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Hermitian Geometry on Resolvent Set

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Hermitian Geometry on Resolvent Set Hermitian geometry on resolvent set (I) Ronald G. Douglas and Rongwei Yang Abstract. For a tuple A = (A1, A2, ..., An) of elements in a unital Banach n algebra B, its projective joint spectrum P (A) is the collection of z ∈ C such that A(z)= z1A1 + z2A2 + · · · + znAn is not invertible. It is known that the −1 B-valued 1-form ωA(z) = A (z)dA(z) contains much topological informa- c tion about the joint resolvent set P (A). This paper studies geometric properties c of P (A) with respect to Hermitian metrics defined through the B-valued fun- ∗ damental form ΩA = −ωA ∧ ωA and its coupling with faithful states φ on B, i.e. φ(ΩA). The connection between the tuple A and the metric is the main subject of this paper. In particular, it shows that the K¨ahlerness of the metric is tied with the commutativity of the tuple, and its completeness is related to the Fuglede-Kadison determinant. Mathematics Subject Classification (2010). Primary 47A13; Secondary 53A35. Keywords. Maurer-Cartan form, projective joint spectrum, Hermitian metric, arXiv:1608.05990v4 [math.FA] 27 Feb 2020 Ricci tensor, Fuglede-Kadison determinant. 0. Introduction In [4], Cowen and the first-named author introduced geometric concepts such as holomorphic bundle and curvature into Operator Theory. This gave rise to complete and computable invariants for certain operators with rich spectral structure, i.e, the Cowen-Douglasoperators. The idea was extended to multivariable cases in the study of Hilbert modules in analytic function spaces, where curvature invariant is defined for some canonical tuples of commuting operators. We refer readers to [8, 9, 10, 24] 2 R.G.DouglasandR.Yang and the references therein for more information. This paper aims to study general (possibly non-commuting) tuples using geometric ideas. The approach is based on the newly emerged concept of projective joint spectrum P (A) introduced by the second-named author (cf. [29]). Some preliminary findings are reported here. They shall help to build a foundation for subsequent and more in-depth studies. Let be a complex Banach algebra with unit I and A = (A , A , , A ) B 1 2 ··· n be a tuple of linearly independent elements in . The multiparameter pencil B A(z) := z A + z A + + z A 1 1 2 2 ··· n n is an important subject of study in various fields, for example in algebraic geometry [28], math physics [1, 21], PDE [2], group theory [13], etc., and more recently in the settlement of the Kadison-Singer Conjecture ([22]). In these studies, the primary interest was in the case when A is a tuple of self-adjoint operators. For general tu- ples, the notion of projective joint spectrum is defined in [29] and its property is investigated in a series of papers (cf. [3], [5], [7], [25]). Definition. For a tuple A = (A , A , , A ) of elements in a unital Banach 1 2 ··· n algebra , its projective joint spectrum is defined as B P (A)= z Cn A(z) is not invertible . { ∈ | } The projective resolvent set refers to the complement P c(A)= Cn P (A). \ Various notions of joint spectra for (commuting) tuples have been defined in the past, for instance the Harte spectrum ([16, 17]) and the Taylor spectrum ([11, 26, 27]), and they are key ingredients in multivariable operator theory. The projective joint spectrum, however, has one notable distinction: it is “base free” in the sense that, instead of using I as a base point and looking at the invertibility of (A z I, A z I, ..., A z I) in various constructions, it uses the much 1 − 1 2 − 2 n − n simpler pencil A(z). This feature makes the projective joint spectrum computable in many interesting noncommuting examples, for example the tuple of general com- pact operators ([25]), the generating tuple for the free group von Neumann algebra Hermitiangeometryonresolventset(I) 3 ([3]), the Cuntz tuple of isometries (S1, S2, ..., Sn) ([6, 19]) satisfying S∗S = δ I for 1 i, j n, i j ij ≤ ≤ n ∗ SiSi = I, Xi=1 where I is the identity. For the generating tuple (1, a, t) of the infinite dihedral group D =<a, t a2 = t2 =1 >, ∞ | the projective spectrum has found convincing applications to the study of group of