−2(...,W (φ ◦ π)(Z )), m m+1 m m+1 m+2 f`,m(...,W ,Z ,... ) := f`,m+1(...,W , σ(Z ),Z ... ) m m+1 m+2 + f`−1,m−1(...,W (φ ◦ π)(Z ),Z ,... ) m m+1 m+2 (7.7) − f`−1,m(...,W , (φ ◦ π)(Z )Z ,... ), 1 1 2 f`,0(Z ,... ) := f`,1(σ(Z ),Z ,... ) 1 2 + (φ ◦ π)(Z )f`−1,0(Z ,... ) 1 2 − f`−1,0((φ ◦ π)(Z )Z ,... ) for 0 < m < ` − 1. Now f` := f`,0 clearly satisfy (7.5). Next, we check IC(Y ) for f` by induction on `. Firstly,
1 2 2 f`([S,Y ],Z ,... ) = φ([S,Y ])f`−1(Z ,... ) − f`−1(φ([S,Y ])Z ,... ) 1 1 2 = Sf`−1(Z ,... ) − f`−1(SZ ,Z ,... ) by the induction hypothesis and (7.6). Next, denote S0 = (φ ◦ π)(Z1). Then
1 2 1 2 f`(Z , [S,Y ],Z ,... ) = f`,1(σ(Z ), [S,Y ],Z ,... ) 0 2 0 2 + (S f`−1([S,Y ],Z ,... ) − f`−1(S [S,Y ],Z ... ) 1 2 1 2 = f`−1,0(σ(Z )φ([S,Y ]),Z ,... ) − f`−1,1(σ(Z ), φ([S,Y ])Z ,... ) 0 2 2 + (S φ([S,Y ])f`−2(Z ,... ) − f`−2(φ([S,Y ])Z ,... ) 0 0 2 − f`−1([S S,Y ] − [S ,Y ]S,Z ... ) 1 2 1 2 = f`−1(σ(Z )S,Z ,... ) − f`−1(σ(Z ),SZ ,... ) 0 2 2 + S Sf`−2(Z ,... ) − f`−2(SZ ,... ) 0 2 0 2 0 2 − S Sf`−2(Z ... ) + f`−2(S SZ ... ) + f`−1([S ,Y ]S,Z ... ) 34 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
1 2 1 2 = f`−1(σ(Z )S,Z ,... ) − f`−1(σ(Z ),SZ ,... ) 0 2 0 2 0 2 − S f`−2(SZ ,... ) + f`−2(S SZ ... ) + f`−1([S ,Y ]S,Z ... ) 1 2 1 2 = f`−1(σ(Z )S,Z ,... ) − f`−1(σ(Z ),SZ ,... ) 0 2 0 2 − f`−1([S ,Y ],SZ ,... ) + f`−1([S ,Y ]S,Z ... ) 1 2 1 2 = f`−1,0(σ(Z )S,Z ,... ) − f`−1,1(σ(Z ),SZ ,... ) 1 2 1 2 − f`−1(π(Z ),SZ ,... ) + f`−1(π(Z )S,Z ... ) 1 2 1 2 = f`−1(Z S,Z ,... ) − f`−1(Z ,SZ ,... ) holds by (7.6) and the induction hypothesis. The rest of IC(Y ) is verified analogously. Now let k = C. Since φ is a linear map between finite-dimensional operator spaces, it is completely bounded; let α = kφkcb. Similarly, let β = kψkcb, where g ψ :Ms(k) → [Ms(k),Y ] × ker π, Z 7→ (π(Z), σ(Z)).
