Matricial Banach Spaces

Algebraic Case Analytic Case References

Matricial Banach Spaces Will Grilliette Texas State University Department of Mathematics

1 November 2018

W. Grilliette

Matricial Banach Spaces


Left Adjoint Functors Free Module/Vector Space

Definition (Reflection along a functor, [2, Definition 3.1.1]) Let D

F

/ C and C ∈ Ob(C). A reflection of C along F consists η

of an object R ∈ Ob(D) and a morphism C

/ F (R) ∈ C such

φ

/ F (D) ∈ C, there is a unique that for any D ∈ Ob(D) and C φˆ / D ∈ D such that F φ ˆ ◦ η = φ. R

F

D ?D

⇒

C C

φ

/ F (D) ;

η

∃!φˆ

R

F (φˆ)

F (R) W. Grilliette




Definition (Left adjoint functor, [2, Definition 3.1.4]) Let D

F

/ C . A left adjoint functor to F consists of a functor

η / D and a natural transformation idC / F ◦ L such C that L(C ) and ηC form a reflection along F for all C ∈ Ob(C). L

Example (Some adjoint pairs) Left Discrete top. space ˇ Stone-Cech compactification Free semigroup Independent set of vertices Independent set of directed edges

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Right Forgetful Top → Set Forgetful Comp → Top Forgetful Sgr → Set Vertex Q → Set Edge Q → Set




How does one construct a free module or vector space? Definition (Free module, [4, p. 335]) Fix a unital ring R and let F : R ModR → Set be the forgetful functor. Given a set S, define L(S) := {f ∈ Set(S, R) : f has finite support} to have pointwise arithmetic: • (f + g )(s) := f (s) + g (s), • (r · f )(s) := r · f (s).

1, t = s, 0, t = 6 s. Also, define ηS : S → FL(S) by ηS (s) := δs . The family (δs )s∈S is the standard basis. For s ∈ S, define the Kronecker delta by δs (t) :=

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Theorem (Universal property, [4, Theorem 10.6]) Given S L(S)

φˆ

φ

/ F (M) ∈ Set, there is a unique

/ M ∈ R ModR such that F φ ˆ ◦ ηS = φ.

Proof. Define φˆ on the standard basis by φˆ (δs ) := φ(s). Theorem (Vector spaces are free, [10, Theorem III.5.1]) If R is a field, then every module is isomorphic to a free module.

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Banach Spaces Weighted Sets Matricially Normed Spaces Array-Weighted Sets

What about normed vector spaces? Definition (Normed vector space, [3, Definitions III.1.1-2]) Let F ∈ {R, C}. Given a vector space V over F, a norm on V is a function k · k : V → [0, ∞) satisfying the following conditions: • kv k = 0 if and only if v = 0, • kλ · v k = |λ| · kv k, • kv + w k ≤ kv k + kw k. A normed vector space is a vector space equipped with a norm. A Banach space is a normed vector space that is complete in the induced metric. d(v , w ) := kv − w k.

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Example (`p -spaces, [3, Exercise III.1.3]) Given n ∈ N and p ≥ 1, then k~v kp :=

n X

!1/p |~v (k)|

p

, k~v k∞ := max {|~v (k)| : 1 ≤ k ≤ n}

k=1

are norms on Fn . Definition (Equivalent norms, [3, p. 64]) Two norms on V are equivalent if they induce the same metric topology on V . Theorem (Finite-dimensional normed spaces, [3, Theorem III.3.1]) If V is finite-dimensional, then all norms on V are equivalent. W. Grilliette




Example (Lp -spaces, [6, Theorem 6.6]) Given a measure space (X , Σ, µ) and p ≥ 1, then Z kf k :=

p

1/p

|f (x)| dµ X

is a norm on Lp (X , Σ, µ). Example (L∞ -spaces, [6, Theorem 6.8]) Given a measure space (X , Σ, µ), then kf k := inf {a ∈ R : µ ({x : |f (x)| > a}) = 0} is a norm on L∞ (X , Σ, µ). W. Grilliette




Definition (Categories, [2, Examples 1.2.5.f & 1.2.5.h]) Let Ban∞ be the category of Banach spaces with continuous linear maps, and Ban1 be the category of Banach spaces with contractive linear maps. Proposition (Continuous linear operators, [3, Proposition III.2.1]) Given normed vector spaces V and W , let φ : V → W be linear. The following are equivalent: 1

φ is uniformly continuous;

2

φ is continuous;

3

φ is continuous at 0;

4

φ is continuous at some point in V ;

