Sphere and projective space of a C *-algebra with a faithful state

Abstract: Let A be a unital C*-algebra with a faithful state φ. We study the geometry of the unit sphere Sφ = {x ∈ A : φ(x*x) = 1} and the projective space Pφ = Sφ/T. These spaces are shown to be smooth manifolds and homogeneous spaces of the group Uφ(A) of isomorphisms acting inAwhich preserve the inner product induced by φ, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in Pφ, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.


Introduction
Let A be a unital C * -algebra with a faithful state φ. There are natural geometric objects associated to this pair: the unit sphere The purpose of this paper is the study of these spaces using the tools of differential geometry. As in the classical setting in finite dimension (the spaces Sφ and Pφ are infinite dimensional), a key feature in this study is the transitive action of a group of movements. In this case, the group Uφ(A) of invertible linear operators acting in A, which preserve the inner product ⟨ , ⟩φ induced by φ (⟨x, y⟩φ = φ(y * x), x, y ∈ A) , i.e.,
Here B(A) denotes the Banach space of bounded linear operators acting in A.
The sphere Sφ is a dense subset of a sphere in a Hilbert space: denote by L = L 2 (A, φ) the GNS Hilbert space of the pair (A, φ), i.e. the completion of the pre-Hilbert space (A, ⟨ , ⟩φ). Then clearly Sφ is dense in the sphere of L. Also it is clear that an element G ∈ Uφ(A) extends uniquely to a unitary operator U G in L. Therefore Uφ(A) can be regarded as a subgroup of the unitary group of L, consisting of all unitaries U acting in L which leave A fixed: U(A) = A. To perform our geometric study, we shall need to introduce topologies in Sφ and Pφ (the ambient topology of A, and its quotient topology, respectively), and also in Uφ(A). Clearly, Uφ(A) is not a Banach-Lie group in the topology that it inherits from the whole unitary group of L: the condition of leaving A ⊂ L fixed is not closed (A is dense in L). To obtain a regular structure for Uφ(A) we shall use the theory of operators in spaces with two norms, developed independently by M.G. Krein [1] and P. Lax [2]. Two norms appear naturally in our context, the usual norm ‖ · ‖ of A and the norm ‖ · ‖φ induced by the φ-inner product.
The projective space Pφ is homeomorphic to a set of projections, the space P 1 (A, φ) of rank one projections acting in A, which are orthogonal for the φ-inner product. We introduce a natural metric in Pφ, and study its geodesics. Though the metric is not complete (and the space is infinite dimensional), using facts from the infinite dimensional Grassmann manifold [3,4], we obtain the existence of minimal geodesics with 1. fixed initial position and initial velocity (Theorem 6.2); 2. fixed endpoints (Theorem 6.3).
The contents of the paper are the following. In Sections 2 and 3 we introduce preliminary facts; Section 3 focuses on basic facts concerning operators in spaces with two norms (our references here are [1] and [2], and also the paper by I. C. Gohberg and M. K. Zambickii [5]), and the local structure of the group Uφ(A). In Section 3 we study the actions of Uφ(A) on Sφ and Pφ. Our main result is that both actions are transitive, and that the sphere and projective space are connected. In Section 4 we examine the local regular structure of Sφ and Pφ: the first space is a complemented submanifold of A, the second is a differentiable manifold, both spaces are homogeneous spaces of the group Uφ(A). In Section 5 we introduce a pre-Riemann-Hilbert metric in Pφ. We do this in two ways, that turn out to coincide: as a quotient metric, and as a trace induced metric. In Section 6 we prove the existence of minimal geodesics for this metric, both in the initial value problem (given initial position and velocity) and in the boundary value problem (given initial and final position).

