Multi-normed spaces and multi-banach algebras. H. G. Dales. Leeds Semester

Transcription

1 Multi-normed spaces and multi-banach algebras H. G. Dales Leeds Semester Leeds, 2 June

2 Motivating problem Let G be a locally compact group, with group algebra L 1 (G). Theorem - B. E. Johnson, 1972 The Banach algebra L 1 (G) is amenable if and only if the group G is amenable. Theorem - Helemski, Johnson Let A be an amenable Banach algebra. Then E is injective for each Banach right A-module. We do not know if the converse holds. If A is a Banach algebra such that E is injective for each, or some, Banach right A-module. The Banach space L p (G) is a Banach left L 1 (G)- module in a canonical way. 2

3 Some results Theorem Suppose that G is an amenable locally compact group. Then L p (G) is an injective Banach left L 1 (G)-module for each p (1, ). We ask if the converse to this holds. For partial results, see a paper of D and Polyakov in Proc. London Math. Soc. Attempts on this question led to a theory of multi-norms, which may have a life of its own. See a proto-memoir of 140 pages, and some Bangalore notes. For a solution in the case where A is L 1 (G) and the module is any L p (G), and more, see the second conference. (Work of Matt Daws, Hung Le Pham, and Paul Ramsden.) Here we give some background on multi-norms; the connections with group algebras and new characterizations of amenability for locally compact groups will come in a talk at the second conference. 3

4 A second motivating problem Again let G be a locally compact group, with measure algebra (M(G), ). For µ M(G), set T µ (f) = µ f (f L p (G)). (Here 1 < p < ; usually, p = 2, so that L 2 (G) is a Hilbert space.) Then T µ B(L p (G)), and the map µ T µ is a representation of M(G). Always σ(t µ ) σ(µ), but maybe σ(t µ ) σ(µ), which can be unfortunate - how can we cure this? 4

5 Basic definitions Let (E, ) be a normed space. Definition A multi-norm on {E n : n N} is a sequence ( n ) such that each n is a norm on E n, such that x 1 = x for each x E, and such that the following hold for all n N and all x 1,..., x n E: (A1) (x σ(1),..., x σ(n) ) = (x n 1,..., x n ) n for each permutation σ of {1,..., n}; (A2) (α 1 x 1,..., α n x n ) n (max i Nn α i ) (x 1,..., x n ) n for each α 1,..., α n C ; (A3) (x 1,..., x n, 0) n+1 = (x 1,..., x n ) n ; (A4) (x 1,..., x n, x n ) n+1 = (x 1,..., x n ) n. 5

6 Another representation of multi-norms Let M m,n be the algebra of m n-matrices over C, and give it a norm by identifying it with B(l n, l m ). Let E be a normed space. Then M m,n acts from E n to E m in the obvious way. Consider a sequence ( n ) such that each n is a norm on E n and such that x 1 = x for each x E. Theorem This sequence of norms is a multinorm if and only if a x m a x n for all m, n N, a M m,n, and x E n. 6

7 Elementary consequences The following hold for all x 1,..., x n+1 E, etc: 1) (x 1,..., x n ) n (x 1,..., x n, x n+1 ) ; n+1 2) max x i (x 1,..., x n ) n n i=1 x i ; 3) (x 1,..., x n, y 1,..., x m ) m+n (x 1,..., x n ) n + (y 1,..., y m ) m. 7

8 Minimum and maximum multi-norms Example 1 Set (x 1,..., x n ) n = max x i. This gives the minimum multi-norm. Example 2 It follows from 2) that there is also a maximum multi-norm, say it is ( max n ). Note that it is not true that n i=1 x i gives the maximum multi-norm because it is not a multi-norm. (It does fit into a more general scenario.) 8

9 Another characterization Let (E, ) be a normed space. Then a c 0 -norm on c 0 E is a norm such that a x a x for all a c 0 and x E and such that T I E is a bounded linear operator on (c 0 E, ) with T I E = T whenever T is a compact operator on c 0. Theorem (Daws) Multi-norms on {E n : n N} correspond to c 0 -norms on c 0 E. The injective tensor product norm gives the minimum multi-norm, and the projective tensor product norm gives the maximum multi-norm [Cf Alexander Helemskii s abstract theory of operator spaces.] 9

