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32
C*-ALGEBRAS IN RECONSTRUCTION OF MANIFOLDS
M. I. Belishev
St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, St. Petersburg, Russia
belishev@pdmi.ras.ru
PACS 02.30.Jr, 02.30.Yy
We deal with two dynamical systems associated with a Riemannian manifold with boundary. The first one is a system governed by the scalar wave equation, the second is governed by Maxwells equations. Both of the systems are controlled from the boundary. The inverse problems are to recover the manifold via the relevant measurements at the boundary (inverse data). We show that that the inverse data determine a C*-algebras, whose (topologized) spectra are identical to the manifold. By this, to recover the manifold is to determine a proper algebra from the inverse data, find its spectrum, and provide the spectrum with a Riemannian structure. This paper develops an algebraic version of the boundary control method (M.I.Belishev'1986), which is an approach to inverse problems based on their relations to control theory.
Keywords: inverse problems on manifolds, C*-algebras, boundary control method.
1. Setup 1.1. Acoustics
We deal with a compact C^-smooth Riemannian manifold Q with the boundary r, dimQ = n ^ 2; A is the (scalar) Beltrami-Laplace operator on Q; H := L2(Q). Forward problem of acoustics is to find a solution u = uf (x,t) of the system
utt - Au = 0 in (Q\r) x (0,T) (1.1)
u|t=o = ut|t=o = 0 in Q (1.2)
u = f on r x [0,T] , (1.3)
where f e FT := L2 (r x [0,T]) is a (given) boundary control.
With the system one associates a response operator RT : FT ^ FT,
,T „ duf
RTf :=
dv
rx[0,T ]
(for smooth enough f), v is the outward normal to r.
Inverse problem is: given for a fixed T > diam Q the operator R2T, to recover Q. 1.2. Electrodynamics
Let Q be oriented, dimQ = 3. The definitions of the vector analysis operations A, curl, div on a manifold see, e.g., in [9].
Forward problem Find a solution e = ef (x,t), h = hf (x,t) of the Maxwell system
et = curl h, ht = -curl e in Q x (0,T) (1.4)
e|t=o = 0, h|t=o = 0 in Q (1.5)
v A e = f in r x [0,T] , (1.6)
f G FT := L2 ([0,T]; Tr) is a boundary control (time-dependent tangent field on T). With the system, one associates a response operator RT : FT m FT,
RT f := v A hf
rx[Q,T ]
(for smooth enough f).
Inverse problem is: given for a fixed T > diamQ the operator R2T, to recover Q. 1.3. Nonuiqueness
Let Q' be such that dQ' = dQ = r and there is an isometry i : Q m Q' provided i|r = id. Then, for the response operators of the systems (1.1)-(1.3) and (1.4)-(1.6) one has R'T = RT for all T > 0.
Hence, the map "manifold m its response operator" in not injective. By this, to determine Q uniquely is impossible, and we have to clarify the setup of the inverse problems as follows [3]. The only reasonable setup is: given R2T for a fixed T > diamQ, to construct a Riemannian manifold Q such that dQ = dQ = r and R'2T = R2T. Philosophical question: From what "material" can such an Q be constructed? Answer in advance: Q is a spectrum of a relevant C*-algebra determined by R2T.
2. Eikonal algebra in Acoustics 2.1. Reachable sets
Return to the system (1.1)-(1.3). Controllability For an open a C r, define a reachable set
U := {uf (■ ,T) | supp f C a x [T - s,T]} CH (0 <s ^ T)
of delayed controls acting from a. Denote
• Qs[a] := {x G Q | dist (x, a) < s} (the metric neighborhood of a)
• H(QS[a]) := {y G H | supp y C Qs[a]} (the subspace of functions supported in Qs[a]).
A finiteness of the wave propagation speed in Q implies U C H(QS[a]). The Holngren-John-Tataru uniqueness theorem leads to the relation
US = H(Qs[a]) (2.1)
(closure in H), which is referred to as a local approximate boundary controllability of the system (1.1)-(1.3) [1]. For T > diam Q, one has UT = H. Eikonals Let PS be the projection in H onto U*. By (2.1) one has
Piv=hin Qsa,,,, (2.2)
10 in Q\QS[a] i.e., PS cuts off functions on Qs[a]. An operator
:= / sdPj
Q
is called an eikonal. If T > diamQ, then (2.2) implies
(r^y) (x) = dist (x, a) y(x), x G Q , i.e., is a multiplication by the distant Unction. It is a bounded self-adjoint operator in H.
2.2. Algebra T
Recall that a spectrum A of a commutative Banach algebra A is the set of its maximal ideals endowed with the Gelfand topology [7], [8]. If A and B are two isometrically isomorphic algebras (we write A ls=m B), then their spectra are homeomorphic (as topological spaces; we write A = B). For the algebra of real continuous functions C(Q), one has C(Q) = Q [7], [8].
For a set S C A, by VS we denote the minimal norm-closed subalgebra of A, which contains S. Let B(H) be the (normed) algebra of bounded operators in H. By T := V{ra | a C r} C B(H) we denote the (sub)algebra generated by eikonals.
Theorem 1. If T > diamQ then T ls=m C(Q) and hence T h==m C(Q) h== Q.
2.3. Solving IP
Connecting operator With the system (1.1)-(1.3) one associates a connecting operator CT : FT ^ FT defined by the relation
(CTf, g) ft = K (■, T), U (■, T)) h , f, g e FT.
It is a positive bounded operator. The following is a key fact of our approach (the Boundary Control method).
Proposition 1. The operator CT is determined by the response operator R2T via a simple and explicit formula [1], [3].