Here [Ms(k),Y ] × ker π is viewed as the `1-direct sum of operator spaces [Ms(k),Y ] and ker π [Pis03, Section 2.6]. Given ε1, ε2 ∈ C, the map k k χε1,ε2 : [Ms( ),Y ] × ker π → [Ms( ),Y ] × ker π, (X,W ) 7→ (ε1X, ε2W ) satisfies k(χε1,ε2 )kcb = max{|ε1|, |ε2|}. By looking at χε1,ε2 ◦ ψ we thus obtain
(7.8) ε1kπ(Z)kns + ε2kσ(Z)kns ≤ β max{ε1, ε2}kZkns g for all ε1, ε2 ≥ 0, Z ∈ Mns(C) and n ∈ N. By (7.7) we have m m+1 m m+1 ` kf`,m(...,W ,Z ,... )k ≤ kf`,m+1k · · · kW kkσ(Z )k · · · kZ k m m+1 ` + kf`−1,m−1k · · · kW (φ ◦ π)(Z )k · · · kZ k m m+1 m+2 ` + kf`−1,mk · · · kW kk(φ ◦ π)(Z )Z k · · · kZ k m m+1 ` ≤ kf`,m+1k · · · kW kkσ(Z )k · · · kZ k m m+1 ` + αkf`−1,m−1k · · · kW kπ(Z )k · · · kZ k m m+1 ` + αkf`−1,mk · · · kW kkπ(Z )k · · · kZ k 1 ` m+1 = kf`,m+1kkW k · · · kZ k · kσ(Z )k 1 ` m+1 + α(kf`−1,m−1k + kf`−1,mk)kW k · · · kZ k · kπ(Z )k ` > L and m < `, and thus
1 ` 1 ` kf`,m(W ,...,Z )k ≤ β max {kf`,m+1k, α(kf`−1,m−1k + kf`−1,mk)} kW k · · · kZ k by (7.8). Actually, the same conclusion holds for all ampliations of f`,m. Therefore
kf`,mkcb ≤ β max {kf`,m+1kcb, α(kf`−1,m−1kcb + kf`−1,mk)cb} .
If fL = 0, then we have f` = 0 for all ` > L. Otherwise, c`,m := kf`+L,mkcb/kfLkcb satisfies p` the assumptions of Lemma 7.1, so lim sup`→∞ kf`kcb < ∞. By applying the “minimal” propagation (depending on the choice of projection π) of Theorem 7.2 to one-term sequences (i.e., matrices), we obtain the following.
g 1 t Theorem 7.3. Let Y ∈ Ms(k) be a semisimple point and a , . . . , a ∈ S(Y ). Then there exist nc functions f 1, . . . , f t, uniformly analytic on an nc ball about Y if k = C, such that LOCAL THEORY OF FREE NC FUNCTIONS 35
(i) f i(Y ) = ai for i = 1, . . . , t;
(ii) for every p ∈ k
i 1 t Proof. Since [a , C(Y )] = {0}, the one-term sequences (a ),..., (a ) satisfy IC0(Y ), so i there exist (f` )` for i = 1, . . . , t as in Theorem 7.2. Therefore there are uniformly analytic nc functions f 1, . . . , f t on an nc ball about Y given by
∞ i X i n n f (X) = f` (X − ⊕ Y,...,X − ⊕ Y ) `=0
i i i by Theorem 2.5. In particular, they satisfy f (Y ) = a and f` |(ker π)` = 0 for ` > 0. 1 t Suppose p(a , . . . , a ) = 0 for p ∈ k
` ` X i `−i (7.9) ∆Y (h1h2) = ∆Y h1∆Y h2. i=0
1 t i Since F is an an polynomial in f , . . . , f and f` |(ker π)` = 0 for all ` > 0, (7.9) implies 1 t F`|(ker π)` = 0 for all ` > 0. Moreover, F0 = p(a , . . . , a ) = 0. Therefore (F`)` is a propagation of (0) as in Theorem 7.2. On the other hand, (0)` is another such propagation, 1 t so F` = 0 for all ` ∈ N by uniqueness. Therefore p(f , . . . , f ) = 0. Remark 7.4. If Y is semisimple and not similar to a direct sum of scalar points, then we can choose a nonzero nilpotent matrix a ∈ S(Y ), so by Theorem 7.3 there exists a nontrivial nilpotent uniformly analytic function on Bε(Y ). Note however that ε is small g enough so that Bε(Y ) ∩ (C · I) = ∅. The following restatement of Theorem 7.3 gives us some further information about the structure of the germ algebras.