10 And another characterization of multi-norms There is a paper La structure des sous-espaces de trellis by J. L. Marcolino Nhani, apparently a student of a student of Pisier, in Dissertationes Math., He introduces condition (P) : for a normed space E, there is a norm α on E c 0 such that, for each T B(c 0 ) and each x E c 0, α((i E T )(x)) T α(x). This condition is equivalent to Daws condition, and so characterizes multi-norms. A theorem of Pisier shows that, in this case, E can be identified with a subspace of a certain Banach lattice X. The structure on X gives exactly what I had already called a Banach lattice multi-norm; see below. This relates our theory to that of operator sequence spaces of Volker Runde etc. 10

11 An associated sequence Let ( n ) be a multi-norm on {E n : n N}. Define ϕ n (E) = sup { (x 1,..., x n ) n : x i 1}. Trivially, 1 ϕ n (E) n for all n N and ϕ m+n (E) ϕ m (E) + ϕ n (E) for all m, n N. What is the sequence (ϕ n (E))? In particular (ϕ max n (E)) is the sequence associated with the maximum multi-norm. It can be shown quite easily that ϕ max (E) is sup where λ 1,..., λ n E and n j=1 n λ j, j=1 x, λ j 1 (x E [1] ). n 11

12 Summing norms - I Let E be a normed space, and take p [1, ). For x 1,..., x n E, set µ p,n (x 1,..., x n ) = sup λ E [1] Then µ 1,n (x 1,..., x n ) = sup n j=1 n j=1 For λ 1,..., λ n E, we have µ 1,n (λ 1,..., λ n ) = sup n j=1 x j, λ p 1/p. ζ j x j : ζ 1,..., ζ n T x, λ j : x E [1] Let E and F be Banach spaces, and take T B(E, F ) and n N. Then π (n) (T ) is sup n j=1 T x j p 1/p p : µ p,n (x 1,..., x n )

13 Summing norms - II Definition π p (T ) = lim n π (n) p,n (T ) is the p- summing norm of T. The p-summing operators form an operator ideal. We write π p (n) (E) for π p (n) (I E ) and π p (E) for π p (I E ). Theorem Let E be a normed space, and let n N. Then If E = F, then ϕ max n (E) = π (n) 1 (E ). ϕ max n (E) = π (n) 1 (F ). 13

14 Another constant Let F be a normed space, and let S F unit sphere of F. It is useful to define be the c n (F ) = inf{µ 1 (λ 1,..., λ n ) : λ 1,..., λ n S F }. Easy fact: π (n) 1 (F )c n(f ) n. In fact π (n) 1 (F )c n(f ) = n, where π (n) 1 (F ) is the version of π(n) 1 (F ) with x 1 = = x n in the definition of µ 1 (x 1,..., x n ). Guess: I think that there should be a large class of Banach spaces F with the property that there is a constant C F such that π (n) 1 (F ) C F π (n) 1 (F ) for all n N. Is this correct? Is it true for F = l q whenever q [1, 2]? In the latter case, F is an Orlicz space, and there is a constant C q such that c n (l q ) C q n (n N); what is C q in the complex case? 14

15 A lower bound We shall use the famous theorem of Dvoretzky, sometimes called the theorem on almost spherical sections. Theorem For each n N and ε > 0, there exists m = m(n, ε) in N such that, for each normed space E with dim E m, there is an n-dimensional subspace L of E such that d(l, ln 2 ) < 1 + ε. Theorem Let E be an infinite-dimensional normed space. Then ϕ max n (E) n for each n N. Corollary Let E be an infinite-dimensional normed space. Then ϕ max n (E), and there is a multi-norm on E not equivalent to the minimum multi-norm. 15