5

there is some L ≥ 0 such that kφ(v )kW ≤ Lkv kV for all v ∈ V. W. Grilliette




Corollary (Internal hom, [3, Exercise III.2.1]) If V , W are Banach spaces, then [V , W ] = Ban∞ (V , W ) is a Banach space under pointwise operations and the operator norm defined below. kφk[V ,W ] := inf {L ∈ R : kφ(v )kW ≤ Lkv kV ∀v ∈ V } Is there a free Banach space? NO! Proposition (Folklore) Let S 6= ∅ and C be a subcategory of Ban1 . If Ob(C) contains V ∼ 6 O, then S has no associated free object in C. = Proposition (Bounded version) Let S be an infinite set and C be a subcategory of Ban∞ . If Ob(C) contains V ∼ 6 O, then S has no associated free object in C. = W. Grilliette




Proof of bounded case. Let FC : C → Set be the forgetful functor. For purposes of η

/ FC (W ) ∈ Set contradiction, say that there is W ∈ C and S satisfy the free mapping property. Let ws := η(s). Fix a unit vector u ∈ V and define φ : S → V by φ(s) := u. There is a φˆ / V ∈ Ban∞ such that FC φˆ ◦ η = φ. For s ∈ S, unique W

1 = kukV = kφ(s)kV = FC φˆ ◦ η (s) V

= φˆ (ws ) ≤ φˆ · kws kW , V

[W ,V ]

so kws kW 6= 0. Let (sn )n∈N ⊆ S be distinct and define ψ : S → V n kwsn kW · u, t = sn , by ψ(t) := 0, otherwise. W. Grilliette




Proof of bounded case, continued. There is a unique W For n ∈ N, n kwsn kW

ψˆ

/ V ∈ Ban∞ such that FC ψˆ ◦ η = ψ.

= kn kwsn kW · ukV = kψ (sn )kV

= FC ψˆ ◦ η (sn ) = ψˆ (wsn ) V V

ˆ ≤ ψ · kwsn kW , [W ,V ]

so n ≤ ψˆ

[W ,V ]

. Hence, ψˆ

[W ,V ]

= ∞, contradicting that ψˆ

was continuous. W. Grilliette




Can one replace Set? YES! Definition (Weighted sets, [7]) A weighted set is a set S equipped with a weight function wS : S → [0, ∞). Given two weighted sets S and T , a function φ : S → T is bounded if there is L ≥ 0 such that for all s ∈ S, wT (φ(s)) ≤ L · wS (s). Let bnd(φ) := inf {L ∈ R : wT (φ(s)) ≤ L · wS (s)∀s ∈ S} , the bound constant of φ. If bnd(φ) ≤ 1, φ is contractive. Let WSet1 denote the category of weighted sets with contractive maps, and WSet∞ denote the category of weighted sets with WSet∞ bounded maps. Let FBan : Ban∞ → WSet∞ the forgetful ∞ functor stripping all linear structure. W. Grilliette




The failure of freeness arises in weighted sets as well. Proposition (Contractive map failure, [9, Proposition 2.2.13]) Let S 6= ∅ and C be a subcategory of WSet1 . Assume that for each n ∈ N, there is a weighted set Tn ∈ Ob(C) with an element tn ∈ Tn satisfying that wTn (tn ) ≥ n. Then, S has no associated free object in C. Proposition (Bounded map failure, [9, Proposition 2.3.12]) Let S be an infinite set and C be a subcategory of WSet∞ . Assume there is a weighted set T ∈ Ob(C) with elements (tn )n∈N ⊆ T satisfying that wT (tn ) ≥ n for all n ∈ N. Then, S has no associated free object in C.

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Definition (Scaled-free Banach space, [7, p. 281]) −1 ˆ For a weighted set the discrete S, let S := S \ wS (0). Define X ˆ measure µS : P S → [0, ∞] by µS (T ) := wS (s). The s∈T

scaled-free Banach space of S is ˆ P Sˆ , µS , BanSp(S) := L1 S, a weighted `1 -space over C. Define ζS : S → BanSp(S) by ζS (s) :=

ˆ 0, s 6∈ S, ˆ δs , s ∈ S,

ˆ where δs is the point mass at s ∈ S. W. Grilliette




Theorem (Universal property, [7, Theorem 3.1.1]) If S

φ

/ F WSet∞ (W ) ∈ WSet∞ . Then, there is a unique Ban∞

BanSp(S)

φˆ

/ W ∈ Ban∞ such that F WSet∞ φ ˆ ◦ ζS = φ. Ban∞

Moreover, bnd(φ) = φˆ

[BanSp(S),W ]

.