Preliminaries and notation
We shall consider A represented in the Hilbert space L = L 2 (A, φ), via the GNS representation induced by φ. Elements x ∈ A will also be regarded as elements of L with norm ‖x‖φ = φ(x * x) 1/2 . As usual, if x, y ∈ A, x ⊗ y will denote the rank one operator acting in L: x ⊗ y(ξ ) = ⟨ξ , y⟩x, and in particular if a ∈ A, x ⊗ y(a) = φ(y * a)x. Denote by P(L) the space of selfadjoint projections of L, and by P 1 (L) the subset of rank one projections. Let the map π : Sφ → P 1 (L) , π(x) = x ⊗ x whose range is P 1 (A, φ) the set of rank one projections onto lines generated by elements in A ⊂ L. This map induces a bijection We shall identify these spaces (we shall see that the bijection above is a homeomorphism between the quotient topology and the norm topology of bounded operators acting in A). Therefore the map π can also be regarded as the quotient map from Sφ onto Pφ.
Part of the material in this section is either well known or follows from well known facts. It falls in the context of the theory of symmetrizable and proper linear operators in spaces with two norms, developed by I. Gohberg and M.K. Zambickii [5], M.G. Krein [1] and P. Lax [2]. In the space A we can consider two norms, the usual norm ‖ · ‖∞ and the norm ‖ · ‖φ induced by φ. These norms are comparable: ‖a‖φ ≤ ‖a‖∞ for all a ∈ A, and only the second norm is complete.
In this case we denote S = T ♯ . For example, if La denotes the left multiplication operator (a ∈ A) then L ♯ a = L a * . Let us denote by Note that Ba(A) is a (non closed) subalgebra of B(A). We shall consider a subset of Ba(A) the adjointable operators which are symmetric with respect to φ. These operators are called usually symmetrizable. M.G. Krein [1] and P. Lax [2] studied this class, in the context of a Banach algebra B (here equal to A) endowed with a positive definite inner product (here, the one induced by the state φ). For instance, they showed that these operators extend to selfadjoint operators in L (we state below a result independently obtained by both authors). Adjointable operators also extend to L. If T is adjointable, T 1 = 1 2 (T + T ♯ ) and T 2 = −i 2 (T − T ♯ ) are symmetrizable, and therefore extend to L, thus T = T 1 + iT 2 extends to L, as well as T ♯ , and clearly the extension of T ♯ is the adjoint in L of the extension of T. This latter result was obtained by I. Gohberg and M.K. Zambickii [5] in the much broader context of Banach spaces with two norms (none of them given by inner products).
Let F(A) be the linear span of a ⊗ b, a, b ∈ A in B(A), which can be regarded also as finite rank operators acting in L, with symbols in A. These operators are adjointable, the subset of operators in F(A) which are symmetric for the φ-inner product. Namely, We introduce a norm in Ba(A): This norm was introduced by Gohberg and Zambickii in [5]. It is easy to verify that Ba(A) is complete with this norm. Also that ‖TS‖a ≤ ‖T‖a‖S‖a, i.e. Ba(A) is a Banach algebra, with involution ♯. Also it is clear that ‖T ♯ ‖a = ‖T‖a. Though Ba(A) is not a C * -algebra. For instance, pick a ∈ A with ‖a‖ = 1 and a * a ≠ 1. Elementary computations show that where φ(a * a) < 1. Indeed, since a * a ≤ 1 and φ is faithful, φ(a * a) = 1 would imply a * a = 1.

Let us denote by Gla(A) the group of invertible operators in
Consider the closed subgroup That is, Uφ(A) consists of the invertible operators acting in A which preserve the inner product given by the state φ. Namely, if G ∈ Uφ(A) then G −1 = G ♯ . Elements G in Uφ(A) need not be isometric, and it is clear that ‖G‖a ≥ 1. Clearly, for a ∈ A, ‖La‖a = ‖a‖∞. Let Qa the set of idempotents in Ba(A), In particular, Qa is an analytic submanifold of Ba(A) (see [6]). We shall consider a subset of Qa: the idempotents which are orthogonal with respect to φ (and extend to selfadjoint projections in L). Recall that P 1 (A, φ) denotes the subset of rank one projections onto lines generated by elements in A ⊂ L. Note that Uφ(A) acts in Pa: G · P = GPG −1 ∈ Pa, for G ∈ Uφ(A) and P ∈ Pa.
Before we finish this section, let us state the following elementary result. Note that P 1 (A, φ) is considered with the ‖ ‖a-topology.