Corollary 7.5. Let Y be a semisimple point. The one-term propagation of Theorem 7.2 ua gives an homomorphism S(Y ) → OY that is a left inverse of the evaluation at Y . g ∼ In particular, if Y ∈ Mn(C) is irreducible, then OY = Mn(eOY e) and analogously for ua (uniformly) analytic germs, where e ∈ OY is the one-term propagation of e11 ∈ Mn(C).
If Y is irreducible, then it follows by Proposition 6.2 that the “corner” algebra eOY e in Corollary 7.5 is prime. A finer structure of this algebra is not known to the authors. ∼ L Also note that if Y is semisimple, non-irreducible and S(Y ) = i Mni (C), then one L cannot conclude that OY is isomorphic to i Mni (Ai) for some algebras Ai. This is further supported by Corollary 7.7 below. Namely, we construct examples of nc functions ua ua ua demonstrating that the canonical homomorphism OY 0⊕Y 00 → OY 0 × OY 00 from Remark 6.16 is not injective.
0 g 00 g Lemma 7.6. Let Y ∈ Ms0 (C) and Y ∈ Ms00 (C) be separated semisimple points, and 0 00 Y = Y ⊕ Y . Let f1 be a Y -admissible linear map such that
g (Ms0 (C) ⊕ Ms00 (C)) ⊂ ker f1. 36 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
Then there exists an nc function f, uniformly analytic on an nc ball about Y , such that 1 ∆Y f = f1 and f vanishes on G g Mn(C) ⊗ (Ms0 (C) ⊕ Ms00 (C)) . n 0 00 Proof. Since Y and Y are separated semisimple points, we have C(Y ) ⊆ Ms0 (C)⊕Ms00 (C). g Therefore (Ms0 (C)⊕Ms00 (C)) is a C(Y )-bimodule, so we can choose the projection π from the beginning of the section in such a way that
g g π(Ms0 (C) ⊕ Ms00 (C)) ⊆ (Ms0 (C) ⊕ Ms00 (C)) .
Since (0, f1) satisfies IC1(Y ), there exists (f`)` as in Theorem 7.2, so there is a uniformly analytic nc function on an nc ball about Y given by ∞ X n n f(X) = f`(X − ⊕ Y,...,X − ⊕ Y ). `=0 By Theorem 2.5 it suffices to show that
01 001 0` 00` (7.10) f`(Z ⊕ Z ,...,Z ⊕ Z ) = 0
0j g 00j g for all ` ≥ 1 and Z ∈ Ms0 (C) , Z ∈ Ms00 (C) . For ` = 1, (7.10) holds by the assumption. Since C(Y ) ⊆ Ms0 (C) ⊕ Ms00 (C), we have φ Ms0 (C) ⊕ Ms00 (C),Y ⊆ Ms0 (C) ⊕ Ms00 (C) by (7.6), where φ is a right inverse of S 7→ [S,Y ]. Therefore
g (7.11) (φ ◦ π)(Ms0 (C) ⊕ Ms00 (C)) ⊆ Ms0 (C) ⊕ Ms00 (C) by the choice of π. Moreover,
g g (7.12) σ(Ms0 (C) ⊕ Ms00 (C)) ⊆ (Ms0 (C) ⊕ Ms00 (C)) , g where σ :Ms(C) → ker π is the projection onto ker π along [Ms(C),Y ]. Now (7.10) follows by induction on ` using the recursive relations (7.7), (7.11) and (7.12). Corollary 7.7. If Y 0 and Y 00 are separated semisimple points, then the canonical homo- ua ua ua morphism OY 0⊕Y 00 → OY 0 × OY 00 is not injective.
