MINIMALITY of GEODESICS in GRASSMANN MANIFOLDS HORACIO PORTA and LÁZARO RECHT Dedicated to L

PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 100, Number 3, July 1987

MINIMALITY OF GEODESICS IN GRASSMANN MANIFOLDS HORACIO PORTA AND LÁZARO RECHT Dedicated to L. A. Santaló on his 75th birthday

ABSTRACT. In the Grassmann manifold of an arbitrary C*-algebra, the geodesies of length less than it are curves of minimal length.

The classical Grassmann manifolds Gqspace of symmetries, which turns out to be an analytic (disconnected) submanifold of the space of n x n matrices. If in this interpretation we choose the operator norm of matrices, C?9i„ is not a Riemannian manifold. However a natural connection arises and the theorem below implies that short geodesies in Gq¡n are minimal curves for the arclength derived from the operator norm. This works equally well in the infinite-dimensional setting where no Riemannian structure is in general available (similar to tr(AB*) for example). Let A be a C*-algebra with identity 1. The Grassmannian Gr(y?) of A is the set of "symmetries" of A, i.e., the elements s G A with s* = s and s2 = 1. Equivalently, Gr(i?) can be identified to the set of selfadjoint idempotents p if s = 2p — 1. Given a symmetry s = 2p — 1 denote by Projs : A —>■A the real-linear norm- bounded projection projs(2) = (1 -p)zp + pz*(l -p). Let Ts and Ns be the range and kernel of Projs. Observe that (1) x GTS if and only if x is selfadjoint and anticommutes with s. Also define the analytic map E9 : A —*A by

Es(z)=ezs/2se~zs/2.

The following is an explicit form of "close symmetries are conjugate" : (2) Let s and r be symmetries with \s — r\ = d < 2 and set x = 2slog(a|a|_1), where a — (1 -j- sr)/2. Then x 6 Ts and Es(x) = r. From ||1-a|| = ||s(s-r)/2|| = d/2 < 1 and aa* = ((s + r)/2)2 =a*a, it follows that a is an invertible normal element. Then u = a|a|_1 = |a|_1a is unitary with ||1 — w|| < d/2 < 1 (again from ||1 — o|| < d/2 and the spectral theorem). Hence w = logu is (well defined if the principal branch of log is used and) skewsymmetric: —w — w*. Further (since sa = ar = (s + r)/2 is selfadjoint) so = ar — a* s — ra* and therefore s and r commute with |oj. In particular

sus = sa]a\~1s = a*s\a\~ s = o*|a|~ = u_1

Received by the editors December 25, 1985 and in revised form April 1, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L05.

464

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whence (recall that s = s l and use the series of log, for example) sws — —w or x = 2sw = —2ws. Thus x = x* and sx = —xs, and (1) implies x € Ts. Also

Es(x) = exal2se-xs/2 = e~wsew = u*su = H-VscM-1 = |a|-1«a*a|o|_1 = r,

which completes the proof of (2). Next consider the map $ : TS®NS -* A, $(x,y) = Es(x)+y. Clearly $(0,0) = s and the derivative of $ at x = 0, y = 0 is the "identity" map (h, k) —►h + k from Tx © yVa to Ts + Ns = A. According to (2) above Gr(A) is locally the image of Ts © {0} under the local diffeomorphism $. Hence (3) Gr(A) is a real-analytic submanifold of A. The inclusion Gv(A) c—>A iden- tifies the tangent space to Gr(y?) at s with the real Banach subspace Ts of A. The normal bundle {Ns} of Gr(y!) induces the canonical connection of Gr(yî) defined by D.y = Proja(|»(c(t)) _ )

where x E Ts, y is a tangent field near s, and the curve c(t) € Gr(A) satisfies c(0) = s, ¿(0) = x. (4) The geodesies through s € Gr(y?) have the form

7(f) = etxs/2se-txs/2

and in particular the exponential of the canonical connection is given by the maps Es above. It suffices to verify that d^/dt(0) = x and Proj7/tj d2^i/dt2(t) = 0; both follow from routine calculations using d^/dt = etxsl2Xe~txs/2 and d2^/dt2 — —x2^ = —7a;2. Observe that this implies

(5) d21/dt2 + ]\x\]21 = 1(\\x\\2-x2).