Lemma 2.1. The bijection
is a homeomorphism.
Proof. First note that an element x ∈ A defines a projection x ⊗ x in A if and only if φ(x * x) = 1. Therefore, it is clear that the map above is a bijection. It is continuous: if [xn] → [x] in Pφ, then there exist zn ∈ T such that zn xn → x in A. Then clearly xn ⊗ xn = zn xn ⊗ zn xn → x ⊗ x in Ba(A). Suppose now that xn , x ∈ Pφ satisfy xn ⊗ xn → x ⊗ x in Ba(A). Then In order to study the structure of Pa, we shall need the following elementary facts, which are consequences of the holomorphic functional calculus in the Banach algebra Ba(A). These facts hold in the broader frame of Banach algebras with involution.
In particular, this fact above enables one to obtain a local chart near 1 for Uφ(A), defined on a neighbourhood of the origin in via the exponential map, in a standard fashion, as with the usual unitary group of a C * -algebra.

Corollary 2.4. The group Uφ(A) is a Banach-Lie C ∞ -group, and a complemented submanifold of Ba(A). Its Banach-Lie algebra is Bas(A).
Note that, as any smooth Banach Lie group, Uφ(A) turns out to be a real analytic Banach Lie group (see e.g. [7]). We can use these facts and notations to prove that the symmetric part Pa of Qa is a complemented submanifold of Ba(A): Pa is a C ∞ submanifold of Ba(A). The action of Uφ(A) on Pa is locally transitive and has C ∞ local cross sections. In particular, for any fixed P 0 ∈ Pa, the map We abbreviate these facts in the statement, by saying that the Pa is a C ∞ -homogeneous space of Uφ(A). Along this line, Porta and Recht [6] proved that Qa is a homogeneous space of Gla(A). To prove this Proposition, we shall need the following result from [8]: Lemma 2.6. Let G be a Banach-Lie group acting smoothly on a Banach space X. For a fixed x 0 ∈ X, denote by πx 0 : G → X the smooth map πx 0 (g) = g · x 0 . Suppose that 1. πx 0 is an open mapping, regarded as a map from G onto the orbit {g · x 0 : g ∈ G} of x 0 (with the relative topology of X). 2. The differential d(πx 0 ) 1 : (TG) 1 → X splits: its nullspace and range are closed complemented subspaces.
Then the orbit {g · x 0 : g ∈ G} is a smooth submanifold of X, and the map the orbit of P 0 . To prove that the action is locally transitive and that π P0 is an open mapping, we shall construct continuous local cross sections for π P0 defined on a neighbourhood of P 0 in Pa. Consider i.e. s P0 is a cross section for the action. Cross sections on neighbourhoods around other points in Pa are obtained by translation with the group action. Let us check condition 2. of Lemma 2.6. These computations are very similar to the case of the Grassmann manifold of B(H) (see [3]), we include them. We differentiate the map π P0 at 1 ∈ Uφ(A), regarding it as a map to Bs(A). It is clear that Thus the nullspace of d(π P0 ) 1 consists of Z in Bas(A) which commute with P 0 . Written as 2 × 2 matrices in terms of P 0 , they are the anti-symmetric P 0 -diagonal matrices: A natural supplement for this nullspace is the space of P 0 -co-diagonal anti-symmetric matrices, i.e, P 0 YP 0 = 0 = (1 − P 0 )Y(1 − P 0 ): the subspace of P 0 -co-diagonal symmetric matrices. Indeed, it is clear that the range of d(π P0 ) 1 is contained in this subspace. Conversely, pick Y ∈ C P0,s and note that Then Therefore the range of d(π P0 ) 1 is complemented in Bs(A): a natural supplement is the space of P 0 -diagonal symmetric matrices. Then, by the Lemma 2.6, the orbit O P0 is a smooth submanifold of Ba(A), the map π P0 is a smooth submersion, and Pa is a discrete union of orbits O P , P ∈ Pa.