16 Special spaces Take p with 1 p, and write q for the conjugate index to p. Take E = l p. Thus ϕ max n (l p ) = π (n) 1 (l q ). We write ln p for the n-dimensional space C n with the usual l p -norm. Direct calculations of (l p ) using Banach Mazur distance give: ϕ max n Theorem (i) For each p [1, 2], we have ϕ max n (ln) p = ϕ max n (l p ) = n 1/p (n N). (ii) For each p [2, ], there is a constant C p such that n ϕ max n (ln) p ϕ max n (l p ) C p n (n N). In general, I do not know the best constant C p in the above inequality. 16

17 Example 1 : Standard (p, q)-multi-norm Let Ω be a measure space, and take p, q with 1 p q <. We consider the Banach space E = L p (Ω), with the usual L p -norm. For each family X = {X 1,..., X n } of pairwisedisjoint measurable subsets of Ω such that X 1 X n = Ω, we set r X ((f 1,..., f n )) = ( PX1 f 1 q + + PXn f n q ) 1/q where P X : L p (Ω) L p (X) is the natural projection. Finally, (f 1,..., f n ) n = sup X r X ((f 1,..., f n )). This is the standard (p, q)-multi-norm. Remark Let q = p.then (f 1,..., f n ) n = f 1 f n. 17

18 Example 2 : Measures Let Ω be a non-empty, locally compact space. Then M(Ω) is the Banach space of all regular Borel measures on Ω. Take q 1. For each partition X = {X 1,..., X n } of Ω into measurable subsets and each µ 1,..., µ n M(Ω), take r X ((µ 1,..., µ n )) to be ( µ 1 X 1 q + + µ n X n q ) 1/q, so that r X is a seminorm on M(Ω) n Then define (µ 1,..., µ n ) (1,q) n = sup X r X ((µ 1,..., µ n )), where the supremum is taken over all such families X. Then n is a norm on M(Ω) n, and it is again easily checked that ( n : n N) is a multi-norm on {M(Ω) n : n N}. It is the standard (1, q)-multi-norm. 18

19 Example 3 - Banach lattice multi-norms Let (E, ) be a complex Banach lattice. [Thus there are lattice operations on E R, the modulus x of an element x E R is defined, and the norm is such that x y whenever x y in E. Now E = E R ie R is a complex Banach lattice.] Example L p (Ω), L (Ω), or C(Ω) with the usual norms and the obvious lattice operations. Definition Let (E, ) be a Banach lattice. For n N and x 1,..., x n E, set (x 1,..., x n ) n = x 1 x n. Then ((E n, n ) : n N) is a multi-banach space. It is the Banach lattice multi-norm. It generalizes the standard (p, p)-multi-norm on L p (Ω) and the minimum multi-norms on L (Ω) and C(Ω). 19

20 Example 4 - The Schauder multi-norm Let E be a Banach space with an unconditional Schauder basis (e n ). Set n=1 α n e n = sup n=1 α n β n e n : β n 1 This norm is equivalent to the original one.. Let X = {X 1,..., X n } be a partition of N, and define r X ((x 1,..., x n )) = P X1 x P Xn x n, where P Xi are the obvious projections, and then set (x 1,..., x n ) n = sup X r X ((x 1,..., x n )). We again obtain a multi-norm ( n ). 20

21 Example 5 : The weak (p 1, p 2 ) multi-norm Again E is a Banach space. weak p -summing norm is µ p,n (x 1,..., x n ) = sup λ E [1] Here x 1,..., x n E. n j=1 Recall that the x j, λ Now take p 1, p 2 with 1 p 1 p 2 <. Define x (p 1,p 2 ) n = sup n j=1 x j, λ j p 1/p. p 1/p 2 2 taking the sup over all λ 1,..., λ n E with µ p1,n(λ 1,..., λ n ) 1. Fact {(E n, (p 1,p 2 ) n ) : n N} is a multi-normed space. It is the weak (p 1, p 2 ) multi-norm over E. 21

22 The weak (p 1, p 2 ) multi-norm, continued Fact The canonical embedding of E into E is a multi-isometry (see later) when we consider the weak (p 1, p 2 ) multi-norms over E and E. Fact We can work out the dual of this multinorm quite explicitly; there is pleasing duality theory. Fact There are relations between these. For example, take 1 p 1 p 2 < and 1 r 1 r 2 <. Suppose that p 2 r 2 and 1 p r 1 1 r p 1. Then (p 1,p 2 ) n (r 1,r 2 ) n on E n. 22