0 g 00 g Proof. Let Y ∈ Ms0 (C) and Y ∈ Ms00 (C) . Since they are separated semisimple points, 0 00 g we have C(Y ⊕Y ) ⊆ Ms0 (C)⊕Ms00 (C). Therefore (Ms0 (C)⊕Ms00 (C)) is a C(Y )-bimodule. Furthermore,
0 00 g 0 00 2 0 2 00 2 dim[Ms0+s00 (C),Y ⊕ Y ] + dim(Ms0 (C) ⊕ Ms00 (C)) < (s + s ) + g((s ) + (s ) ) and so g. 0 00 g Ms0+s00 (C) [Ms0+s00 (C),Y ⊕ Y ] + (Ms0 (C) ⊕ Ms00 (C)) is a nonzero C(Y )-bimodule. Therefore there exists a nonzero C(Y )-bimodule homomor- g phism f1 :Ms0+s00 (C) → Ms0+s00 (C) such that 0 00 g [Ms0+s00 (C),Y ⊕ Y ] + (Ms0 (C) ⊕ Ms00 (C)) ⊆ ker f1 by Lemma 6.9. Hence the assumptions on Lemma 7.6 are satisfied, and let f be the ua ua ua resulting nc function. Then f ∈ ker(OY 0⊕Y 00 → OY 0 × OY 00 ). LOCAL THEORY OF FREE NC FUNCTIONS 37
References [AK-V15] G. Abduvalieva, D. S. Kaliuzhnyi-Verbovetskyi: Implicit/inverse function theorems for free noncommutative functions, J. Funct. Anal. 269 (2015) 2813–2844.2, 31 [AM15] J. Agler, J. E. McCarthy: Global holomorphic functions in several noncommuting variables, Canad. J. Math. 67 (2015) 241–285.2, 19 [AM16] J. Agler, J. E. McCarthy: Aspects of non-commutative function theory, Concr. Oper. 3 (2016) 15–24.1,4,5,7, 28 [AM16’] J. Agler, J. E. McCarthy: The implicit function theorem and free algebraic sets, Trans. Amer. Math. Soc. 368 (2016) 3157–3175.2 [Ami66] S. A. Amitsur: Rational identities and applications to algebra and geometry, J. Algebra 3 (1966) 304–359.9, 13 [Art68] M. Artin: On the solutions of analytic equations, Invent. Math. 5 (1968) 277–291. 15 [Aug19] M. Augat: The free Grothendieck theorem, Proc. London Math. Soc. 118 (2019) 787–825.2, 31 [BGM05] J. A. Ball, G. Groenewald, T. Malakorn: Structured noncommutative multidimensional linear systems, SIAM J. Control Optim. 44 (2005) 1474–1528.1, 18 [BMV18] J. A. Ball, G. Marx, V. Vinnikov: Interpolation and transfer-function realization for the non- commutative Schur–Agler class, Operator theory in different settings and related applications, 23– 116, Oper. Theory Adv. Appl. 262, Birkh¨auser/Springer,Cham, 2018.4 [Ber70] G. M. Bergman: Skew fields of noncommutative rational functions, after Amitsur, in S´eminaire M.P. Sch¨utzenberger, A. Lentin et M. Nivat 1969/70, Probl. Math. Theor. Automates 16, Paris, 1970. 10 [Bla06] B. Blackadar: Operator algebras. Theory of C∗-algebras and von Neumann algebras, Encyclopae- dia of Mathematical Sciences 122, Operator Algebras and Non-commutative Geometry III, Springer- Verlag, Berlin, 2006.3, 13 [BS99] M. Breˇsar,P. Semrl:ˇ On locally linearly dependent operators and derivations, Trans. Amer. Math. Soc. 351 (1999) 1257–1275. 24 [Coh06] P.M. Cohn: Free ideal rings and localization in general rings, New Mathematical Monographs 3, Cambridge University Press, Cambridge, 2006.2,3,9, 13, 14, 17, 18, 19, 20 [CR94] P.M. Cohn, C. Reutenauer: A normal form in free fields, Canad. J. Math. 46 (1994) 517–531. 18 [DK+] K. R. Davidson, M. Kennedy: Noncommutative Choquet theory, preprint arXiv:1905.08436.2 [Dic82] W. Dicks: A commutator test for two elements to generate the free algebra of rank two, Bull. London Math. Soc. 14 (1982) 48–51. 31 [GMS18] E. Griesenauer, P. S. Muhly, B. Solel: Boundaries, bundles and trace algebras, New York J. Math. 24a (2018) 136–154. 