Group actions in S φ and P φ
In this section, we shall define natural C ∞ homogeneous structures, induced on Sφ and Pφ by the group action of Uφ(A). Let x ∈ Sφ, and G ∈ Uφ(A), then the action is given by G · x = Gx. Indeed, Operators in Uφ(A) extend to unitary operators in L. Thus one can also regard this subgroup as consisting of unitary operators U acting in L which satisfy U(A) = A.
Clearly, Uφ(A) contains U A , the unitary group of A, acting by left multiplication on A: For many computations (for instance, to show that the above action is transitive), it will suffice to consider special elements in Uφ(A). For instance, if X ∈ F(A)s, we have e iX ∈ Uφ(A).
Additionally, in matrix form (in term of P 0 = 1 ⊗ 1), since then, one has that where sinc(t) = sin(t) t is the cardinal sine, defined for t ≥ 0.
In the above result it was shown that the invertibles in Uφ(A) linking 1 to y are exponentials. Thus

Differentiable structure of S φ and P φ
We shall see that Sφ is a C ∞ complemented submanifold of A, and that Pφ is a C ∞ differentiable manifold, presenting both spaces as homogeneous spaces of Uφ(A).
To prove the assertion on Sφ, we shall use again Lemma 2.6: is a C ∞ -submersion.
Proof. First, recall from Corollary 3.4 that πx 0 is onto. In the frame of Lemma 2.6, consider Sφ ⊂ A with the relative topology, and fix x 0 ∈ Sφ. To prove that πx 0 is open, we exhibit a continuous local cross section near x 0 (local cross sections near other points of Sφ are obtained by translating this one with the group action). By Proposition 2.5, there exists rx 0 such that if P ∈ Pa satisfies then there exists V P ∈ Uφ(A), which is a smooth function of P, such that and V x0⊗x0 = 1. Note that if a, b ∈ A, then (by the Cauchy-Schwarz inequality) Thus in particular Note that since rx 0 2‖x0‖∞+1 ≤ 1, one has that for such y Then by Proposition 2.5, there exists Vy ∈ Uφ(A), depending continuously on y ⊗ y (and therefore on y) such that and similarly Uy U ♯ y = 1. It is clear that also Uy depends continuously on y. Moreover Then µx 0 defined on y such that ‖y − x 0 ‖∞ < In particular, R(δx 0 ) is the nullspace of a bounded real functional in A, thus it is a (real) complemented subspace of A. Thus Lemma 2.6 applies and the proof follows.

Remark 4.3. The facts that
Pφ is a differentiable manifold with the quotient topology, and that the map π [x0] is a submersion for any [x 0 ] ∈ Pφ are easier to prove. Note that here we do not claim that Pφ is a submanifold (it does not lie in a Banach space). Indeed, in the quotient topology, Pφ identifies (is homeomorphic) with P 1 (A, φ), the manifold of rank one projections of Ba(A), which coincides with the Uφ(A)-orbit of x 0 ⊗ x 0 , as seen before. By the results earlier in this section, this space is a differentiable manifold and the projection map a submersion.
Note that b ⊗ 1 − 1 ⊗ b belongs to Fas(A) and thus e b⊗1−1⊗b ∈ Uφ(A) and x b = e b⊗1−1⊗b (1) ∈ Sφ. Then It is clear that µ is C ∞ . By elementary computations similar to that of Remark 3.1, if b ≠ 0 one has which makes sense even if b = 0. Note that e a ∈ U A and that x b is a selfadjoint element (in Sφ).
Let a(t), b(t) be smooth curves in A ah and N(φ)s with a(0) = b(0) = 0,ȧ(0) = z andḃ(0) = y, where x = z + y is an arbitrary element of (TSφ) 1 . Then it is clear that (using that the differential of the exponential map at the origin is the identity map) Therefore, using the inverse function theorem, one has the following result.   Proof. Consider the C ∞ map q : As → R >0 , q(a) = φ(a 2 ).
It is clearly a retraction: s : R >0 → As, s(t) = t 1/2 · 1 is a smooth cross section for q. In particular, q is a submersion. Then is a submanifold of As.
Let us prove that µs : N(φ)s → Sφ,s is a covering space. We have to show that every point in Sφ,s has a neighborhood V such that µ −1 (V) = ⋃︀ Uα where Uα are disjoint open subsets of N(φ)s and µ| Uα is a homeomorphism of Uα onto V. It is clearly that this is verified in x ≠ 1 or x ≠ −1.
Suppose that a(t), t ∈ (−r, r) is a smooth curves in N(φ)s with a(0 In both cases, there exists a ball V in Sφ,s such that