23 The weak (p 1, p 2 ) multi-norm, connections with L 1 (Ω) Fact For 1 q <, the weak (1, q) multinorm on the family {L 1 (Ω) n : n N} is the same as the standard (1, q) multi-norm described above. Fact There are several other useful identifications using measures and second duals. Fact Various nice multi-norms that I mentioned (and others) have canonical extensions - and these we now know are suitable weak (p 1, p 2 ) multi-norms. 23

24 Example 6 :The Hilbert multi-norm Let H = l 2 (S) be a Hilbert space. For each family H = {H 1,..., H n } of closed subspaces of H such that H = H 1 H n, set r H ((x 1,..., x n )) = ( P 1 x P n x n 2) 1/2 where P i : H H i for i = 1,..., n is the projection, and then set (x 1,..., x n ) H n = sup H r H ((x 1,..., x n )) ; we obtain multi-norms H n. We immediately have (x 1,..., x n ) n (x 1,..., x n ) H n, where ( n ) is the standard (2,2)-multi-norm on l 2 (S). 24

25 Maximality of the Hilbert multi-norm Question Is the Hilbert multi-norm the maximum multi-norm on the family {H n : n N}? This seemed to be very likely because I could not think of a bigger one. However it seems to be rather a hard question. In fact it can be reduced to a question about Hilbert spaces that does not mention multinorms. Let H be a Hilbert space. Then the closed unit ball of the dual of (H n, H n ) is described as follows. Set S := (α 1e 1,..., α n e n ) : n j=1 α j 2 1, where the union is taken over all orthonormal subsets {e 1,..., e n } of H. The required unit ball is the weak- -closed convex hull of S, call it K. 25

26 On the other hand, the closed unit ball of the dual of (H n, max n ) is {y = (y 1,..., y n ) H n : µ 1,n (y 1,..., y n ) 1} ; this set, temporarily called L, is equal to the set of y = (y 1,..., y n ) H n such that for all ζ 1,..., ζ n T. ζ 1 y ζ n y n 1 Since H n max n, necessarily K L. To establish the equality of the two multinorms, we need to show that L K for each (implicit) n N. In fact, we need ex L ex K = S (n N), where ex denotes the set of extreme points of a convex set. Is this always the case? 26

27 Towards this, I know the following. Theorem Let H be a Hilbert space of dimension n. (i) Suppose that n = 2. Then ex L S. (ii) Suppose that n = 3 and H is a real Hilbert space. Then this fails. (iii) (Pham) Suppose that n = 3 and H is complex. Then ex L S. (iv) (Daws) There is a universal constant C with C H n n max, and so the Hilbert multinorm is equivalent to the maximum multi-norm. [At present C is K G, Grothendieck s constant. Maybe we have C = 1.] 27

28 Multi-topological linear spaces Let E be a linear space, and let F be a subspace in E N such that, for each x E, we have (x, 0, 0,... ) F. A subset B of F is basic if: (1) for each permutation σ of N and (x n ) B, also (x σ(n) ) B; (2) for each (x n ) B and α n with α n 1, also (α n x n ) B; (3) for each (x n ) B, also (x 1, x 1, x 2, x 2, x 3,... ) B; (4) (x n ) B if and only if (x 1,..., x k, 0,... ) B for each k N. Suppose that F has a basis (in the usual sense of topological linear spaces) consisting of basic sets. Then F is a multi-topological linear space. 28

29 Examples of multi-topological linear spaces (1) Let E be a multi-normed space. Set F = {(x n ) : (x 1,..., x k ) k < (k N)}. Then F is a multi-topological linear space. The basic sets are {(x n ) F : (x 1,..., x k ) k C}. (2) Let E be a topological linear space, and set F = E N. Set B = U 1 U 2, with U i open in E, a basic set. Then F is a multi-topological linear space. (Box topology.) Definition Let F be a multi-topological linear space. Then (x n ) is multi-null if, for each basic set B, there exists n 0 N such that (x n, x n+1,... ) B (n n 0 ). There is a version of Kolmogorov s theorem. 29