10, 14 [HKM11] J. W. Helton, I. Klep, S. McCullough: Proper analytic free maps, J. Funct. Anal. 260 (2011) 1476–1490.1 [HKM12] J. W. Helton, I. Klep, S. McCullough: Free analysis, convexity and LMI domains, Mathe- matical methods in systems, optimization, and control, 195–219, Oper. Theory Adv. Appl. 222, Birkh¨auser/SpringerBasel AG, Basel, 2012.2 [HMS18] J. W. Helton, T. Mai, R. Speicher: Applications of realizations (aka linearizations) to free probability, J. Funct. Anal. 274 (2018) 1–79. 10, 14, 18 [HMV06] J. W. Helton, S. McCullough, V. Vinnikov: Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240 (2006) 105–191. 12 [K-VV12] D. S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov: Noncommutative rational functions, their difference-differential calculus and realizations, Multidimens. Syst. Signal Process. 23 (2012) 49–77. 3, 17 [K-VV14] D. S. Kalyuzhnyi-Verbovetskyi, V. Vinnikov: Foundations of free noncommutative function theory, Mathematical Surveys and Monographs 199, American Mathematical Society, Providence RI, 2014.1,2,5,6,7,8, 29 [KS17]ˇ I. Klep, S.ˇ Spenko:ˇ Free function theory through matrix invariants, Canad. J. Math. 69 (2017) 408–433. 14 [KVV+] I. Klep, V. Vinnikov, J. Volˇciˇc: Multipartite rational functions, preprint arXiv:1509.03316.3 38 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
[KV17] I. Klep, J. Volˇciˇc: Free loci of matrix pencils and domains of noncommutative rational functions, Comment. Math. Helv. 92 (2017) 105–130. 26 [Kra92] S. G. Krantz: Function theory of several complex variables, 2nd edition, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. 16 [Lam91] T. Y. Lam: A first course in noncommutative rings, Graduate Texts in Mathematics 131, Springer-Verlag, New York, 1991. 17 [Lum91] D. Luminet: Functions of several matrices, Boll. Un. Mat. Ital. B (7) 11 (1997) 563–586. 14, 16 [Mat89] H. Matsumura: Commutative ring theory, 2nd edition, Cambridge Studies in Advanced Mathe- matics 8, Cambridge University Press, Cambridge, 1989. 10 [MS11] P. S. Muhly, B. Solel: Progress in noncommutative function theory, Sci. China Math. 54 (2011) 2275–2294.1 [Pas14] J. E. Pascoe: The inverse function theorem and the Jacobian conjecture for free analysis, Math. Z. 278 (2014) 987–994.2, 31 [Pau02] V. Paulsen: Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2002.6, 18 [Pis03] G. Pisier: Introduction to operator space theory, London Mathematical Society Lecture Note Series 294, Cambridge University Press, Cambridge, 2003.6, 18, 34 [Pop02] G. Popescu: Central intertwining lifting, suboptimization, and interpolation in several variables, J. Funct. Anal. 189 (2002) 132–154.4 [Pop06] G. Popescu: Free holomorphic functions on the unit ball of B(H )n, J. Funct. Anal. 241 (2006) 268–333.1 [Pop08] G. Popescu: Free holomorphic functions and interpolation, Math. Ann. 342 (2008) 1–30.2,4 [Pro76] C. Procesi: The invariant theory of n × n matrices, Adv. Math. 19 (1976) 306–381. 14, 16 [Pro07] C. Procesi: Lie groups. An approach through invariants and representations, Universitext, Springer, New York, 2007. 15, 22, 23, 24, 25 [Rei93] Z. Reichstein: On automorphisms of matrix invariants, Trans. Amer. Math. Soc. 340 (1993) 353–371. 31 [RLL00] M. Rørdam, F. Larsen, N. Laustsen: An introduction to K-theory for C∗-algebras, London Mathematical Society Student Texts 49, Cambridge University Press, Cambridge, 2000.3, 13 [Row80] L. H. Rowen: Polynomial identities in ring theory, Pure and Applied Mathematics 84, Academic Press, Inc., New York-London, 1980. 