intermediate growth (finitely generated group whose growth is faster than polyno- mial growth but slower than exponential growth (cf. [14])). For a general tuple A, it can happen that P (A) = Cn and hence P c(A) is empty. But in this paper we shall always assume that P c(A) is non-empty. It is known that every path-connected component of P c(A) is a domain of holomorphy. This fact is a consequence of a result in [30], and it is also proved independently in [20]. From this point of view, P c(A) has good analytic properties. On P c(A), the holomorphic Maurer-Cartan type -valued (1, 0)-form B n −1 −1 ωA(z)= A (z)dA(z)= A (z)Aj dzj Xj=1 played an important role in [29], where it is shown to contain much information about the topologyof P c(A). For example,the de Rham cohomology H∗(P c(A), C) can be computed by coupling ωA with invariant multilinear functionals or cyclic co- cycles ([7, 29]). Two simple facts about ωA are useful here. 1) By differentiating the equation A(z)A(z)−1 = I, one easily verifies that dA−1(z)= A−1(z)(dA(z))A−1(z), − and consequently, dω = dA−1(z) dA(z)= ω ω . (0.1) A ∧ − A ∧ A Here d is complex differentiation (see Section 1) and is the wedge product. ∧ 2) ωA is invariant under left multiplication on the tuple A by invertible ele- ments. To be precise, we let GL( ) denote the set of invertible elements in . Then B B 4 R.G.DouglasandR.Yang for any L GL( ) and B = LA = (LA , LA , , LA ), we have for the ∈ B 1 2 ··· n multiparameter pencils B(z)= LA(z), and hence P (B)= P (A). Moreover, −1 −1 −1 ωB = B (z)dB(z)= A (z)L LdA(z)= ωA. (0.2) This shows that ω is invariant under the left action of GL( ) on the tuple A. A B For simplicity, we shall use Einstein convention for summation in many places in this paper. For example, we shall write ω (z)= A−1(z)A dz . When is a C∗- A j j B algebra, the adjoint of ω (z) is the -valued (0, 1)-form A B ∗ −1 ∗ ωA(z) = (A (z)Aj ) dzj . Now we define the fundamental form for the tuple A as Ω = ω∗ ω = (A−1(z)A )∗A−1(z)A dz dz . (0.3) A − A ∧ A j k k ∧ j Here the negative sign exists in the first equality because it is conventional to write a (1, 1)-form as linear combinations of dz dz¯ . Often, the factor i is used to make j ∧ k 2 i 2 ΩA a self-adjoint form (cf. (1.3)), but it is not important in this paper. For a suitable choice of linear functional φ, such as a faithful state on , the B evaluation φ(Ω )= φ (A−1(z)A )∗A−1(z)A dz dz A j k k ∧ j induces a positive definite bilinear form on the holomorphic tangent bundle of P c(A), thus giving a Hermitian metric on P c(A). The connection between the metric and the tuple A shall be the primary concern of this paper. In particular, it shows that the K¨ahlerness of the metric is tied with the commutativity of the tuple (cf. Theorem 2.4). A notable feature of this metric is that it has singularities at the joint spectrum P (A). So completeness of the metric is an important issue. 1. -valued differential forms B Let M be a complex manifold of dimension n. If z = (z ,z , ,z ) is the coordi- 1 2 ··· n nate in a local chart, then ∂ stands for ∂ , and ∂¯ stands for ∂ . As a convention, i ∂zi i ∂z¯i ¯ ¯ ¯ we let ∂ = i ∂i, ∂ = i ∂i, and d = ∂ + ∂. Consider a globally defined smooth (1, 1)-formPΦ(z) = g P(z)dz dz¯ expressed in the local chart with the Einstein jk j ∧ k Hermitiangeometryonresolventset(I) 5 summation convention. Φ induces a bilinear form Φ(ˆ z) = g (z)dz dz¯ on the jk j ⊗ k holomorphic tangent bundle T (M) over the local chart such that Φ(ˆ z)(∂j , ∂k)= gjk(z). We say that Φ defines a Hermitian metric on M if the n n matrix function × g(z) = (g (z)) is positive definite for each z M. In this case g(z) is called jk ∈ i the associated metric matrix, and 2 Φ(z) is called the fundamental form of the met- i ric. Here, the constant 2 is to normalize the fundamental form so that i i i (a) 2 Φ(z) is real i.e. 2 Φ(z)= 2 Φ(z); (b) in the one variable case z = x + iy, we have i dz dz¯ = dx dy. 2 ∧ ∧ Some geometric concepts are relevant to the study here. 1) A Hermitian metric induced by Φ(z) is said to be K¨ahler if dΦ=0, i.e. Φ(z) is closed. 2) The Ricci curvature tensor is the n n matrix R(z) = (R (z)), where × jk R = ∂ ∂¯ log det g(z), and the Ricci form is Ric (z)= R (z)dz dz¯ .
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