Traces in Functional Analysis and Operator Algebras Matthias Neufang Carleton University, Ottawa I. Traces in Banach space theory 1 1. The projective tensor product Let E be a Banach space. Then the dual pair (E∗;E) determines the trace Tr on E∗ E via ⊗ Tr(φ x) = φ, x : ⊗ h i Definition. Let E; F be Banach spaces. The projective tensor product E γ F is the completion of E F w.r.t. ⊗ ⊗ the norm n n z γ = inf xi yi n N; z = xi yi : k k fX k kk k j 2 X ⊗ g i=1 i=1 Alternatively: (E γ F )∗ = (E; F ∗) ⊗ B Putting on operator glasses, we have a natural map j : E∗ E (E) ⊗ !B given by j(φ x)(y) = φ, y x. Consider its extension ⊗ h i J : E∗ γ E (E): ⊗ !B Grothendieck defined the image E∗ γ E=Ker(J) with the ⊗ quotient norm, as the space of nuclear operators (E). N 2 Since φ, x φ x , the trace Tr extends to a functional jh ij ≤ k k k k on E γ E∗. ⊗ If J is injective, then the trace is defined for all nuclear operators. When is this the case? Theorem. J is injective iff E has the approximation property (AP). Definition. E has the AP if for any compact set K E ⊆ and " > 0 there is a finite-rank map S : E E such that ! S(x) x < " for all x K. k − k 2 ; All \usual" spaces { such as c0, C(K), Lp { have the AP; but NOT ( )!! B H 3 2. The Schatten ideals The nuclear operators on a Hilbert space are usually called H trace class operators ( ). By the above, T H ( ) = γ : T H H ⊗ H Alternative description: Every S ( ) has the form 2 K H 1 S = si ; ξi ηi X h· i i=1 for appropriate ONS (ξi), (ηi). Here, si(S) 0 are the 1=2 ≥ eigenvalues of S = (S∗S) (= singular values). j j Now, S ( ) is trace class iff (si(S)) `1. Then S γ = 2 K H 2 k k (si(S)) 1. k k Example: Operator on `2 of multiplication with `1-function The trace Tr(ρ) for ρ ( ) can be expressed as 2 T H 1 Tr(ρ) = ρξi; ξi Xh i i=1 with any ONB of . H One defines the Schatten class Sp( ) as the collection of H S ( ) with (si(S)) `p (1 < p < ), with the norm 2 K H 2 1 S p := (si(S)) p. One sets S ( ) = ( ). k k k k 1 H K H 4 S1( ) Sp( ) ( ) ( ) • H ⊆ H ⊆ K H ⊆ B H Sp( ) form 2-sided ideals in ( ) • H B H 1 1 Sp( )∗ = Sq( ) where + = 1 • H H p q the trace gives the dualities ( )∗ = ( ) • K H T H and ( )∗ = ( ) T H B H ( )∗ = ( ) 1 ( )? •B H T H ⊕ K H 3. Convolution of trace class operators commutative world: L1( ) noncommutative world: (L2( )) G T G pointwise product in (`1; ) composition in ( (`2( )); ) · T G · convolution in (L1( ); ) ??? G ∗ Let be a locally compact group. Then G L ( ) , (L2( )) 1 G !B G as multiplication operators. The pre-adjoint of this embedding gives a canonical quotient map π : (L2( )) = L2( ) γ L2( ) L1( ): T G G ⊗ G ! G We have π(ξ η) = ξη. ⊗ 5 Definition. [N.] For trace class operators ρ, τ (L2( )) 2 T G we define ρ τ = Z LxρL 1π(τ)(x)dλ(x): ∗ x− G Then we obtain a new associative product on the space (L2( )). This defines a convolution of trace class T G operators! Your (legitimate) objection: Does this indeed extend classical convolution to the non-commutative context? Let's convolve matrices! If A = [Ai;j] and B = [Bk;l] are matrices, then their convolution product C = A B is given by ∗ Ci;j = A 1 1 Bt;t: X t− i;t− j t 2G Thus, in case A and B are just functions (i.e., diagonal matrices), C as well is a diagonal matrix, namely Ci = A 1 Bt: X t− i t 2G So (`1; ) becomes a subalgebra of ( (`2); ). ∗ T ∗ 6 Some \amuse-gueules" for the study of The Non-Commutative Convolution Algebra ( ) = ( (L ( )); ) S1 G T 2 G ∗ If 1( ) = 1( 0), then = 0. • S G ∼ S G G ∼ G amenable 1( ) right amenable (`ala Lau) •G () S G The dual space ( ) = 1( )∗ becomes a Hopf{von • S1 G S G Neumann algebra; its comultiplication extends the one of L ( ) = L1( )∗. 