Proof. Define φˆ on the standard basis by φˆ (δs ) := φ(s). Observe that

!

X X

ˆ

λs δs ≤ |λs | · kφ(s)kW

φ

s∈E s∈E W

X

X

≤ |λs | · bnd(φ)wS (s) = bnd(φ) λs δ s .

s∈E

s∈E

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BanSp(S)



Definition (Matrix-normed spaces, [1, p. 264]) For n ∈ N, equip Cn with the Euclidean norm. For m, n ∈ N, let Mm,n be equipped with the operator norm from Cn to Cm . For a vector space V , a matrix-norm on V is a net (k · kV ,m,n )m,n∈N such that 1

k · kV ,m,n is a norm on Mm,n (V ),

2

kABC kV ,k,l ≤ kAkMk,m kBkV ,m,n kC kMn,l

for all k, m, n, l ∈ N, A ∈ Mk,m , B ∈ Mm,n (V ), C ∈ Mn,l . A vector space V equipped with such a matrix-norm is a matrix-normed space.

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Definition (Completely bounded maps, [8, Definition 2.2.4]) Given matrix-normed spaces V and W , a linear map φ : V → W is completely bounded if 1 2

Mm,n (φ) is continuous for all m, n ∈ N, n o kφk[V ,W ]cb := sup kMm,n (φ)k[Mm,n (V ),Mm,n (W )] : m, n ∈ N < ∞.

The map φ is completely contractive if kφk[V ,W ]cb ≤ 1. The standard definition given in [1, 5, 11, 12] uses square matrices, but was referenced in [5, p. 246].

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Definition (Matricial Banach space, [8, Definition 2.2.5]) A complete matrix-normed space is a matricial Banach space. Let MBan∞ be the category of matricial Banach spaces with completely bounded linear maps, and MBan1 be the category of matricial Banach spaces with completely contractive linear maps. Ban∞ : MBan∞ → Ban∞ be the forgetful functor stripping Let FMBan ∞ all matrix-norm structure except the underlying norm.

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Definition (Minimal operator space structure, [5, Theorem 2.1]) Given a Banach space V , let MIN(V ) be V equipped with o n kAkMIN(V ),m,n := sup kMm,n (φ)(A)kMm,n : φ ∈ V ∗ , kφkV ∗ ≤ 1 , the injective tensor norm on V ⊗ Mm,n . Definition (Abs. max. matrix-norm structure, [5, Theorem 2.1]) Recall that M∗n,m can be identified as Mm,n equipped with the trace norm. Given a Banach space V , let AMAX(V ) be V equipped with ( p ) p X X kAkAMAX(V ),m,n := inf kvl kV kCl kM∗n,m : A = vl ⊗ Cl , l=1

l=1

the projective tensor norm on V ⊗ M∗n,m W. Grilliette




Theorem (Universal property of MIN) Ban∞ If FMBan (W ) ∞

ϕ

/ V ∈ Ban∞ , then there is a unique

ϕ ˆ

/ MIN(V ) ∈ MBan∞ such that F Ban∞ (ϕ) W MBan∞ ˆ = ϕ. Moreover, kϕk ˆ [W ,MIN(V )]cb = kϕk[F Ban∞ (W ),V ] . MBan∞

Theorem (Universal property of AMAX) If V

φ

/ F Ban∞ (W ) ∈ Ban∞ . Then, there is a unique MBan∞ φˆ

AMAX(V )

Moreover, φˆ

/ W ∈ MBan∞ such that F Ban∞ ˆ MBan∞ φ = φ.

[AMAX(V ),W ]cb

= kφk[V ,F Ban∞

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].

MBan∞ (W )




How should one build a matricial Banach space? MBan∞

/ Ban∞

/ Vec

/ Set

WSet∞

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Definition (Array-weighted set, [8, Definition 3.1.2]) An array-weight on a set X is a net (wX ,m,n )m,n∈N such that 1

wX ,m,n is a weight function for all m, n ∈ N;

2

wX ,j,k (A ◦ (α × β)) ≤ wX ,m,n (A) for all 1 ≤ j ≤ m, 1 ≤ k ≤ n, A ∈ Mm,n (X ), and one-to-one functions α : [j] → [m] and β : [k] → [n];

3

wX ,m,n (A) ≤ wX ,j,n A ◦ α × id[n] + wX ,m−j,n A ◦ γ × id[n] for all 1 ≤ j < m and one-to-one functions α : [j] → [m] and γ : [m − j] → [m] satisfying ran(α) ∩ ran(γ) = ∅;

4

wX ,m,n (A) ≤ wX ,m,k A ◦ id[m] × β + wX ,m,n−k A ◦ id[m] × δ for all 1 ≤ k < n and one-to-one functions β : [k] → [n] and δ : [n − k] → [n] satisfying ran(β) ∩ ran(δ) = ∅.