30 Multi-convergence Proposition Let ((E n, n ) : n N) be a multi-normed space, and let (x i ) be a sequence in E. Then Lim i x i = 0 if, for each ε > 0, there exists n 0 N with (x n1,..., x nk ) k < ε (n 1,..., n k n 0 ). These are exactly the multi-null sequences. Theorem Multi-null sequences in a multi-normed space (E n, n ) are the null sequences of some topology on E if and only if the multi-norm is equivalent to the minimum multi-norm. 30

31 Multi-convergence - Examples 1) Let {E n : n N} have the minimum multinorm. Then Lim i x i = 0 if and only if lim i x i = 0 in (E, ). 2) Let E = l p with the standard (p, q)-multinorm, and let x i = α i δ i (i N). Then Lim i x i = 0 if and only if α i q <. Here δ i is the sequence (δ i,j : j N). 3) Let E = L p (Ω) with the standard (p, p)- multi-norm. A sequence (f n ) is multi-null iff (f n ) is order-bounded and f n 0 almost everywhere. More generally 4) Let (E, ) be an order-continuous Banach lattice, and consider the Banach lattice multi-norm on {E n : n N}. Then a sequence is a multi-null sequence if and only if it converges to 0 in order. 31

32 Multi-bounded sets and operators Let (E n, n ) be a multi-normed space. subset B of E is multi-bounded if A c B := sup { (x 1,..., x n ) n : x 1,..., x n B} <. n N Let (E n, n ) and (F n, n ) be multi-banach spaces. An operator T B(E, F ) is multibounded if T (B) is multi-bounded in F whenever B is multi-bounded in E. The set of these is a linear subspace M(E, F ) of B(E, F ). For T M(E, F ), set T mb = sup {c T (B) : c B 1}. Theorem Now ((M(E, F ), mb ) is a Banach space, and M(E) is a Banach operator algebra. [Recall that these depend on the multi-norm structure, and not just on the Banach space, despite the notation.] 32

33 The multi-bounded norm More generally, for n N and T 1,..., T n M(E, F ), set (T 1,..., T n ) mb,n = sup {c T1 (B) T n (B) : c B 1}. Theorem Now ((M(E, F ) n, mb,n ) : n N) is a multi-banach space. 33

34 Examples of M(E, F ) - I Throughout, {(E n, n ) : n N} and {(F n, n ) : n N} are multi-normed spaces. Fact Suppose that E and F are operator sequence spaces (see Lambert, Neufang, and Runde). Then the multi-bounded operators are just the sequentially bounded operators. Theorem Always N (E, F ) M(E, F ) B(E, F ), where N (E, F ) denotes the space of nuclear operators. Theorem Suppose that F has the minimum or E the maximum multi-norm structure. Then (M(E, F ), mb ) = (B(E, F ), ). Question When exactly do we get the above equality? 34

35 Examples of M(E, F ) Theorem We can have M(E, F ) = B(E, F ) and M(F, E) = N (F, E). Theorem Let E = l p and F = l q, where p, q 1. Regard them as multi-normed spaces with the standard (p, p) and (q, q) multi-norms, respectively. Then M(E, F ) consists of the regular operators. [An operator is regular if it is the difference of two positive operators.] 35

36 Another example Theorem We can have K(E) M(E). Proof Let H be the Hilbert space l 2 (N), with the standard (2, 2)-multi-norm. Consider the system of vectors (x s r : r = 1,... s, s N) defined as follows: x s r(k) = 0 except when k {2 s 1,..., 2 s 1} ; at the 2 s 1 numbers k in this set, x s r(k) = ±1/ 2 s 1, the values ±1 being chosen so that [x s r 1, x s r 2 ] = 0 when r 1, r 2 = 1,..., s and r 1 r 2. Such a choice is clearly possible. Then S := {x s r : r = 1,..., s, s N} is an orthonormal set in H. Order the set S as (y n ) by using the lexicographic order on the pairs (s, r). 36