1 G G Thus, 1( ) is an operator convolution algebra in the sense S G of Effros–Ruan. Z (f g) dλ = Z fdλ Z gdλ ∗ Tr(ρ τ) = Tr(ρ) Tr(τ) ∗ 1( ) yields a Banach algebra extension of L1( ): •S G G ι π 0 ( ) ( (L2( )); ) (L1( ); ) 0 ! ML1 G ? ! T G ∗ ! G ∗ ! The ideal I := ( ) satisfies I2 = (0). ML1 G ? In case is discrete, ( 1( ); π; ι) is a singular extension G S G of the group algebra (`1( ); ). G ∗ 7 Some more homology theory Talk at International Conference on Banach Algebras 2001 in Odense, Denmark: Presentation of my 1( ) to the Banach algebra community S G homological properties of ( ) S1 G ! properties of G The following results are due in part to Prof. Pirkovskii { a former student of Prof. Helemskii and active member of his \Moscow school" { and in part to myself. is discrete the above extension splits G , L1( ) is projective in mod- 1( ) , G S G is compact C is projective in mod- 1( ) G , S G is finite 1( ) is biprojective G ,S G is amenable 1( ) is biflat G ,S G C is flat in mod- 1( ) , S G 8 Alternative view through Hopf algebra glasses Definition. A Hopf{von Neumann algebra is a pair (M; Γ) where M is a von Neumann algebra; • Γ: M M M is a co-multiplication, i.e., a normal, • ! ⊗ unital, isometric -homomorphism satisfying (id Γ) Γ = ∗ ⊗ ◦ (Γ id) Γ (co-associativity). ⊗ ◦ Extend comultiplication of the Hopf{vN (even Kac) algebra L ( ) to (L2( )) by setting 1 G B G ∆(x) = W (1 x)W ∗ (x (L2( ))) ⊗ 2 B G where W is the fundamental unitary for L ( ): W ξ(s; t) = 1 G ξ(s; st). Then our product in 1(G) := ( (L2( )); ) is S T G ∗ precisely ∆ . ∗ 9 Consider dual Kac algebra VN (G) = λ(x) x 00 = f j 2 Gg group vN algebra. Its predual is the Fourier algebra A( ) = G λ( )ξ; η ξ; η L2( ) . fh · i j 2 G g Then using W yields a \pointwise" product on (L2( )). T G c We recover (L1(G); ) and (A(G); ) as complete quotient ∗ · algebras of ( (L2); ) resp. ( (L2); ): T ∗ T • ( (L2); ) (L1(G); ) T ∗ ∗ ( (L2); ) (A(G); ) T • · (L2( )) carries 2 dual structures: a \convolution" and a •T G \pointwise" product! We have: Tr(ρ τ) = Tr(ρ τ) = Tr(ρ)Tr(τ) • ∗ • ; This can even be done over locally compact quantum groups (Junge{N.{Ruan)! 10 II. Traces in operator space theory 11 1. What are operator spaces? Modern answer: www.math.uni-sb.de/~ag-wittstock/projekt2001.html Short answer: Definition. A concrete operator space is a subspace X of ( ) for some Hilbert space . B H H Simple observation: X inherits the structure of ( ) on each B H matrix level, i.e., we have isometrically n Mn(X) Mn( ( )) = ( ::: ) = ( ) ⊆ B H B H ⊕ ⊕ H B H for every n N. 2 The norms n obtained on Mn(X) satisfy Ruan's axioms: k · k (R 1) a x b n a M x n b M k · · k ≤ k k n k k k k n for all x Mn(E), a; b Mn 2 2 x 0 (R 2) = max x n; y m 0 y fk k k k g n+m for all x Mn(E); y Mm(E) 2 2 12 Definition. [Ruan '88] An abstract operator space is a Banach space which carries a sequence of (matrix) norms satisfying (R1) and (R2). Theorem. [Ruan '88] Every abstract operator space is a concrete one. For any Banach space E, we have an isometric embedding ι : E, C(Ω) ! where Ω = BALL(E∗). Here, (ι(x))(ξ) = ξ; x : h i But C(Ω) is the prototype of a commutative C∗-algebra. Hence we obtain the following scheme: Banach spaces Operator spaces E, C∗-algebra E, C∗-algebra !A !A commutative non-commutative A A Another nice way of distinguishing both concepts is the following. E Banach space norm E ! k · k E ( ) operator space sequence of matrix ⊆ B H ! norms ( ) k · kMn(E) 13 We now define, for each x Mn(E): 2 Sn x := lim x Mn(E) k k1 n k k K(E) := M (E)k·k1: [ n n It is crucial to note that even if dimE < , we have 1 sup dim Mn(E) = ; i:e:; dim K(E) = : n 1 1 Thus, the \non-commutative" unit ball is NON compact. The reason for this is that we replace the scalars C by the n n-matrices (i.e., the \scalars" of quatum mechanics) { or, × equivalently, by the elements of = K(C) = M (C)k·k1: [ n K n Summarizing we obtain the following picture: E Banach space λixi E ! k P k where λi C, xi E 2 2 E ( ) operator space λi xi K(E) ⊆ B H ! k P ⊗ k where λi , xi E 2 K 2 14 Philosophy: Operator space theory studies the many ways a Banach space can \sit" in ( ).