Isometry UT By the definitions, the map
UT : UT 3 (■ , T) ^ (CT) 2 f e FT
is an isometry. For T > diam Q, one has UT = H, and UT is a unitary operator from H onto
(CT)1FT.
Let PS := UTPS(UT)* be the projection in F^ onto the subspace
I2f | suppf C a x [T - s,T]) = UTUs .
By Proposition 1, PS is determined by the response operator R2T. By the latter, the operators
T := UTtv(UT)* = / sd [UTP;(UT)*] = / sdP; (2.3)
are also determined by R2T. We define an algebra T := UTT(UT)* c B ((CT)1FT) . By the definition, we have
T = UT [V{rCT | a C r}] (UT )* = V{Q | a C r} . (2.4)
By the aforesaid, this algebra and its spectrum T =: Q are determined by the response operator R2T. Since T ls=m T, with regards to Theorem 1 one has
Q h=m T =m T =: T (2.5)
as T > diamQ.
0
0
Reconstruction The response operator R2T (provided T > diam Q) determines the manifold Q up to a homeomorphism by the following scheme:
r2T P^pi CT ^ |(CT)2f | supp f C a x [T - s,T]} ^ ^ {pP; | a C r} (2I) {Q | a C r} ^ T ^
^ T (2=5) P h=m Q.
Then, one can endow p with a proper Riemannian metric and identify dp with r (see, e.g., [5]).
As a result, we get a Riemannian manifold p, which is isometric to the original (unknown) Q by construction, and R2T = R2T does hold. The inverse problem for the system (1.1)-(1.3) is solved.
3. Eikonal algebra in Electrodynamics
3.1. Maxwell system
Turn to the system (1.4)-(1.6). The Hilbert space L2(Q) of the square-summable vector fields (sections of the tangent bundle TQ) contains the subspace of curls
C := jcurl h | h, curl h G L2(Q)j.
Electric reachable sets For an open a C r, define
ES := {ef (■ ,T) | suppf C a x [T - s,T]} cC (0 <s ^ T).
We denote C(QS[a]) := {y G C | suppy C QS[a]}. The finiteness of the electromagnetic wave propagation speed in Q implies ES C C(QS[a]).
Controllability The Eller-Isakov-Nakamura-Tataru uniqueness theorem leads to
ES = C(QS[a]) (3.1)
(the local boundary controllability). For T > diam Q, one has EJ = C. Projections Let ES be the projection in C onto Es. This projection acts in more complicated way than its acoustic analog: its action is not reduced to cutting off fields. Moreover, in the general case, for the different a and a' the projections ES and ES/ do not commute. Eikonals An operator
£o := f sdES
a Q
acts in the space C and is called an eikonal. Since diamQ < to, £o is a bounded positive self-adjoint operator. In the general case, for a = a' the eikonals E T and E T/ do not commute. The following fact plays a key role.
Lemma 1. (M.N.Demchenko [6]) The representation
(eo-y) (x) = dist (x,a) y(x) + (KTy) (x), x G Q holds with a compact operator KT : Cm L 2 (Q).
3.2. Algebra E
Let B(C) be the (normed) algebra of bounded operators in C. It contains the two-side ideal K(C) of compact operators. We denote by
E := V{eo | a C r}
the algebra generated by (electric) eikonals. Also, we denote K[E] := E if K(C) and introduce the factor-algebra
E := E/K[E]
Theorem 2. (M.N.Demchenko [6]) The factor-algebra E is a commutative Banach algebra. The relation E C(Q) holds and implies E h= Q.
3.3. Solving IP
Connecting operator A Maxwell connecting operator CT : FT m Ft is introduced by the relation
(CTf,g)^T = (ef (■ ,T),eg(■ ,T))c
for smooth controls f, g G FT vanishing near t = 0 [3]. In contrast to the scalar (acoustic) case, this CT is an unbounded operator. However, the following principal fact of the BC-method remains valid.
Proposition 2. The operator CT is determined by the response operator R2T via a simple and explicit formula [3], [5].
Isometry UT By the definitions, the map
UT : EJ 9 ef (■ , T) m (C t ) 2 f gFt
is an isometry. For T > diam Q, by (3.1) one has EJ = C, and UT is a unitary operator from C onto (CT) 2FT C FT.
By Proposition 2, the projection ES := UTES(UT)* in (CT) 2FT onto the subspace
f | supp f c ä x [T - s, T]} = UTEs
is determined by the response operator R2T. An operator
£ := UT4(UT)* = [T sd [UTES(UT)*] = [T sdES
.7 0 .7 0
acts in (CT)1FT and is determined by the response operator R2T. An algebra
E := UTE(UT)* = UT [V{eo | a C r}] (UT)* = V{eo | a C r}
is a subalgebra of B (JCT) 2 FTJ. By the aforesaid, this algebra, the factor-algebra E := E/K[E] and its spectrum
E =: Q
are determined by the response operator R2T.
The isometry E ls= E implies the isometry of the factors E ls= E. Theorem 2 leads to
Q h=m E h=m E =: ft.
Reconstruction The response operator R2T (provided T > diam Q) determines the manifold Q up to a homeomorphism by the following scheme:
R2T ^ CT ^ |(CT)1 /| supp / c a x [T - s,T]| ^ {EE^ | a c r} ^ {ft | a c r} ^ E ^ (5 ^ E =: ft h=m Q .
Then, one can endow ft with a proper Riemannian metric and identify dft with r (see, e.g., [5]).
As a result, we get a Riemannian manifold ft, which is isometric to the original (unknown) Q by construction, and R2T = R2T does hold. The inverse problem for the Maxwell system (1.4)-(1.6) is thus solved.
Acknowledgments
The work is supported by grants RFBR 11-01-00407A and SPbGU 11.38.63.2012, 6.38.670.2013.
References
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