37 Let (α i ) l. We define T B(H) by setting T x s r = α s δ n when x s r = y n. It is clear that, in the case where (α i ) c 0, we have T K(H). For k N, set N k = k(k + 1)/2. We see that However (y 1, y 2,..., y Nk ) 2 = k. N k (T y 1, T y 2,..., T y Nk ) 2 = k N i=1 i α i 2. k Now take γ (0, 1/2), and set α i = i γ. Then (T y 1, T y 2,..., T y Nk ) (y 1, y 2,..., y Nk ) Nk Nk ck (1 2γ)/2 for a constant c > 0. Since γ < 1/2, we have T M(H). We have shown that K(H) M(H). 37

38 M(E, F ) for Banach lattices Let (E, ) be a Banach lattice. Then E is monotonically bounded if each increasing, - bounded net in E has an (order) upper bound. Thus each Banach lattice L p (Ω) (for p [1, ]) and C(Ω) (for Ω compact) is monotonically bounded, but c 0 is not monotonically bounded. Theorem Let (E, ) and (F, ) be two monotonically bounded Banach lattices, each with the lattice multi-norms. Then M(E, F ) is the space of all order-bounded operators from E to F. 38

39 Multi-bounded and multi-continuous operators Let E and F be multi-topological linear spaces. An operator T : E F is multi-continuous if (T x i ) is multi-null in F whenever (x i ) is multinull in E. Theorem Let (E n, n ) and (F n, n ) be multi-normed spaces, and take T B(E, F ). Then T is multi-continuous if and only if T is multi-bounded. 39

40 Banach lattices: examples Return to the embedding µ T µ of M(G) into B(L p (G)). In fact, it is a mapping into M(L p (G)), when L p (G) has the standard (p, p)- multi-norm, and now we do get σ M (T µ ) = σ(µ) always, where σ M (T µ ) is the spectrum of T µ in the Banach algebra M(L p (G)). A failure of Banach s isomorphism theorem: Let E be a Banach lattice and consider the Banach lattice multi-norm. Then M(E) consists of the regular operators; this Banach algebra is B r (E). As mentioned, there are examples of T B r (E) such that T is invertible in B(E), but the inverse is not in B r (E). 40

41 The problem of duality Let E be a Banch space, and let ( n ) be a multi-norm on {E n : n N}. We might expect that the dual of the multinormed space is M(E, C). But this gives just E, and forgets the multi-norm structure. We could try: n is the norm on (E ) n which is the dual of the norm n on E n. We obtain a sequence n that satisfies (A1), (A2), and (A3), but not (A4). Rather it satisfies: (B4) (x 1,..., x n, x n ) n+1 = (x 1,..., x n 1, 2x n ) n. We have characterizations of these dual multinorms analogous to the above for example we replace B(l n, l m ) by B(l 1 n, l 1 m) and c 0 E by l 1 E. [The second duals of a multi-norm sequence do give a multi-norm.] So this does not work either. 41

42 A consequence of duality In fact we have a duality theory that involves a long detour through an orthogonality theory for multi-normed spaces that generalizes that of Banach lattices. This gives the concepts of multi-dual and multi-reflexive spaces. We have the following, which was the point of the definitions. Theorem For 1 < p <, let the families {(l p ) n : n N} have the standard (p, p)-multinorm. Then the multi-dual of the multi- normed space {(l p ) n : n N} is {(l q ) n : n N} with the standard (q, q)-multi-norm, where q is the conjugate index to p. Hence these multi-normed spaces are multi-reflexive. 42

43 Decompositions - definitions Definition Let (E, ) be a normed space. A direct sum decomposition E = E 1 E k is valid if ζ 1 x ζ k x k x x k for all ζ 1,..., ζ k D and x 1 E 1,..., x k E k. Definition Let ((E n, n ) : n N) be a multinormed space, and let E = E 1 E k be a direct sum decomposition of E. Then the decomposition is small if P 1 x P k x k (x 1,..., x k ) k for all x 1,..., x k E. Fact A small decomposition is valid. 43