A set equipped with such an array-weight is an array-weighted set. W. Grilliette




Definition (Completely bounded function, [8, Definition 3.1.3]) Given two array-weighted sets X and Y , a function φ : X → Y is completely bounded if there is L ≥ 0 such that for all m, n ∈ N and A ∈ Mm,n (X ), wY ,m,n (Mm,n (φ)(A)) ≤ L · wX ,m,n (A). Let wY ,m,n (Mm,n (φ)(A)) ≤ L · wX ,m,n (A) cbnd(φ) := inf L ∈ R : , ∀m, n ∈ N, A ∈ Mm,n (X ) the complete bound constant of φ. If cbnd(φ) ≤ 1, φ is completely contractive. Let AWSet1 denote the category of array-weighted sets with completely contractive maps, and AWSet∞ denote the category of array-weighted sets with completely bounded maps. WSet∞ Let FAWSet : AWSet∞ → WSet∞ be the forgetful functor ∞ stripping all weight functions except the underlying weight AWSet∞ function, and let FMBan : MBan∞ → AWSet∞ be the forgetful ∞ functor stripping all linear structure. W. Grilliette




MBan∞

/ Ban∞

/ Vec

/ WSet∞

/ Set

AWSet∞

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Definition (Minimum array-weight structure, [8, Definition 3.2.1]) Given a weighted set S, let mA(S) be S equipped with the weight functions wmA(S),m,n (A) := max {wS (A(j, k)) : 1 ≤ j ≤ m, 1 ≤ k ≤ n} , the maximum weight of an entry in A. Definition (Maximum array-weight structure, [8, Definition 3.2.2]) Given a weighted set S, let MA(S) be S equipped with the weight functions m X n X wMA(S),m,n (A) := wS (A(j, k)) , j=1 k=1

the sum of the weights of the entries in A. W. Grilliette




Theorem (Universal property of mA, [8, Theorem 3.2.3]) WSet∞ If FAWSet (Y ) ∞

Y

φˆ

φ

/ S ∈ WSet∞ , then there is a unique

/ mA(S) ∈ AWSet∞ such that F WSet∞ φ ˆ = φ. AWSet∞

Moreover, cbnd φˆ = bnd(φ). Theorem (Universal property of MA, [8, Theorem 3.2.4]) If S

ϕ

MA(S)

/ F WSet∞ (Y ) ∈ WSet∞ . Then, there is a unique AWSet∞ ϕ ˆ

/ Y ∈ AWSet∞ such that F WSet∞ φ ˆ = φ. AWSet∞

Moreover, cbnd φˆ = bnd(φ).

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Definition (Array-free, [8, Definition 3.5.1]) For an array-weighted set X , an element x ∈ X is array-free in X if the characteristic function of x is completely bounded when regarded as a map from X to MIN(C). Example Let V be a matrix-normed space and X ⊂ V a finite, linearly independent subset equipped with the inherited array-weight from V . For x ∈ X , consider the characteristic function χ : X → C of x. Letting W := span(X ), define a linear map φ : W → C on the basis X by φ(y ) := χ(y ) for all y ∈ X . As W is finite-dimensional, φ is bounded. Letting ι : W → V be the inclusion map, there is a bounded linear map ϕ : V → C such that ϕ ◦ ι = φ by the Hahn-Banach Theorem. Then, ϕ is completely bounded from V to MIN(C). Letting : X → W be the inclusion of generators, then ϕ ◦ ι ◦ = χ is completely bounded. Thus, x is array-free in X . W. Grilliette




Proposition (Array-free & finite support, [8, Proposition 3.5.3]) Given an array-weighted set X , let Y be a subset of X . All functions from X to MIN(C) with finite support contained in Y are completely bounded if and only if x is array-free in X for all x ∈ Y .

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Definition (Bounded range number, [8, Definition 3.5.4]) Given an array-weighted set X , the bounded range number of X is wX ,m,n (A) √ brn(X ) := inf : m, n ∈ N, A ∈ Mm,n (X ) . mn Theorem (Bounded range maps & brn, [8, Theorem 3.5.5]) Given an array-weighted set X , the following are equivalent: 1

all bounded range maps from X to MIN(C) are completely bounded;

2

the constant map to 1 regarded as a map from X to MIN(C) is completely bounded;

3

brn(X ) > 0.