A pre-Hilbert-Riemann metric for P φ
As written in Remark 4.3, we shall identify Pφ ≃ P 1 (A, φ). Therefore the tangent space of Pφ Also P 1 (A, φ) ⊂ P 1 (L), and this last manifold has a well behaved Hilbert-Riemann structure induced by the Frobenius norm (it shall be recalled in Section 5).
We have pointed out though that P 1 (A, φ) is not a submanifold of P 1 (L), the differentiable structure of both spaces is quite different (the inclusion is dense in the topology of B(L)). Nevertheless this inclusion has the remarkable property to be locally geodesically complete: if two elements in P 1 (A, φ) lie close, the minimal geodesic of P 1 (L) which joins them lies inside P 1 (A, φ). This property would suggest to consider in P 1 (A, φ) the metric induced by this inclusion. We shall present below an intrinsic metric in Pφ, and will show thereafter that it is (a multiple) of the metric induced by P 1 (L).
Let us first characterize the tangent spaces of Pφ as quotient spaces.
i.e. a, b define the same tangent vector at [x 0 ] if φ(a * x 0 ), φ(b * x 0 ) ∈ iR and a − b = irx 0 for some r ∈ R. By naturally isomorphic we mean the following: if one chooses another representative x ′ 0 for [x 0 ], i.e. x ′ 0 = wx 0 for some w ∈ T, then the mapping
Note thatẇ(0) ∈ iR. Then it is clear that b − a ∈ iR · x 0 . It follows that the tangent space is contained in the quotient (2).
Let us prove that any element in this quotient can be realized as a velocity vector. Pick a ∈ A with φ(x * 0 a) ∈ iR. Note that a − φ(x 0 a * ) ∈ (TSφ)x 0 and [a − φ(x 0 a * )] is the same as the class of a in the quotient (2). Again, using i.e. the quotient norm in the quotient (2) induced by the norm ‖ · ‖φ in A.
Clearly the metric is well defined (it does not depend on the representative of [x 0 ]).
Note that since A is not complete with the norm ‖ · ‖φ, this quotient norm is non complete in (TPφ) [x0] . Also note that the orthogonal projection is given by the state φ: P(a) = φ(x * 0 a)x 0 . Therefore the infimum at the quotient norm is in fact a minimum, given by This quantity is positive: φ(a * a) = |φ(x * 0 a)| 2 means equality in the Cauchy-Schwarz inequality and therefore a = λx 0 , then φ(x * 0 a) = λ ∈ iR, and thus [a] = 0. From these observations, it follows that this metric coincides with (a multiple of) the Frobenius norm in P 1 (A, φ): where Tr denotes the usual trace in B(L) and ‖ · ‖ HS denotes the Hilbert-Schmidt norm.
Proof. Straightforward computations show that Thus (using that Note that φ(a * x) = φ(x * a) ∈ iR, and thus Therefore