10, 13, 16, 21 [Rya02] R. A. Ryan: Introduction to Tensor Products of Banach Spaces, Springer Monographs in Math- ematics, Springer-Verlag London, Ltd., London, 2002. 18 [SSS18] G. Salomon, O. Shalit, E. Shamovich: Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball, Trans. Amer. Math. Soc. 370 (2018) 8639–8690. 2, 30 [Sal99] D. J. Saltman: Lectures on division algebras, CBMS Regional Conference Series in Mathematics 94, American Mathematical Society, Providence RI, 1999. 14 [Shi19] Y. Shitov: An improved bound for the length of matrix algebras, Algebra Number Theory 6 (2019) 1501–1507. 28 [Sch85] A. H. Schofield: Representation of rings over skew fields, London Mathematical Society Lecture Note Series 92, Cambridge University Press, Cambridge, 1985.3 [Tak67] M. Takesaki: A duality in the representation theory of C∗-algebras, Ann. Math. 85 (1967) 370– 382.2 [Tay72] J. L. Taylor: A general framework for a multi-operator functional calculus, Adv. Math. 9 (1972) 183–252.1 [Tay73] J. L. Taylor: Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973) 1–34.1,8 [Voi10] D.-V. Voiculescu: Free analysis questions II: the Grassmannian completion and the series expan- sions at the origin, J. Reine Angew. Math. 645 (2010) 155–236.1,2,5,8 [Vol18] J. Volˇciˇc: Matrix coefficient realization theory of noncommutative rational functions, J. Algebra 499 (2018) 397–437. 18 LOCAL THEORY OF FREE NC FUNCTIONS 39
Igor Klep, Department of Mathematics, University of Ljubljana E-mail address: [email protected]
Victor Vinnikov, Department of Mathematics, Ben-Gurion University of the Negev E-mail address: [email protected]
Jurij Volciˇ c,ˇ Department of Mathematics, Texas A&M University E-mail address: [email protected] 40 I. KLEP, V. VINNIKOV, AND J. VOLCIˇ Cˇ
Contents 1. Introduction...... 1 Main results and guide to the paper...... 2 2. Preliminaries...... 4 2.1. Noncommutative functions...... 5 2.2. Differential operators...... 5 2.2.1. Ampliations...... 5 2.2.2. Canonical intertwining conditions...... 6 2.3. Topologies on a noncommutative space and analyticity of noncommu- tative functions...... 6 2.3.1. Disjoint union topology...... 7 2.3.2. Uniformly open topology...... 7 2.4. Noncommutative germs...... 8 2.5. Universal skew field of fractions...... 9 3. Universal skew field of fractions of formal power series and the Amitsur-Cohn theorem for meromorphic identities...... 9 3.1. Meromorphic expressions and identities...... 9 3.2. Completion of the ring of generic matrices...... 10 3.3. Construction of the skew field M and its universality...... 12 3.4. Amitsur-Cohn theorem for meromorphic identities...... 13 4. Meromorphic GLn(C)-invariants...... 14 5. Universal skew field of fractions of analytic germs...... 16 5.1. Semifir property and inertness...... 17 5.2. Meromorphic noncommutative germs...... 18 5.3. Local-global rank principle...... 19 5.4. Level-wise meromorphic functions...... 21 6. Germs about semisimple points and Hermite interpolation...... 21 6.1. Truncated canonical intertwining conditions...... 23 6.2. Noncommutative algebra intermezzo...... 23 6.3. Hermite interpolation...... 25 6.4. Completions of the free algebra...... 29 7. Minimal propagation and nilpotent noncommutative functions...... 31 References...... 37