In this case, x is array-free in X for all x ∈ X . W. Grilliette




Corollary (Array-free & finite sets, [8, Corollary 3.5.6]) Given a finite array-weighted set X , the following are equivalent: 1

all maps from X to MIN(C) are completely bounded;

2

the constant map to 1 regarded as a map from X to MIN(C) is completely bounded;

3

brn(X ) > 0;

4

x is array-free in X for all x ∈ X .

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Example Let V be `∞ with any matrix-norm. Letting (~ ej )j∈N ⊂ V be the 1 standard basis, define xj := e~j and X := {xj : j ∈ N} ⊂ V with j the inherited array-weight from V . Then, brn(X ) = 0, and xj is array-free in X for all j ∈ N. Example Let S be a weighted set and s ∈ S. • s is array-free in MA(S) if and only if wS (s) 6= 0. • s is not array-free in mA(S).

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Example Consider X := {1, −1} with the inherited array-weight from MIN(C). Let Jm,n be the m × n-matrix with all entries 1. Inductively construct the following sequence of matrices. A0 := 1 , J1,2m −J1,2m Am+1 := ∀m ∈ W. Am Am Computing directly, wX ,m+1,2m (Am ) = 2m/2 and wX ,m+1,2m (Am ) 2m/2 1 = m/2 √ =√ brn(X ) ≤ p m 2 m+1 m+1 (m + 1)2 for all m ∈ W. Consequently, brn(X ) = 0. As the inclusion ι = χ1 − χ−1 is completely bounded, neither 1 nor −1 is array free. W. Grilliette




Let X be an array-weighted set. 1 Construct the free vector space VX over C on X and let θX : X → VX be the map x 7→ δx . 2 Quotient by the subspace φ(v ) = 0∀φ : VX → MIN(C) linear, NX := v : φ ◦ θX completely contractive 3

4

to form QX := VX /NX with quotient map qX : VX → QX . Equip QX with the matrix-norm below. n o kAkQX ,m,n := sup kMm,n (φ)(A)kW ,m,n , where the supremum is over all linear φ : QX → W such that φ ◦ qX ◦ θX is completely contractive. Complete QX into a matricial Banach space MBanSp(X ). Let κQX : QX → MBanSp(X ) be the inclusion into the completion, and ηX := κQX ◦ qX ◦ θX . W. Grilliette




Proposition (Array-free & nullspace, [8, Proposition 4.8]) If all x ∈ X are array-free in X , then NX = {0}. Theorem (Finite array-free & nullspace, Part II, [8, Theorem 4.9]) Given a finite array-weighted set X , then NX = {0} if and only if brn(X ) > 0.

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Theorem (Universal property of MBanSp, [8, Theorem 4.4]) If X

φ

/ F AWSet∞ (W ) ∈ AWSet∞ , there is a unique MBan∞

completely bounded linear map MBanSp(X ) AWSet∞ ˆ such that FMBan φ ◦ ηX = φ. Moreover, ∞

cbnd(φ) ≥ φˆ

[MBanSp(X ),W ]cb

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φˆ

.


/ W ∈ MBan∞



Corollary (Maximal array-weight, [8, Corollary 4.6]) Given a weighted set S, then MBanSp(MA(S)) ∼ =MBan1 ∼ =MBan1

AMAX(BanSp(S)) AMAX `1 ({s : wS (s) 6= 0}) .

Proposition (Failure of mA, [8, Proposition 4.7]) Given a weighted set S, then MBanSp(mA(S)) ∼ = 0. Theorem (Characterization of singletons, [8, Theorem 4.10]) Let X be a singleton array-weighted set. Then, AMAX(C), brn(X ) > 0, ∼ MBanSp(X ) =MBan1 0, brn(X ) = 0. W. Grilliette




Example Consider X := {1, −1} with the inherited array-weight from MIN(C). Then, MBanSp(X ) ∼ = MIN(C). Proof. Let wx := θX (x) for x ∈ X . Given a linear map φ : VX → MIN(C), φ ◦ θX = φ (w1 ) ι + (φ (w1 ) + φ (w−1 )) χ−1 , so φ ◦ θX is completely bounded if and only if φ (w1 ) = −φ (w−1 ). Thus, φ (λw1 + µw−1 ) = (λ − µ)φ (w1 ) , which is 0 if and only if λ = µ. Hence, NX = span {w1 + w−1 }. / MIN(C) ∈ MBan1 arising from the Then, MBanSp(X ) inclusion ι is shown to be completely contractive. ˆ ι

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