Minimality of geodesics in P φ
The results in this section refer to minimal curves in Pφ, i.e. curves δ(t) = [α(t)], α(t) ∈ Sφ, with minimal length joining fixed endpoints. The length of the curve δ(t), 0 ≤ t ≤ 1, is given by denotes the quotient norm as in the Definition 5.2. Note that if P 0 ∈ Pa and Z ∈ Bas(A), then the geodesic δ(t) = e tZ P 0 e −tZ remains inside Pa. We shall call these curves geodesics in Pa, though we have not defined a linear connection in Pa. In the case when n (the rank of P 0 ) equals 1, we shall call them geodesics of Pφ. Note also that if δ(t) = e tZ P 0 e −tZ is a geodesic of Pa and U ∈ Uφ(A), then Uδ(t)U ♯ is also a geodesic, with exponent UZU ♯ . If n = 1, this means that [Ue tZ (x)] is a geodesic of Pφ.
Elements P in Pa extend to orthogonal projectionsP in L. Conversely, any orthogonal projection E in L which leaves A ⊂ L invariant, i.e. E(A) ⊂ A, induces an element E| A in Pa. Also, P 1 (L) ⊂ Gr(L) the Grassmann manifold of L which is just the set of orthogonal projections of L. In [3,4] a reductive structure was introduced in the Grassmann manifold of a Hilbert space (parametrized by selfadjoint projections). In [9] the geometry of the restricted or Sato Grassmannian was studied. Later in [10] it was characterised when there are unique geodesics between projections. Let us summarize this information, in the case of L, in the following remark. The orbits of the action coincide with the connected components of P(L). In particular, the projections of rank one form a connected component and coincide the space P 1 (L). 2. There is a natural linear connection in P(L). If dim(L) < ∞, it is the Levi-Civita connection of the Riemannian metric which consists of considering the Frobenius inner product at every tangent space. It is based on the diagonal -codiagonal decomposition of B(L). To be more specific, given P 0 ∈ P(L), the tangent space of P(L) at P 0 consists of all selfadjoint codiagonal matrices (in terms of P 0 ). The linear connection in P(L) is induced by a reductive structure, where the horizontal elements at P 0 (in the Lie algebra of U(L): the space of antihermitian elements of B(L)) are the codiagonal antihermitian operators. The geodesics of P(L) which start at P 0 are curves of the form with Z * = −Z, codiagonal with respect to P 0 . 3. It was proved in [4] that if P 0 , P 1 ∈ P(L) satisfy ‖P 0 − P 1 ‖ < 1, then there exists a unique geodesic (up to reparametrization) joining P 0 and P 1 . This condition is not necessary for the existence of a unique geodesic. 4. In [10] a necessary and sufficient condition was found, in order that there exists a unique geodesic joining two projections P and Q. This is the case if and only if 5. If dim(L) = ∞, and one endows each tangent space of P(L) with the usual norm of B(L), one obtains a continuous (non regular) Finsler metric. In [4] it was shown that the geodesics (3) remain minimal among their endpoints for all t such that |t| ≤ π 2‖Z‖ .
If dim(L) < ∞, the Frobenius metric is available to measure lengths of curves. In [9] it was shown that the geodesics remain minimal in the Frobenius norm as long as |t| ≤ π 2‖Z‖ , which is a condition on the usual operator norm. 6. It is sometimes useful to parametrize projections using symmetries S (S * = S, S 2 = 1), via the affine map P ←→ S P = 2P − 1.
Some algebraic computations are simpler with symmetries. For instance, the condition that the exponent Z (of the geodesic) is P 0 -codiagonal means that Z anti-commutes with S P0 . Thus the geodesic (3), in terms of symmetries, can be expressed S δ (t) = e itZ S P0 e −itZ = e 2itZ S P0 = S P0 e −2itZ .

It is clear that
Lx ∩ L ⊥ y = {0} and L ⊥ x ∩ Ly = {0} because < x, y >= φ(y * x) ≠ 0. It follows that there exists a unique geodesic in P(L) joining x ⊗ x and y ⊗ y.
Let Lx and Ly the complex lines generated by x and y in L. Since Lx ⊥ Ly, Lx ∩ L ⊥ y = Lx and L ⊥ x ∩ Ly = Ly , and therefore there exist infinitely many geodesics joining [x] and [y].