REMARK. All these notions coincide with the usual ones in the case of Gq,„ (see, for example, [W]). THEOREM. Short geodesies in the Grassmann manifold of a C*-algebra have minimal length. More precisely, a geodesic of length less that w is shorter than any other curve joining its endpoints. PROOF. Suppose 7(i) = etxs/'2se-txa/2. Since x = x* there is a state / on A such that f(x2) = \\x\\2 (for C*-algebra concepts see, for example, [A]). Let p: A —►£(M) be the associated cyclic representation with cyclic vector f £ M, so that f(a) = (p(a)£, £) for all a G A. In this situation £ is an eigenvector of p(x2) with eigenvalue ||x||2. In fact, since 0 < p(x2) < \\x\\2 there is a hermitian operator H e Z()i) with ||x||2 - p(x2) = H2. Then ll^ll2 = <(NI2- P(x2)K,0 = IN2 - f(x2) = 0

or Ht] — 0 whence p(x2)t¡, = ||z||2£- Consider the curve g(t) in the unit sphere E of )i defined by g(t) — p("l(t))Ç. From (5) it follows that

d2g/dt2 + \\x]\2g = p(d21/dt2 + ||x||37)€ = P(l)p(\]xf - x2)t = 0

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so that g is a geodesic in S for the Riemannian structure induced by U. Suppose p(t) is another curve in Gr(y?) joining ¡i(0) = 7(0) and h(t) = 7(7) and let m(t) = p(p,(t))£. The curve m in S joins g(0) to g(r) and since ||dm/di|| < ||d/i/di||, its length L(m) = f* \\dm/dt\\dt does not exceed the length L(p) of \x. Also \\dg/dt\\ = \\x\\2 = ||d7/df||, so L(g) = ¿(7). Now if ¿(7) < n then 0 is a "short" geodesic in E, hence minimal, and therefore ¿(7) = L(g) < L(m) < L(p). This proves the theorem. According to (2) if ||s — r\] = d < 2, then a geodesic Es(tx) joining s = Es(0) to r = Es(x) can be explicitly described; its length L is L — \\(dEs(tx) / dt)(0)\\ = ||i||. Using the definition of x in (2) ||x|| = ||2slogo|a|_1|| = 2|| loga - (l/2)logaa*|| = || logo - logo*|| and then L = ||log(l + rS)/2-log(l + sr)/2||. The estimate ||1 — u\\ < d/2 (see (2)) gives in particular L < 2arcsin(d/2). Suppose that Ea(ty) (y ETS) also joins stor = Es(y) and abbreviate v = eys¡2. Then sus = esxl2 — u* so usu — s and similarly vsv = s. But usu* = vsv* = r so ru2 = usu = vsv = rv2 and u2 — v2, i.e., (exs/2)2 — (eys/2)2. This shows that uniqueness of minimizing goedesics is tied to uniqueness of square roots and logarithms. It can be proved that (6) If ]\s = r\\ = d < 1, then the minimizing geodesic joining s and r is unique. We are grateful to Norberto Sardinas for several valuable suggestions. In'particular (2) and (6) are due to him.

REFERENCES

[A] W. Arveson, An invitation to C-algebra, Springer, 1976. [W] J. Wolf, Spaces of constant curvature, Berkeley, 1972.

Department of Mathematics, University of Illinois, Urbana, Illinois 6I8OI- 2975

MATEMÁTICAS Y CIENCIAS DE LA COMPUTACIÓN, UNIVERSIDAD SIMÓN BOLIVAR, CARACAS, VENEZUELA

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