44 Orthogonality Let (E n, n ) be a multi-normed space. A family {E 1,..., E k } of linear subspaces of E is orthogonal if, for each partition {S 1,..., S j } of {1,..., k} and each x i E i, we have (x 1,..., x k ) k = (y 1,..., y j ), j where y i := {x r : r S i } (i = 1,..., j). In particular, we require that (x 1,..., x k ) k = x x k whenever x i E i for i = 1,..., k. A set {x 1,..., x k } is orthogonal if the family {Cx 1,..., Cx k } is orthogonal. Remark It is possible that {x 1, x 2, x 3 } is not orthogonal, but each of {x 1, x 2 }, {x 1, x 3 }, and {x 2, x 3 } is orthogonal. Remark Every orthogonal decomposition is valid. 44

45 Orthogonality - Examples (1) Let H be a Hilbert space with the Hilbert multi-norms Then subspaces are orthogonal if and only if they are orthogonal in the classical sense if and only if they give a valid decomposition. (2) Let E = l p (S) have the standard (p, p)- multi-norms. Then E has the orthogonal decomposition E = E 1 E k if and only if there is a partition {S 1,..., S k } of S such that E j = l p (S j ) (j = 1,..., k). (3) Let E = l p (S) have the standard (p, q)- multi-norm, with q p. Then there are no non-trivial orthogonal decompositions of E. 45

46 Continuous functions Let E = C(Ω) for a compact space Ω, and let {E n : n N} have the minimum multi-norm. Then the only orthogonal decompositions of E have the form C(Ω) = C(Ω 1 ) C(Ω k ) for a partition {Ω 1,..., Ω k } of Ω into clopen subspaces. 46

47 Decompositions - some facts Fact An orthogonal decomposition is valid. Fact A small decomposition is not necessarily orthogonal. Fact An orthogonal decomposition is not necessarily small. [Maybe the definitions need tweaking to bring small into the definition of orthogonality?] 47

48 Orthogonality - Examples (1) Let H be a Hilbert space with the Hilbert multi-norms. Then subspaces are orthogonal if and only if they are classically orthogonal if and only if they give a valid decomposition. (2) Let E = l p (S) have the standard (p, p)- multi-norms. Then E has the orthogonal decomposition E = E 1 E k if and only if there is a partition {S 1,..., S k } of S such that E j = l p (S j ) (j = 1,..., k). (3) Let E = l p (S) have the standard (p, q)- multi-norm, with q p. Then there are no non-trivial orthogonal decompositions of E. (4) Let E = C(Ω) for a compact space Ω, and let {E n : n N} have the minimum multi-norm. Then the only orthogonal decompositions of E have the form C(Ω) = C(Ω 1 ) C(Ω k ) for a partition {Ω 1,..., Ω k } of Ω into clopen subspaces 48

49 Decompositions and Banach lattices Let E be a Banach lattice. Recall that the lattice multi-norms are defined by (x 1,..., x n ) n = x 1 x n for x 1,..., x n E. Recall from that E = E 1 E n is a classically orthogonal decomposition if x i x j = 0 whenever x i E i, x j E j, and i j. Easy: a classically orthogonal decomposition is orthogonal for the lattice multi-norm. Let E = E 1 E k be orthogonal with respect to the lattice multi-norm. Then x 1 x k = x x k whenever x 1 E 1,..., x k E k. Theorem - Nigel Kalton This already implies that the decomposition is classically orthogonal, and so the new concept of orthogonal coincides with the old one in this case. 49

50 Families of decompositions Definition Let (E, ) be a normed space, and consider a family K = {(E 1,α,..., E nα,α) : α A}, where A is an index set, n α N (α A), and E = E 1,α E nα,α is a direct sum decomposition of E for each α A. The family K is closed provided that the following conditions are satisfied: (C1) (E σ(1),α,..., E σ(nα ),α ) K whenever (E 1,α,..., E nα,α) K and σ S nα ; (C2) (E 1,α E 2,α, E 3,α,..., E nα,α) K whenever (E 1,α,..., E nα,α) K and n α 2; (C3) K contains all trivial direct sum decompositions. 50

51 Orthogonal multi-norms The families of all direct sum decompositions, of all valid decompositions, of all small decompositions, and of all orthogonal decompositions are closed families of decompositions. Let (E n, n ) be a multi-normed space. Consider a closed family K = {{E 1,α,..., E nα,α} : α A} of orthogonal decompositions of E. Definition Let (E n, n ) be a multi-normed space. This space is orthogonal with respect to K if }, n { (P1,α (x 1,..., x n ) n = sup x 1,..., P n,α x n ) α for x 1,..., x n E, where the supremum is taken over all α A with n α = n. 51

52 Multi-norms from families of decompositions Let (E, ) be a normed space, and consider a closed family K = {{E 1,α,..., E nα,α} : α A} of valid decompositions of E. For n N and x 1,..., x n E, set (x 1,..., x n ) n,k { P1,α = sup x P nα,αx n : nα = n }. α A Then ((E n, n,k ) : n N) is a multi-normed space, each member of K is an orthogonal decomposition of E with respect to this multinorm, and the multi-normed space is orthogonal with respect to K. This is the multi-norm generated by K. Query: what are the conditions on a multinorm that ensure that it is orthogonal with respect to some closed family of valid decompositions? 52

53 Duals of valid decompositions Fact Let E = E 1 E k be a valid decomposition of a Banach space E. Then E = E 1 E k is a valid decomposition of the dual space E. Consider a closed family K = {{E 1,α,..., E nα,α} : α A} of valid decompositions of E. The dual family is K = {{E 1,α,..., E n α,α} : α A}, and it generates a multi-norm ( n,k : n N) on the family {(E ) n : n N}. 53

54 Dual multi-norms Definition Let (E, ) be a normed space, and let K be a closed family of valid decompositions of E. Then the multi-norm on {(E ) n : n N} generated by K is denoted by ( n,k : n N). The multi-normed space (((E ) n, n,k ) : n N) is the multi-dual space with respect to K. Orthogonal decompositions for Banach lattices For example, the family K of all orthogonal decompositions is closed, and the Banach lattice multi-norm is orthogonal with respect to this family. Moreover, if E is order-continuous, the dual multi-norm is exactly the Banach lattice multi-norm of the dual space E. 54

55 The theorem on duality Theorem Let (E, ) be a normed space, and let K be a closed family of valid decompositions of E. Then (((E ) n, n,k ) : n N) is a multi-normed space, each member of K is an orthogonal decomposition of E, and this multi-normed space is orthogonal with respect to K. In very many (but not all) cases, the dual multinorms are independent of the defining family K, as we would wish. These definitions make the earlier theorems on duality correct (I think!). Is the above the correct duality theory, or is there a simpler one? 55

56 Reduced valid decompositions I A valid de- Let (E, ) be a normed space. composition E = E 1 E k is reduced if there is a function θ : R +n R + such that θ( x 1,..., x k ) = whenever x i E i. k x i i=1 A closed family K of valid decompositions is reduced if each member is reduced and the corresponding θ depends only on the value of k. 56

57 Reduced valid decompositions II Suppose that K is a closed family of valid decompositions which contains non-trivial decompositions of length at least 3. For s, t R +, set s t = θ 2 (s, t). Theorem (R +,,, ) is a topological ordered semiring, and hence the only possibilities for the binary operation are s t = max{s, t} (s, t R + ), s t = (s p + t p ) 1/p (s, t R + ), for some p 1. The two possibilities are realised for the normed spaces C(Ω) and l p, respectively. 57

58 Multi-Banach algebras Let (A, ) be a Banach algebra, and let ((A n, n ) : n N) be a multi-normed space. Then (A n, n ) is a multi-banach algebra if multiplication is a multi-bounded bilinear operator, and so (a 1 b 1,..., a n b n ) n (a 1,..., a n ) n (b 1,..., b n ) n. Examples (1) Each Banach algebra is a multi- Banach algebra with respect to both the minimum and maximum multi-norms. (2) Take 1 p q <. Then (l p (S), ) is a multi-banach algebra with respect to the standard (p, q)-multi-norm. (3) Let G be a locally compact group. Then the group algebra (L 1 (G), ) with the standard (1, 1)-multi-norm is a multi-banach algebra. (4) For each multi-banach space (E n, n ), (M(E) n, mb,n ) is a multi-banach algebra. 58